LieBN: Batch Normalization over Lie Groups

# LieBN: Batch Normalization over Lie Groups
Source: [https://arxiv.org/html/2607.08783](https://arxiv.org/html/2607.08783)
Ziheng Chen, Yue Song, Rui Wang, Xiao\-Jun Wu, and Nicu SebeZiheng Chen and Nicu Sebe are with the Department of Information Engineering and Computer Science, University of Trento, Trento, Italy\. Yue Song is with Computing and Mathematical Sciences, Caltech, CA, USA\. Rui Wang and Xiao\-Jun Wu are with the School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China\. E\-mail: ziheng\_ch@163\.com, yuesong@caltech\.edu, niculae\.sebe@unitn\.it, \{cs\_wr, wu\_xiaojun\}@jiangnan\.edu\.cn\.

###### Abstract

Manifold\-valued measurements are prevalent in various machine learning tasks\. Recent advances have extended Deep Neural Networks \(DNNs\) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization\. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold\-valued sample distributions\. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization \(RBN\) over Lie groups\. Our approach leverages the theoretically convenient left\- and right\-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance\. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite \(SPD\) manifold, one on the group of rotation matrices, and four on the manifold of full\-rank correlation matrices\. Notably, among the SPD metrics, we introduce a novel right\-invariant metric and extend three existing Lie group structures via matrix power deformation\. Extensive experiments on different manifolds validate the effectiveness of our framework\. The code is available at[https://github\.com/GitZH\-Chen/LieBN\.git](https://github.com/GitZH-Chen/LieBN.git)\.

###### Index Terms:

Riemannian batch normalization, Lie groups, symmetric positive definite matrices, rotations, correlation matrices\.

## 1Introduction

![Refer to caption](https://arxiv.org/html/2607.08783v1/x1.png)Figure 1:Illustration of LieBN on the SPD, rotation, and correlation Lie groups\. The2×22\\times 2SPD,3×33\\times 3rotation, and3×33\\times 3correlation manifolds can be embedded intoℝ3\\mathbb\{R\}^\{3\}as an open cone\[[88](https://arxiv.org/html/2607.08783#bib.bib86)\], a closed ball with antipodal points identified\[[39](https://arxiv.org/html/2607.08783#bib.bib70)\], and an open elliptope\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\], respectively\. LieBN is illustrated by \(1\) the left\-invariant AIM and the proposed right\-invariant CRIM geometry on the SPD manifold, \(2\) left or right translation under a bi\-invariant metric on the rotation manifold, and \(3\) the bi\-invariant ECM and LSM geometry on the correlation manifold\. On the SPD and correlation manifolds, the batch mean and variance of the same input samples differ under different geometries\. In all sub\-figures, the black, blue, green, and red dots denote the boundary of the space, the input Lie group samples, the normalized samples, and the batch mean, respectively\. As illustrated, our LieBN effectively normalizes the Lie group distribution\.Over the past decade or so, Deep Neural Networks \(DNNs\) have achieved significant progress across various scientific fields\[[42](https://arxiv.org/html/2607.08783#bib.bib129),[53](https://arxiv.org/html/2607.08783#bib.bib94),[40](https://arxiv.org/html/2607.08783#bib.bib38),[78](https://arxiv.org/html/2607.08783#bib.bib93)\]\. Traditionally, DNNs have been developed under the assumption that the latent space of the input data is Euclidean\. However, many applications involve non\-Euclidean structures, such as manifolds\[[12](https://arxiv.org/html/2607.08783#bib.bib136)\]\. To address this challenge, researchers have extended various types of DNNs to manifolds, based on the theories of Riemannian geometry\[[43](https://arxiv.org/html/2607.08783#bib.bib41),[44](https://arxiv.org/html/2607.08783#bib.bib28),[45](https://arxiv.org/html/2607.08783#bib.bib52),[34](https://arxiv.org/html/2607.08783#bib.bib50),[16](https://arxiv.org/html/2607.08783#bib.bib60),[14](https://arxiv.org/html/2607.08783#bib.bib87),[84](https://arxiv.org/html/2607.08783#bib.bib11),[29](https://arxiv.org/html/2607.08783#bib.bib17),[83](https://arxiv.org/html/2607.08783#bib.bib10),[26](https://arxiv.org/html/2607.08783#bib.bib8),[19](https://arxiv.org/html/2607.08783#bib.bib105),[89](https://arxiv.org/html/2607.08783#bib.bib16),[22](https://arxiv.org/html/2607.08783#bib.bib3),[20](https://arxiv.org/html/2607.08783#bib.bib6),[82](https://arxiv.org/html/2607.08783#bib.bib12),[81](https://arxiv.org/html/2607.08783#bib.bib13),[62](https://arxiv.org/html/2607.08783#bib.bib56),[24](https://arxiv.org/html/2607.08783#bib.bib4),[23](https://arxiv.org/html/2607.08783#bib.bib2)\]\.

Motivated by the great success of normalization techniques\[[46](https://arxiv.org/html/2607.08783#bib.bib37),[4](https://arxiv.org/html/2607.08783#bib.bib77),[77](https://arxiv.org/html/2607.08783#bib.bib76),[87](https://arxiv.org/html/2607.08783#bib.bib55)\], researchers have sought to devise normalization layers tailored for manifold\-valued data\. Brooks*et al*\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\]introduced Riemannian Batch Normalization \(RBN\) designed specifically for the Symmetric Positive Definite \(SPD\) manifold, with the ability to normalize the Riemannian mean\. Kobler*et al*\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]extended this approach to further control the Riemannian variance\.However, the above methods are constrained within the Affine\-Invariant Metric \(AIM\) on the SPD manifold, limiting their applicability\.On the other hand, Chakraborty\[[17](https://arxiv.org/html/2607.08783#bib.bib137)\]proposed two distinct Riemannian normalization frameworks: one for Riemannian homogeneous spaces\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 1\-2\]and another for matrix Lie groups\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 3\-4\]\.Nonetheless, the normalization designed for Riemannian homogeneous spaces cannot normalize mean nor variance, while the one for matrix Lie groups is confined to a specific type of distance\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Sec\. 3\.2\]\.Meanwhile, Luo\[[58](https://arxiv.org/html/2607.08783#bib.bib99), Alg\. 2\]proposed an RBN layer for general geometries\. However, similar to\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 1\-2\], it lacks theoretical guarantees for normalizing sample statistics\. Therefore, a principled Riemannian normalization framework capable of controlling both Riemannian mean and variance remains unexplored\.

Given that Batch Normalization \(BN\)\[[46](https://arxiv.org/html/2607.08783#bib.bib37)\]serves as the foundational prototype for various types of normalization, our paper focuses on RBN, with the potential to be extended to other normalization variants\. Since several manifold\-valued measurements form Lie groups, such as Symmetric Positive Definite \(SPD\) manifolds\[[3](https://arxiv.org/html/2607.08783#bib.bib140),[56](https://arxiv.org/html/2607.08783#bib.bib126),[72](https://arxiv.org/html/2607.08783#bib.bib79)\], special orthogonal groupsSO​\(n\)\\mathrm\{SO\}\(n\)\[[11](https://arxiv.org/html/2607.08783#bib.bib74)\], and full\-rank correlation matrices\[[72](https://arxiv.org/html/2607.08783#bib.bib79),[75](https://arxiv.org/html/2607.08783#bib.bib68)\], we further direct our attention to Lie groups\. As each Lie group naturally admits left\- and right\-invariant metrics\[[32](https://arxiv.org/html/2607.08783#bib.bib143), Ch\. 1\.2\], we propose a principled framework for RBN over Lie groups under invariant metrics, referred to as LieBN\. Compared to previous work, our framework provides a theoretical guarantee for normalizing the Riemannian sample mean and variance across general Lie groups\.

Empirically, we focus on the SPD, special orthogonal, and full\-rank correlation manifolds\. On SPD manifolds, we generalize three existing Lie group structures into parameterized ones by matrix power deformation\. Additionally, we propose a novel right\-invariant metric, which, to the best of our knowledge, is thefirstnon\-trivial right\-invariant SPD metric111Although some metrics are bi\-invariant, the associated group structures are commutative\[[3](https://arxiv.org/html/2607.08783#bib.bib140),[56](https://arxiv.org/html/2607.08783#bib.bib126)\]\. Therefore, their bi\-invariance is reduced to left\-invariance\., referred to as Cholesky Right Invariant Metric \(CRIM\)\. We then instantiate our LieBN framework on SPD manifolds under these four Lie group structures\. For rotation matrices, we adopt the popular bi\-invariant metric\[[11](https://arxiv.org/html/2607.08783#bib.bib74)\], which will induce two types of LieBN: one w\.r\.t\. left\-invariance and another w\.r\.t\. right\-invariance\. On the correlation manifold, we manifest our LieBN under four recently developed correlation geometries\[[72](https://arxiv.org/html/2607.08783#bib.bib79),[75](https://arxiv.org/html/2607.08783#bib.bib68)\]\. Besides, we discuss the optimization of the involved correlation\-valued parameters\. To facilitate usage, we provide a LieBN toolbox compatible with PyTorch, which can be used as a drop\-in module\.[Fig\.1](https://arxiv.org/html/2607.08783#S1.F1)illustrates our LieBN on different geometries, while[Fig\.2](https://arxiv.org/html/2607.08783#S1.F2)illustrate a minimal demo\. Extensive experiments on SPD, rotation, and correlation manifolds across three tasks—radar recognition, human action recognition, and electroencephalography \(EEG\) classification—demonstrate the effectiveness of our methods\.

fromLieBNimportLieBNSPD,LieBNRot,LieBNCor

fromLieBN\.Geometry\.SPDimportSPDMatrices

fromLieBN\.Geometry\.RotationsimportRotMatrices

fromLieBN\.Geometry\.CorrelationimportCorrelation

P\_spd=SPDMatrices\(n=5\)\.random\(4,2,5,5\)

liebn\_spd=LieBNSPD\(\[2,5,5\],metric="LEM",batchdim=\[0\]\)

output\_spd=liebn\_spd\(P\_spd\)

P\_so3=RotMatrices\(\)\.random\(4,2,3,3,3\)

liebn\_so3=LieBNRot\(\[3,3,3\],batchdim=\[0,1\],is\_left=False\)

output\_so3=liebn\_so3\(P\_so3\)

P\_cor=Correlation\(n=5\)\.random\(4,2,5,5\)

liebn\_cor=LieBNCor\(\[2,5,5\],metric="ECM",batchdim=\[0\]\)

output\_cor=liebn\_cor\(P\_cor\)

Figure 2:Minimal examples of applying LieBN\.We emphasize that our work is fundamentally distinct from\[[13](https://arxiv.org/html/2607.08783#bib.bib96),[51](https://arxiv.org/html/2607.08783#bib.bib61),[58](https://arxiv.org/html/2607.08783#bib.bib99)\]in theory and more general than\[[17](https://arxiv.org/html/2607.08783#bib.bib137)\]\. Previous RBN methods are either designed for specific geometries\[[13](https://arxiv.org/html/2607.08783#bib.bib96),[51](https://arxiv.org/html/2607.08783#bib.bib61),[17](https://arxiv.org/html/2607.08783#bib.bib137)\]or fail to control both the mean and variance\[[58](https://arxiv.org/html/2607.08783#bib.bib99)\]\. In contrast, our LieBN ensures the normalization of both the mean and variance across general Lie groups\. In summary, our main contributions are:

- •A general Lie group batch normalization framework with controllable first\- and second\-order moments;
- •A novel right\-invariant metric on the SPD manifold, which is the first non\-trivial right\-invariant SPD metric;
- •Concrete instantiations of our LieBN framework on different geometries: four on SPD manifolds, one on the rotation matrices, and four on the correlation manifold;
- •Validation of effectiveness of our LieBN framework by extensive experiments on different geometries\.

This paper extends our previous conference paper\[[21](https://arxiv.org/html/2607.08783#bib.bib5)\]in both theory and implementation\. Theoretically, the original LieBN framework was restricted to Lie groups under a left\-invariant metric\. However, Lie groups also naturally admit right\-invariant metrics, which share many theoretical properties with left\-invariant ones\. Therefore, we generalize LieBN to all natural invariant metrics, including left\-, right\-, and bi\-invariant metrics, providing a more comprehensive framework for RBN\. Additionally, we propose a novel non\-trivial right\-invariant metric on the SPD manifold\. In terms of implementation, beyond the original applications to SPD and rotation matrices, we further manifest LieBN on four correlation geometries\. Besides, the previous LieBN onSO​\(3\)\\mathrm\{SO\}\(3\)was based solely on left\-invariance and validated on a small dataset\. In contrast, this journal submission expands the implementation to both left\- and right\-invariance and conducts extensive experiments across multiple datasets\.

## 2Preliminaries

This section briefly reviews Lie groups, as well as the concrete Lie groups of SPD, rotation, and full\-rank correlation matrices\. For more in\-depth discussions, we refer the reader to\[[76](https://arxiv.org/html/2607.08783#bib.bib146)\]for smooth manifolds,\[[32](https://arxiv.org/html/2607.08783#bib.bib143),[54](https://arxiv.org/html/2607.08783#bib.bib155)\]for Riemannian manifolds, and\[[38](https://arxiv.org/html/2607.08783#bib.bib156)\]for Lie groups\.

### 2\.1Lie Groups

###### Definition 2\.1\(Lie Groups\[[76](https://arxiv.org/html/2607.08783#bib.bib146)\]\)\.

A manifoldℳ\\mathcal\{M\}is a Lie group, if it forms a group with a group operation⊙\\odotsuch thatm​\(⋅,⋅\):ℳ×ℳ∋\(x,y\)→x⊙y∈ℳm\(\\cdot,\\cdot\):\\mathcal\{M\}\\times\\mathcal\{M\}\\ni\(x,y\)\\to x\\odot y\\in\\mathcal\{M\}and the group inversei​\(⋅\):ℳ∋x→x⊙−1∈ℳi\(\\cdot\):\\mathcal\{M\}\\ni x\\to x\_\{\\odot\}^\{\-1\}\\in\\mathcal\{M\}are both smooth\.

###### Definition 2\.2\(Invariance\[[32](https://arxiv.org/html/2607.08783#bib.bib143)\]\)\.

A Riemannian metricgLg^\{\\mathrm\{L\}\}over a Lie group\{ℳ,⊙\}\\\{\\mathcal\{M\},\\odot\\\}is left\-invariant, if for anyx,y∈ℳx,y\\in\\mathcal\{M\}andV1,V2∈Tx​ℳV\_\{1\},V\_\{2\}\\in T\_\{x\}\\mathcal\{M\}, it satisfiesgyL​\(V1,V2\)=gLx⁡\(y\)L​\(Lx⁣∗,y⁡\(V1\),Lx⁣∗,y⁡\(V2\)\)g^\{\\mathrm\{L\}\}\_\{y\}\(V\_\{1\},V\_\{2\}\)=g^\{\\mathrm\{L\}\}\_\{\\operatorname\{L\}\_\{x\}\(y\)\}\\left\(\\operatorname\{L\}\_\{x\*,y\}\(V\_\{1\}\),\\operatorname\{L\}\_\{x\*,y\}\(V\_\{2\}\)\\right\), withLx⁡\(y\)=x⊙y\\operatorname\{L\}\_\{x\}\(y\)=x\\odot yas the left translation byxx, andLx⁣∗,y\\operatorname\{L\}\_\{x\*,y\}as the differential map ofLx\\operatorname\{L\}\_\{x\}atyy\. Similarly, a right\-invariant metricgRg^\{\\mathrm\{R\}\}satisfiesgyR​\(V1,V2\)=gRx⁡\(y\)R​\(Rx⁣∗,y⁡\(V1\),Rx⁣∗,y⁡\(V2\)\)g^\{\\mathrm\{R\}\}\_\{y\}\(V\_\{1\},V\_\{2\}\)=g^\{\\mathrm\{R\}\}\_\{\\operatorname\{R\}\_\{x\}\(y\)\}\\left\(\\operatorname\{R\}\_\{x\*,y\}\(V\_\{1\}\),\\operatorname\{R\}\_\{x\*,y\}\(V\_\{2\}\)\\right\), withRx⁡\(y\)=y⊙x\\operatorname\{R\}\_\{x\}\(y\)=y\\odot xas the right translation byxx, andRx⁣∗,y\\operatorname\{R\}\_\{x\*,y\}as the differential atyy\.

A Lie group is both a group and a manifold\. The most natural Riemannian metric on a Lie group is the left\- or right\-invariant metric222Invariant metric always exists for every Lie group\[[32](https://arxiv.org/html/2607.08783#bib.bib143), Ch\. 1\.2\]\.\. A bi\-invariant metric has both left\- and right\-invariance\. In this paper,\{ℳ,⊙,g\}\\\{\\mathcal\{M\},\\odot,g\\\}, abbreviated asℳ\\mathcal\{M\}, always denotes a Lie group with an invariant metric\.

The idea of pullback is ubiquitous in differential geometry and can be viewed as a natural counterpart of the bijection\.

###### Definition 2\.3\(Pullback Metrics\[[54](https://arxiv.org/html/2607.08783#bib.bib155)\]\)\.

Supposeℳ1,ℳ2\\mathcal\{M\}\_\{1\},\\mathcal\{M\}\_\{2\}are smooth manifolds,ggis a Riemannian metric onℳ2\\mathcal\{M\}\_\{2\}, andf:ℳ1→ℳ2f:\\mathcal\{M\}\_\{1\}\\rightarrow\\mathcal\{M\}\_\{2\}is a diffeomorphism\. The pullback ofggbyffis defined point\-wisely as\(f∗​g\)p​\(V,W\)=gf​\(p\)​\(f∗,p​\(V\),f∗,p​\(W\)\)\(f^\{\*\}g\)\_\{p\}\(V,W\)=g\_\{f\(p\)\}\(f\_\{\*,p\}\(V\),f\_\{\*,p\}\(W\)\), wheref∗,p​\(⋅\)f\_\{\*,p\}\(\\cdot\)is the differential map offfatp∈ℳ1p\\in\\mathcal\{M\}\_\{1\}, andV,W∈Tp​ℳ1V,W\\in T\_\{p\}\\mathcal\{M\}\_\{1\}\.f∗​gf^\{\*\}gis a Riemannian metric onℳ1\\mathcal\{M\}\_\{1\}, called the pullback metric ofggbyff\.

Although pullback metrics can also be defined by smooth map\[[54](https://arxiv.org/html/2607.08783#bib.bib155)\], we focus on the diffeomorphism\. Additionally, if\{ℳ2,⊙2\}\\\{\\mathcal\{M\}\_\{2\},\\odot\_\{2\}\\\}forms a Lie group, the diffeomorphismffcan pull back the group operation⊙2\\odot\_\{2\}to⊙1\\odot\_\{1\}onℳ1\\mathcal\{M\}\_\{1\}:

P⊙1Q=f−1​\(f​\(P\)⊙2f​\(Q\)\),∀P,Q∈ℳ1\.P\\odot\_\{1\}Q=f^\{\-1\}\(f\(P\)\\odot\_\{2\}f\(Q\)\),\\forall P,Q\\in\\mathcal\{M\}\_\{1\}\.\(1\)
###### Definition 2\.4\(Weighted Fréchet Mean & Variance\[[33](https://arxiv.org/html/2607.08783#bib.bib58)\]\)\.

Let\{w1​…​N\}\\\{w\_\{1\\ldots N\}\\\}be weights satisfying a convexity constraint,*i\.e\.*,∀i,wi\>0\\forall i,w\_\{i\}\>0and∑iwi=1\\sum\_\{i\}w\_\{i\}=1\. The weighted Fréchet mean \(WFM\) of a set of manifold\-valued points\{Pi​…​N∈ℳ\}\\\{P\_\{i\\ldots N\}\\in\\mathcal\{M\}\\\}is defined as

WFM⁡\(\{wi\},\{Pi\}\)=argminM∈ℳ​∑i=1Nwi​d2⁡\(Pi,M\),\\operatorname\{WFM\}\\left\(\\\{w\_\{i\}\\\},\\\{P\_\{i\}\\\}\\right\)=\\underset\{M\\in\\mathcal\{M\}\}\{\\operatorname\{argmin\}\}\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\operatorname\{d\}^\{2\}\\left\(P\_\{i\},M\\right\),\(2\)whered⁡\(⋅,⋅\)\\operatorname\{d\}\(\\cdot,\\cdot\)denotes the geodesic distance\. Whenwi=1/Nw\_\{i\}=\\nicefrac\{\{1\}\}\{\{N\}\}for allii,[Eq\.2](https://arxiv.org/html/2607.08783#S2.E2)is reduced to the Fréchet mean, denoted asFM⁡\(\{Pi\}\)\\operatorname\{FM\}\(\\\{P\_\{i\}\\\}\)\. The Fréchet variancev2v^\{2\}is the attained value at the minimizer of the Fréchet mean\.

On Riemannian manifolds, WFM uniquely exists when samples are locally distributed\[[1](https://arxiv.org/html/2607.08783#bib.bib72)\], which are detailed in[Sec\.B\.1](https://arxiv.org/html/2607.08783#A2.SS1)for completeness\. In this paper, we always assume the Fréchet batch mean exists\. As we focus on Riemannian manifolds, we will use the terms ”Riemannian mean” and ”Fréchet mean” interchangeably, as well as ”Riemannian variance” and ”Fréchet variance\.”

Basic notations\.For Euclidean spaces, we denote⟨⋅,⋅⟩\\left\\langle\\cdot,\\cdot\\right\\rangleas the canonical inner product overℝn×n\\mathbb\{R\}^\{n\\times n\}, with∥⋅∥F\\left\\\|\{\\cdot\}\\right\\\|\_\{\\mathrm\{F\}\}as the induced norms,*i\.e\.*,FF\-norm for matrices\. For manifoldℳ\\mathcal\{M\}, we denoteLogP\\operatorname\{Log\}\_\{P\},ExpP\\operatorname\{Exp\}\_\{P\}, and⟨⋅,⋅⟩P=gP​\(⋅,⋅\)\\left\\langle\\cdot,\\cdot\\right\\rangle\_\{P\}=g\_\{P\}\(\\cdot,\\cdot\)as the Riemannian logarithm, exponentiation, and metric atP∈ℳP\\in\\mathcal\{M\}, respectively\. Besides,γ\(P,Q\)​\(t\)\\gamma\_\{\(P,Q\)\}\(t\)is the geodesic connectingPPandQQwith geodesic distance asd⁡\(⋅,⋅\)\\operatorname\{d\}\(\\cdot,\\cdot\)\. We provide a complete table of notations in[App\.A](https://arxiv.org/html/2607.08783#A1)\.

### 2\.2SPD Lie Groups

The SPD manifold has shown great success in diverse applications\[[43](https://arxiv.org/html/2607.08783#bib.bib41),[14](https://arxiv.org/html/2607.08783#bib.bib87),[80](https://arxiv.org/html/2607.08783#bib.bib131),[57](https://arxiv.org/html/2607.08783#bib.bib91),[27](https://arxiv.org/html/2607.08783#bib.bib9),[51](https://arxiv.org/html/2607.08783#bib.bib61)\]\. We denoten×nn\\times nSPD matrices as𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}andn×nn\\times nreal symmetric matrices as𝒮n\\mathcal\{S\}^\{n\}\. As shown by Arsigny*et al*\[[3](https://arxiv.org/html/2607.08783#bib.bib140)\],𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}forms an open submanifold of the Euclidean space𝒮n\\mathcal\{S\}^\{n\}, known as the SPD manifold\. SPD manifolds exhibit three Lie group structures, each associated with an invariant metric\. These metrics include the Log\-Euclidean Metric \(LEM\)\[[3](https://arxiv.org/html/2607.08783#bib.bib140)\], Affine\-Invariant Metric \(AIM\)\[[65](https://arxiv.org/html/2607.08783#bib.bib123)\], and Log\-Cholesky Metric \(LCM\)\[[56](https://arxiv.org/html/2607.08783#bib.bib126)\]\. We denote LEM and AIM as\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}and\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}, as they are induced by the followingO​\(n\)\\mathrm\{O\}\(\{n\}\)\-invariant inner product on the tangent space at the identity matrix\[[73](https://arxiv.org/html/2607.08783#bib.bib135)\]:

⟨V,W⟩\(α,β\)=α​⟨V,W⟩\+β​tr⁡\(V\)​tr⁡\(W\),\\langle V,W\\rangle^\{\(\\alpha,\\beta\)\}=\\alpha\\langle V,W\\rangle\+\\beta\\operatorname\{tr\}\(V\)\\operatorname\{tr\}\(W\),\(3\)whereV,W∈TI​𝒮\+\+n≅𝒮nV,W\\in T\_\{I\}\\mathcal\{S\}^\{n\}\_\{\+\+\}\\cong\\mathcal\{S\}^\{n\}, and\(α,β\)∈𝐒𝐓=\{\(α,β\)∈ℝ2∣min⁡\(α,α\+n​β\)\>0\}\(\\alpha,\\beta\)\\in\\mathbf\{ST\}=\\\{\(\\alpha,\\beta\)\\in\\mathbb\{R\}^\{2\}\\mid\\min\(\\alpha,\\alpha\+n\\beta\)\>0\\\}\.[Tab\.I](https://arxiv.org/html/2607.08783#S2.T1)summarizes the Lie structures on SPD manifolds with the following notations\. LetP,Q∈𝒮\+\+nP,Q\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}be SPD matrices\. We denote the matrix logarithm, exponentiation, and Cholesky decomposition bylog⁡\(⋅\)\\operatorname\{log\}\(\\cdot\),exp⁡\(⋅\)\\operatorname\{exp\}\(\\cdot\), andChol⁡\(⋅\)\\operatorname\{Chol\}\(\\cdot\), respectively\. The Cholesky factors ofPPandQQareL=Chol⁡\(P\)L=\\operatorname\{Chol\}\(P\)andK=Chol⁡\(Q\)K=\\operatorname\{Chol\}\(Q\)\.⌊⋅⌋\\lfloor\\cdot\\rfloorreturns the strictly lower triangular part of a square matrix\. Other Riemannian operators are summarized in[Sec\.B\.3](https://arxiv.org/html/2607.08783#A2.SS3)\.

TABLE I:SPD Lie groups and invariant metrics\.MetricQ⊙PQ\\odot PInvariance\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}K​P​K⊤KPK^\{\\top\}Left\-invariance\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}exp⁡\(log⁡\(P\)\+log⁡\(Q\)\)\\operatorname\{exp\}\(\\operatorname\{log\}\(P\)\+\\operatorname\{log\}\(Q\)\)Bi\-invarianceLCMChol−1⁡\(⌊L\+K⌋\+𝕂​𝕃\)\\operatorname\{Chol\}^\{\-1\}\(\\lfloor L\+K\\rfloor\+\\mathbb\{K\}\\mathbb\{L\}\)Bi\-invarianceFor\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}and LCM, the Fréchet mean admits a closed\-form expression\. Moreover, since\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}has non\-positive sectional curvature\[[73](https://arxiv.org/html/2607.08783#bib.bib135), Tab\. 5\], the WFM is a global operation\[[7](https://arxiv.org/html/2607.08783#bib.bib141), Ch\. 6\.1\.5\]and can be computed by the Karcher flow algorithm\[[49](https://arxiv.org/html/2607.08783#bib.bib138)\]\.

### 2\.3Rotation Lie Groups

The set ofn×nn\\times nrotation matrices form a Lie group, known as special orthogonal groups, dented asSO​\(n\)\\mathrm\{SO\}\(n\)\[[76](https://arxiv.org/html/2607.08783#bib.bib146)\]\. Its group operation is the matrix product, with the identity matrix as the neutral element\. Any tangent vectorAAinTR​SO​\(n\)T\_\{R\}\\mathrm\{SO\}\(n\)can be represented asA=R​VA=RV, withV∈𝔰​𝔬​\(n\)V\\in\\mathfrak\{so\}\(n\)\. Here,𝔰​𝔬​\(n\)\\mathfrak\{so\}\(n\)is the Lie algebra ofSO​\(n\)\\mathrm\{SO\}\(n\), which is the tangent space at the identity matrix, formed by the set ofn×nn\\times nskew\-symmetric matrices\. The Fréchet mean can be obtained by Karcher flow\[[59](https://arxiv.org/html/2607.08783#bib.bib18)\]\. Furthermore, if all the rotations lie in a closed ball of radiusr<π/2r<\\nicefrac\{\{\\pi\}\}\{\{2\}\}, then Karcher flow converges to the unique mean\[[59](https://arxiv.org/html/2607.08783#bib.bib18), Thm\. 5\.1\]\. The associated Riemannian operators are summarized in[Sec\.B\.4](https://arxiv.org/html/2607.08783#A2.SS4)\.

### 2\.4Full\-Rank Correlation Lie Groups

The correlation matrix of a covariance matrixΣ\\Sigmais defined asC=Cor⁡\(Σ\)=𝔻​\(Σ\)−1/2​Σ​𝔻​\(Σ\)−1/2C=\\operatorname\{Cor\}\(\\Sigma\)=\\mathbb\{D\}\(\\Sigma\)^\{\-\\nicefrac\{\{1\}\}\{\{2\}\}\}\\Sigma\\mathbb\{D\}\(\\Sigma\)^\{\-\\nicefrac\{\{1\}\}\{\{2\}\}\}, where𝔻​\(⋅\)\\mathbb\{D\}\(\\cdot\)returns a diagonal matrix with diagonal elements ofΣ\\Sigma\. The space ofn×nn\\times nfull\-rank correlation matrices, denoted as𝒞\+\+n\\mathcal\{C\}^\{n\}\_\{\+\+\}, forms a quotient manifold of the SPD manifold𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}\[[28](https://arxiv.org/html/2607.08783#bib.bib67), Thm\. 1\], referred to as the correlation manifold\. This manifold can be interpreted as a compactly normalized SPD manifold that encodes scale\-invariant information\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]\. However, its Riemannian structure has been less studied than SPD matrices\. Recently, Thanwerdas and Pennec\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]developed two convenient Riemannian metrics: the Euclidean\-Cholesky Metric \(ECM\) and Log\-Euclidean\-Cholesky Metric \(LECM\)\. Thanwerdas\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\]further proposed two permutation\-invariant metrics, the Off\-Log Metric \(OLM\) and Log\-Scaled Metric \(LSM\)\. All four geometries above are pullback metrics from simpler Euclidean spaces\. We first review the related prototype spaces, followed by an examination of the four Riemannian metrics\.

Prototype spaces:LT1n\\mathrm\{LT\}\_\{1\}^\{n\}andLT0n\\mathrm\{LT\}\_\{0\}^\{n\}denote the Euclidean spaces ofn×nn\\times nlower triangular matrices with unit and null diagonals, respectively\.Holn\\mathrm\{Hol\}^\{n\}denotes the Euclidean space ofn×nn\\times nsymmetric matrices with null diagonals\. The tangent spaceTC​𝒞\+\+nT\_\{C\}\\mathcal\{C\}^\{n\}\_\{\+\+\}atC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}can be identified withHoln\\mathrm\{Hol\}^\{n\}\.Row0n\\mathrm\{Row\}\_\{0\}^\{n\}denotes the Euclidean space ofn×nn\\times nsymmetric matrices with null row sum\.

ECMis derived fromLT1n\\mathrm\{LT\}\_\{1\}^\{n\}by𝒞\+\+n⇌Θ−1=Cor∘Chol−1ΘLT1n\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\\Theta^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}\]\{\\Theta\}\\mathrm\{LT\}\_\{1\}^\{n\}, whereΘ​\(C\)=𝔻​\(Chol⁡\(C\)\)−1​Chol⁡\(C\)\\Theta\(C\)=\\mathbb\{D\}\(\\operatorname\{Chol\}\(C\)\)^\{\-1\}\\operatorname\{Chol\}\(C\)for anyC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\.

LECMis defined by further pulling back ECM:𝒞\+\+n⇌\(log∘Θ\)−1=Cor∘Chol−1∘explog∘ΘLT0n\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\(\\log\\circ\\Theta\)^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}\\circ\\exp\]\{\\log\\circ\\Theta\}\\mathrm\{LT\}\_\{0\}^\{n\}\. Due to the nilpotency ofLT0n\\mathrm\{LT\}\_\{0\}^\{n\}, the matrix logarithmlog⁡\(⋅\):LT1n→LT0n\\log\(\\cdot\):\\mathrm\{LT\}\_\{1\}^\{n\}\\rightarrow\\mathrm\{LT\}\_\{0\}^\{n\}and its inverseexp⁡\(⋅\)\\exp\(\\cdot\)overLT0n\\mathrm\{LT\}\_\{0\}^\{n\}are free from eigendecomposition\.

OLMis derived from a permutation invariant inner product overHoln\\mathrm\{Hol\}^\{n\}by𝒞\+\+n⇌Exp∘Log∘=Off∘logHoln\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\\operatorname\{Exp\}^\{\\circ\}\]\{\\operatorname\{Log\}^\{\\circ\}=\\mathrm\{Off\}\\circ\\log\}\\mathrm\{Hol\}^\{n\}\. Here,Off​\(⋅\)\\mathrm\{Off\}\(\\cdot\)returns a matrix inHoln\\mathrm\{Hol\}^\{n\}consisting of off\-diagonal elements\. For any symmetric hollow matrixH∈HolnH\\in\\mathrm\{Hol\}^\{n\}, there exists a unique diagonal matrix𝒟\+​\(H\)\\mathcal\{D\}^\{\+\}\(H\), such thatExp∘⁡\(H\)=exp⁡\(𝒟\+​\(H\)\+H\)∈𝒞\+\+n\\operatorname\{Exp\}^\{\\circ\}\(H\)=\\exp\(\\mathcal\{D\}^\{\+\}\(H\)\+H\)\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}is a diffeomorphism, where𝒟\+​\(H\)\\mathcal\{D\}^\{\+\}\(H\)can be computed by the following exponentially converged algorithm:Dk\+1=Dk−log⁡\(𝔻​\(exp⁡\(Dk\+H\)\)\)D\_\{k\+1\}=D\_\{k\}\-\\log\(\\mathbb\{D\}\(\\exp\(D\_\{k\}\+H\)\)\)\[[2](https://arxiv.org/html/2607.08783#bib.bib66), Sec\. 5\]\.

LSMis derived from a permutation invariant inner product overRow0n\\mathrm\{Row\}\_\{0\}^\{n\}by𝒞\+\+n⇌Exp⋆=Cor∘expLog⋆Row0n\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\\operatorname\{Exp\}^\{\\star\}=\\operatorname\{Cor\}\\circ\\exp\]\{\\operatorname\{Log\}^\{\\star\}\}\\mathrm\{Row\}\_\{0\}^\{n\}\. For any correlation matrixC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}, there exists a unique positive diagonal matrix𝒟⋆​\(C\)\\mathcal\{D\}^\{\\star\}\(C\)such thatLog⋆⁡\(C\)=log⁡\(𝒟⋆​\(C\)​C​𝒟⋆​\(C\)\)∈Row0n\\operatorname\{Log\}^\{\\star\}\(C\)=\\log\(\\mathcal\{D\}^\{\\star\}\(C\)C\\mathcal\{D\}^\{\\star\}\(C\)\)\\in\\mathrm\{Row\}\_\{0\}^\{n\}is a diffeomorphism, where𝒟⋆​\(C\)\\mathcal\{D\}^\{\\star\}\(C\)could be solved by damped Newton’s method\[[75](https://arxiv.org/html/2607.08783#bib.bib68), Sec\. 3\.5\]\.

Four group operations are defined as[Eq\.1](https://arxiv.org/html/2607.08783#S2.E1):

P⊙Q=f−1​\(f​\(P\)\+f​\(Q\)\),∀P,Q∈𝒞\+\+n\.P\\odot Q=f^\{\-1\}\(f\(P\)\+f\(Q\)\),\\forall P,Q\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\.\(4\)withffasΘ\\Theta,log∘Θ\\log\\circ\\Theta,Log∘\\operatorname\{Log\}^\{\\circ\}, andLog⋆\\operatorname\{Log\}^\{\\star\}for ECM, LECM, OLM, and LSM, respectively\. The following discusses the invariance and WFM\. More details on Riemannian structures are presented in[Sec\.B\.5](https://arxiv.org/html/2607.08783#A2.SS5)\.

\{mymath\}

###### Proposition 2\.5\(Invariance & WFM\)\.

ECM, LECM, OLM, and LSM are bi\-invariant\. Following the notations in[Def\.2\.4](https://arxiv.org/html/2607.08783#S2.Thmtheorem4), the WFM for a batch of samples\{Pi​…​N∈𝒞\+\+n\}\\\{P\_\{i\\ldots N\}\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\\\}aref−1​\(∑i=1Nwi​f​\(Pi\)\)f^\{\-1\}\\left\(\\sum\_\{i=1\}^\{N\}w\_\{i\}f\(P\_\{i\}\)\\right\), withffasΘ\\Theta,log∘Θ\\log\\circ\\Theta,Log∘\\operatorname\{Log\}^\{\\circ\}, andLog⋆\\operatorname\{Log\}^\{\\star\}for ECM, LECM, OLM, and LSM, respectively\.

###### Proof\.

The proof is presented in[Sec\.H\.1](https://arxiv.org/html/2607.08783#A8.SS1)\. ∎

## 3Revisiting Normalization

TABLE II:Summary of some representative RBN methods\.MethodsInvolvedStatisticsControllableMeanControllableVarianceGeometriesSPDBN\[[13](https://arxiv.org/html/2607.08783#bib.bib96), Alg\. 1\]Mean✓N/ASPD manifolds under AIMSPDBN\[[52](https://arxiv.org/html/2607.08783#bib.bib92), Alg\. 1\]Mean\+Variance✓✓SPD manifolds under AIMSPDDSMBN\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]Mean\+Variance✓✓SPD manifolds under AIMManifoldNorm\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 1\-2\]Mean\+Variance✗✗Riemannian homogeneous spaceManifoldNorm\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 3\-4\]Mean\+Variance✓✓A specific Lie group structure and distanceRBN\[[58](https://arxiv.org/html/2607.08783#bib.bib99), Alg\. 2\]Mean\+Variance✗✗Geodesically complete manifoldsLieBN \(Ours\)Mean\+Variance✓✓General Lie groups### 3\.1Revisiting Euclidean Normalization

In Euclidean DNNs, normalization is a significant technique for accelerating network training by mitigating the issue of internal covariate shift\[[46](https://arxiv.org/html/2607.08783#bib.bib37)\]\. While various normalization methods have been introduced\[[46](https://arxiv.org/html/2607.08783#bib.bib37),[4](https://arxiv.org/html/2607.08783#bib.bib77),[77](https://arxiv.org/html/2607.08783#bib.bib76),[87](https://arxiv.org/html/2607.08783#bib.bib55)\], they all share a common purpose: the normalization of the first and second moments\. This paper focuses on Batch Normalization \(BN\), the prototype of other normalization variants\.

Given a batch of activations\{xi​…​N\}\\\{x\_\{i\\ldots N\}\\\}, the core operations in the standard Euclidean BN can be expressed as:

∀i≤N,xi←γ​xi−μbvb2\+ϵ\+β\\forall i\\leq N,x\_\{i\}\\leftarrow\\gamma\\frac\{x\_\{i\}\-\\mu\_\{b\}\}\{\\sqrt\{v^\{2\}\_\{b\}\+\\epsilon\}\}\+\\beta\(5\)whereμb\\mu\_\{b\}is the batch mean,vb2v^\{2\}\_\{b\}is the batch variance,γ\\gammais the scaling parameter,β\\betais the biasing parameter, andϵ\\epsilonis a small scalar for stability\.

### 3\.2Revisiting Riemannian Batch Normalization

Although endeavors have been made to develop Riemannian normalization approaches tailored for manifolds, none of the existing methods effectively handle the first and second moments in a principled manner\.

Brooks*et al*\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\]introduced RBN over SPD manifolds under AIM\. The core operations are defined as follows:

Centering from mean​M∈𝒮\+\+n:P¯i←M−12​Pi​M−12,\\displaystyle\\text\{Centering from mean \}M\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}:\\bar\{P\}\_\{i\}\\leftarrow M^\{\-\\frac\{1\}\{2\}\}P\_\{i\}M^\{\-\\frac\{1\}\{2\}\},\(6\)Biasing towards parameter​B∈𝒮\+\+n:P^i←B12​P¯i​B12,\\displaystyle\\text\{Biasing towards parameter \}B\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}:\\hat\{P\}\_\{i\}\\leftarrow B^\{\\frac\{1\}\{2\}\}\\bar\{P\}\_\{i\}B^\{\\frac\{1\}\{2\}\},\(7\)where\{Pi​…​N\}\\\{P\_\{i\\ldots N\}\\\}are SPD matrices, andMMare their Fréchet mean under AIM\. LetΓP→Q​\(S\)=ExpQ⁡\[PTP→Q⁡\(LogP⁡\(S\)\)\]\\Gamma\_\{P\\rightarrow Q\}\(S\)=\\operatorname\{Exp\}\_\{Q\}\\left\[\\operatorname\{PT\}\_\{P\\rightarrow Q\}\\left\(\\operatorname\{Log\}\_\{P\}\(S\)\\right\)\\right\], whereP,Q,S∈𝒮\+\+nP,Q,S\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}\. Under AIM,[Eqs\.6](https://arxiv.org/html/2607.08783#S3.E6)and[7](https://arxiv.org/html/2607.08783#S3.E7)can be more generally expressed as

ΓI→B​\[ΓM→I​\(Pi\)\]\.\\Gamma\_\{I\\rightarrow B\}\[\\Gamma\_\{M\\rightarrow I\}\(P\_\{i\}\)\]\.\(8\)However,[Eqs\.6](https://arxiv.org/html/2607.08783#S3.E6)and[7](https://arxiv.org/html/2607.08783#S3.E7)only consider the Riemannian mean333Although not discussed in\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\],[Eqs\.6](https://arxiv.org/html/2607.08783#S3.E6)and[7](https://arxiv.org/html/2607.08783#S3.E7)as congruent actions can transfer the batch mean into desirable one under AIM\.and does not consider the Riemannian variance\. To remedy this limitation, Kobler*et al*\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]further extended the RBN to involve the second\-order statistics\. The key operation is formulated as

∀i≤N,P¯i←ΓI→B​\[\(ΓM→I​\(Pi\)\)sv\],\\forall i\\leq N,\\bar\{P\}\_\{i\}\\leftarrow\\Gamma\_\{I\\rightarrow B\}\[\(\\Gamma\_\{M\\rightarrow I\}\(P\_\{i\}\)\)^\{\\frac\{s\}\{v\}\}\],\(9\)wherev2v^\{2\}is the Fréchet variance,s∈ℝs\\in\\mathbb\{R\}is a scaling factor\. However, this method is still limited to SPD manifolds under AIM\. In parallel, Chakraborty\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 1\-2\]proposed a general framework for Riemannian homogeneous spaces based on[Eq\.8](https://arxiv.org/html/2607.08783#S3.E8), which involves both first and second moments\. However,[Eq\.8](https://arxiv.org/html/2607.08783#S3.E8)does not generally guarantee the control over Riemannian mean, resulting in agnostic Riemannian statistics\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Sec\. 3\.1\]\. To mitigate this limitation, Chakraborty\[[17](https://arxiv.org/html/2607.08783#bib.bib137), Algs\. 3\-4\]further proposed normalization over the matrix Lie group\. However, the discussion is limited to a certain distance, limiting the applicability of their method\. On the other hand, Lou*et al*\[[58](https://arxiv.org/html/2607.08783#bib.bib99), Alg\. 2\]proposed an RBN based on a variance of[Eq\.8](https://arxiv.org/html/2607.08783#S3.E8)\. Similarly, their approach suffers from the same problem of agnostic Riemannian statistics on general manifolds\.

In summary, prevailing Riemannian normalization approaches lack a principled guarantee for controlling the first and second\-order statistics\. In contrast, our method can normalize first and second\-order statistics over general Lie groups\. We summarize the above RBN methods in[Tab\.II](https://arxiv.org/html/2607.08783#S3.T2)\.

## 4Lie Group Batch Normalization

Since every Lie group naturally admits invariant metrics, we propose batch normalization over Lie groups based on invariant metrics, referred to as LieBN\. We first introduce the core operations under left\-invariant metrics and then extend them to right\-invariant metrics\. Finally, we present the theoretical LieBN framework\. In the following, we denote the neutral element in the Lie groupℳ\\mathcal\{M\}asEE444The neutral elementEEis not necessarily the identity matrix\.\.

### 4\.1Ingredients under Left\-invariant Metrics

In this subsection, we always assume that the Lie groupℳ\\mathcal\{M\}admits a left\-invariant metricgLg^\{\\mathrm\{L\}\}\. Recalling the standard Euclidean BN\[[46](https://arxiv.org/html/2607.08783#bib.bib37)\]in[Eq\.5](https://arxiv.org/html/2607.08783#S3.E5), two key points are noteworthy: \(a\) the Euclidean BN implicitly assumes a Gaussian distribution and can effectively normalize the latent Gaussian distribution; \(b\) the centering and biasing operations control the mean, while the scaling controls the variance\. Therefore, extending BN into Lie groups requires the counterparts of Gaussian distribution, centering, biasing, and scaling\.

There are several notions of Gaussian distribution over manifolds\[[66](https://arxiv.org/html/2607.08783#bib.bib158),[86](https://arxiv.org/html/2607.08783#bib.bib109),[15](https://arxiv.org/html/2607.08783#bib.bib108),[5](https://arxiv.org/html/2607.08783#bib.bib24)\]\. We adopt the intrinsic definition from Chakraborty*et al*\[[15](https://arxiv.org/html/2607.08783#bib.bib108)\], which characterizes a Gaussian distribution on the Lie groupℳ\\mathcal\{M\}with a mean parameterM∈ℳM\\in\\mathcal\{M\}and varianceσ2\\sigma^\{2\}\. This distribution is denoted as𝒩​\(M,σ2\)\\mathcal\{N\}\(M,\\sigma^\{2\}\), and its Probability Density Function \(P\.D\.F\.\) is

p​\(X∣M,σ2\)=k​\(σ\)​exp⁡\(−d\(X,M\)22​σ2\),p\\left\(X\\mid M,\\sigma^\{2\}\\right\)=k\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(X,M\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\),\(10\)wherek​\(σ\)k\(\\sigma\)is the normalizing constant andd⁡\(⋅,⋅\)\\operatorname\{d\}\(\\cdot,\\cdot\)is the geodesic distance\. Whenℳ\\mathcal\{M\}isℝ\\mathbb\{R\}with the standard Euclidean metric,[Eq\.10](https://arxiv.org/html/2607.08783#S4.E10)reduces to the Euclidean Gaussian\.

On Lie groups, the natural counterparts of addition and subtraction in[Eq\.5](https://arxiv.org/html/2607.08783#S3.E5)are group operations\. Therefore, centering and biasing on Lie groups can be defined by the left translation\. Additionally, we define scaling via the tangent space\. Specifically, for a batch of activations\{Pi​…​N∈ℳ\}\\\{P\_\{i\\ldots N\}\\in\\mathcal\{M\}\\\}, we define the key operations of LieBN as follows:

Centering from mean​M∈ℳ:P¯i←LM⊙−1​\(Pi\),\\displaystyle\\text\{Centering from mean \}M\\in\\mathcal\{M\}:\\bar\{P\}\_\{i\}\\leftarrow L\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\),\(11\)Scaling:​P^i←ExpE⁡\[sv2\+ϵ​LogE⁡\(P¯i\)\],\\displaystyle\\text\{Scaling: \}\\hat\{P\}\_\{i\}\\leftarrow\\operatorname\{Exp\}\_\{E\}\\left\[\\frac\{s\}\{\\sqrt\{v^\{2\}\+\\epsilon\}\}\\operatorname\{Log\}\_\{E\}\(\\bar\{P\}\_\{i\}\)\\right\],\(12\)Biasing towards parameter​B∈ℳ:P~i←LB​\(P^i\),\\displaystyle\\text\{Biasing towards parameter \}B\\in\\mathcal\{M\}:\\tilde\{P\}\_\{i\}\\leftarrow L\_\{B\}\\left\(\\hat\{P\}\_\{i\}\\right\),\(13\)whereMMis the Fréchet mean,v2v^\{2\}is the Fréchet variance,M⊙−1∈ℳM\_\{\\odot\}^\{\-1\}\\in\\mathcal\{M\}is the group inverse ofMM,LM⊙−1L\_\{M\_\{\\odot\}^\{\-1\}\}andLBL\_\{B\}are left translations \(LB​\(Pi\)=B⊙PiL\_\{B\}\(P\_\{i\}\)=B\\odot P\_\{i\}\), ands∈ℝ/\{0\}s\\in\\mathbb\{R\}/\\\{0\\\}is a scaling parameter\. The following two propositions demonstrate the above operations in normalizing mean and variance: one related to population statistics and the other related to sample statistics\.

\{mymath\}

###### Proposition 4\.1\(Population\)\.

Given a random pointXXover\{ℳ,gL\}\\\{\\mathcal\{M\},g^\{\\mathrm\{L\}\}\\\}, and the Gaussian distribution𝒩​\(M,v2\)\\mathcal\{N\}\(M,v^\{2\}\)defined in[Eq\.10](https://arxiv.org/html/2607.08783#S4.E10), we have the following for the population statistics:

1. 1\.\(MLE ofMM\) Given\{Pi​…​N∈ℳ\}\\\{P\_\{i\\ldots N\}\\in\\mathcal\{M\}\\\}i\.i\.d\. sampled from𝒩​\(M,v2\)\\mathcal\{N\}\(M,v^\{2\}\), the maximum likelihood estimator \(MLE\) ofMMis the sample Fréchet mean\.
2. 2\.\(Gaussian homogeneity\) GivenX∼𝒩​\(M,v2\)X\\sim\\mathcal\{N\}\(M,v^\{2\}\)andB∈ℳB\\in\\mathcal\{M\},LB⁡\(X\)∼𝒩​\(LB⁡\(M\),v2\)\\operatorname\{L\}\_\{B\}\(X\)\\sim\\mathcal\{N\}\(\\operatorname\{L\}\_\{B\}\(M\),v^\{2\}\)

###### Proof\.

The proof is presented in[Sec\.H\.2](https://arxiv.org/html/2607.08783#A8.SS2)\. ∎

\{mymath\}

###### Proposition 4\.2\(Sample\)\.

GivenNNsamples\{Pi​…​N\}\\\{P\_\{i\\ldots N\}\\\}over the Lie group\{ℳ,gL\}\\\{\\mathcal\{M\},g^\{\\mathrm\{L\}\}\\\}, denotingϕs​\(Pi\)=ExpE⁡\[s​LogE⁡\(Pi\)\]\\phi\_\{s\}\(P\_\{i\}\)=\\operatorname\{Exp\}\_\{E\}\\left\[s\\operatorname\{Log\}\_\{E\}\(P\_\{i\}\)\\right\], we have the following for the sample statistics\.

- •Sample mean homogeneity: FM⁡\{LB⁡\(Pi\)\}=LB⁡\(FM⁡\{Pi\}\),∀B∈ℳ\.\\operatorname\{FM\}\\\{\\operatorname\{L\}\_\{B\}\(P\_\{i\}\)\\\}=\\operatorname\{L\}\_\{B\}\(\\operatorname\{FM\}\\\{P\_\{i\}\\\}\),\\forall B\\in\\mathcal\{M\}\.\(14\)
- •Controllable dispersion fromEE: ∑i=1Nwi​d2⁡\(ϕs​\(Pi\),E\)=s2​∑i=1Nwi​d2⁡\(Pi,E\),\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\operatorname\{d\}^\{2\}\(\\phi\_\{s\}\(P\_\{i\}\),E\)=s^\{2\}\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\operatorname\{d\}^\{2\}\(P\_\{i\},E\),\(15\)

where\{w1​…​N\}\\\{w\_\{1\\ldots N\}\\\}are weights satisfying a convexity constraint,*i\.e\.*,∀i,wi\>0\\forall i,w\_\{i\}\>0and∑iwi=1\\sum\_\{i\}w\_\{i\}=1\.

###### Proof\.

The proof is presented in[Sec\.H\.3](https://arxiv.org/html/2607.08783#A8.SS3)\. ∎

[Prop\.4\.1](https://arxiv.org/html/2607.08783#S4.Thmtheorem1)and[Eq\.14](https://arxiv.org/html/2607.08783#S4.E14)imply that our centering and biasing in[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11)and[13](https://arxiv.org/html/2607.08783#S4.E13)can transfer the sample and population mean\. As the post\-centering mean isEE,[Eq\.15](https://arxiv.org/html/2607.08783#S4.E15)implies that[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)can control the sample variance\. More interestingly, the latent Gaussian distribution can be transferred under some geometries, such as SPD manifolds under LEM and LCM, which are discussed in[App\.D](https://arxiv.org/html/2607.08783#A4)\.

### 4\.2Ingredients under Right\-invariant Metrics

The key insight beneath[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11),[13](https://arxiv.org/html/2607.08783#S4.E13),[4\.1](https://arxiv.org/html/2607.08783#S4.Thmtheorem1)and[4\.2](https://arxiv.org/html/2607.08783#S4.Thmtheorem2)is that left translation is an isometry under left\-invariant metrics\. Similarly, right translation is an isometry under right\-invariant metrics\. Therefore, it can be used for centering and biasing under right\-invariant metrics\. Following the previous notations, we define the centering and biasing under a right\-invariant metricgRg^\{\\mathrm\{R\}\}as

centering toE:​P¯i←RM⊙−1⁡\(Pi\),\\displaystyle\\text\{ centering to $E$: \}\\bar\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\),\(16\)biasing towardsB:​P~i←RB⁡\(P^i\)\.\\displaystyle\\text\{ biasing towards $B$: \}\\tilde\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{B\}\(\\hat\{P\}\_\{i\}\)\.\(17\)
Similar to the case under left\-invariant metrics,[Props\.4\.1](https://arxiv.org/html/2607.08783#S4.Thmtheorem1)and[4\.2](https://arxiv.org/html/2607.08783#S4.Thmtheorem2)can be easily extended into the ones under right\-invariant metrics\. Notable, the proofs for MLE ofMMin[Prop\.4\.1](https://arxiv.org/html/2607.08783#S4.Thmtheorem1)and controllable dispersion in[Prop\.4\.2](https://arxiv.org/html/2607.08783#S4.Thmtheorem2)can be directly applied to the right\-invariant metric\. Therefore, we only show the homogeneity in the following proposition\.

\{mymath\}

###### Proposition 4\.5\.

Given a random pointX∼𝒩​\(M,v2\)X\\sim\\mathcal\{N\}\(M,v^\{2\}\)over\{ℳ,gR\}\\\{\\mathcal\{M\},g^\{\\mathrm\{R\}\}\\\},B∈ℳB\\in\\mathcal\{M\}, andNNsamples\{Pi​…​N\}\\\{P\_\{i\\ldots N\}\\\}overℳ\\mathcal\{M\}, we have

1. 1\.Gaussian homogeneity:RB⁡\(X\)∼𝒩​\(RB⁡\(M\),v2\)\\operatorname\{R\}\_\{B\}\(X\)\\sim\\mathcal\{N\}\(\\operatorname\{R\}\_\{B\}\(M\),v^\{2\}\);
2. 2\.Sample homogeneity:FM⁡\{RB⁡\(Pi\)\}=RB⁡\(FM⁡\{Pi\}\)\\operatorname\{FM\}\\\{\\operatorname\{R\}\_\{B\}\(P\_\{i\}\)\\\}=\\operatorname\{R\}\_\{B\}\(\\operatorname\{FM\}\\\{P\_\{i\}\\\}\)\.

###### Proof\.

The proof is presented in[Sec\.H\.4](https://arxiv.org/html/2607.08783#A8.SS4)\. ∎

### 4\.3LieBN under Invariant Metrics

Input :A batch of activations

\{P1​…​N\}\\\{P\_\{1\\ldots N\}\\\}over Lie groups

\{ℳ,⊙,g\}\\\{\\mathcal\{M\},\\odot,g\\\}, a small positive constant

ϵ\\epsilon, and momentum

γ∈\[0,1\]\\gamma\\in\[0,1\], running mean

Mr=EM\_\{r\}=E, running variance

vr2=1v^\{2\}\_\{r\}=1, biasing parameter

B∈ℳB\\in\\mathcal\{M\}, and scaling parameter

s∈ℝ/\{0\}s\\in\\mathbb\{R\}/\\\{0\\\}\.

Output :Normalized activations

\{P~1​…​N\}\\\{\\tilde\{P\}\_\{1\\ldots N\}\\\}\.

1exif*training*then

Compute batch mean

MbM\_\{b\}and variance

vb2v\_\{b\}^\{2\}
Update running statistics:

Mr←WFM⁡\(\{1−γ,γ\},\{Mr,Mb\}\)M\_\{r\}\\leftarrow\\operatorname\{WFM\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{M\_\{r\},M\_\{b\}\\\}\)vr2←\(1−γ\)​vr2\+γ​vb2v^\{2\}\_\{r\}\\leftarrow\(1\-\\gamma\)v^\{2\}\_\{r\}\+\\gamma v^\{2\}\_\{b\}
Use the batch statistics,

M←Mb,v2←vb2M\\leftarrow M\_\{b\},v^\{2\}\\leftarrow v^\{2\}\_\{b\}
else

Use the running statistics,

M←Mr,v2←vr2M\\leftarrow M\_\{r\},v^\{2\}\\leftarrow v^\{2\}\_\{r\}
end if

for*i←1i\\leftarrow 1toNN*do

Centering to the neutral element

EE:

if*ggis left\-invariant*then

P¯i←LM⊙−1⁡\(Pi\)\\bar\{P\}\_\{i\}\\leftarrow\\operatorname\{L\}\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\)
else

P¯i←RM⊙−1⁡\(Pi\)\\bar\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\)
end if

Scaling the variance:

P^i←ExpE⁡\[sv2\+ϵ​LogE⁡\(P¯i\)\]\\hat\{P\}\_\{i\}\\leftarrow\\operatorname\{Exp\}\_\{E\}\\left\[\\frac\{s\}\{\\sqrt\{v^\{2\}\+\\epsilon\}\}\\operatorname\{Log\}\_\{E\}\(\\bar\{P\}\_\{i\}\)\\right\]
Biasing towards parameter

BB:

if*ggis left\-invariant*then

P~i←LB⁡\(P^i\)\\tilde\{P\}\_\{i\}\\leftarrow\\operatorname\{L\}\_\{B\}\(\\hat\{P\}\_\{i\}\)
else

P~i←RB⁡\(P^i\)\\tilde\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{B\}\(\\hat\{P\}\_\{i\}\)
end if

end for

Algorithm 1Lie Group Batch Normalization \(LieBN\)With the above ingredients,[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)presents our theoretical LieBN framework\. Similar to Ioffe*et al*\[[46](https://arxiv.org/html/2607.08783#bib.bib37)\], we use the moving average to update the running statistics\. For a bi\-invariant metric, LieBN can be implemented using either left or right translation\. If the Lie group is commutative, LieBN under left and right translations are equivalent\.[Tab\.III](https://arxiv.org/html/2607.08783#S4.T3)summarizes the LieBN types under different conditions\.

TABLE III:Summary of LieBN types\.CommutativityNon\-commutativeCommutativeInvarianceLeftRightBiLeft = Right = BiLieBN TypesLeftRightLeft & RightLeft = Right

The centering and biasing in Euclidean BN correspond to the group action ofℝ\\mathbb\{R\}\. From a geometric perspective, the standard Euclidean metric is invariant under this group operation\. Consequently, it is not surprising that our LieBN algorithm naturally generalizes the standard Euclidean BN\.

\{mymath\}

###### Proposition 4\.6\.

The LieBN algorithm presented in[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)is equivalent to the standard Euclidean BN whenℳ=ℝn\\mathcal\{M\}=\\mathbb\{R\}^\{n\}, both during the training and testing phases\.

###### Proof\.

The proof is presented in[Sec\.H\.5](https://arxiv.org/html/2607.08783#A8.SS5)\. ∎

## 5Manifestations

This section instantiates our LieBN in[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)on nine different Lie groups, including four on the SPD manifold, one on the rotation, and four on the correlation manifold\.

### 5\.1LieBN on SPD Manifolds

We first extend the current Lie groups on SPD manifolds by the matrix power deformation, resulting in three families of parameterized Lie groups\. Then, we propose a novel right\-invariant metric on the SPD manifold, the first non\-trivial right\-invariant metric on this manifold\. Finally, we construct LieBN layers based on these Lie structures\.

#### 5\.1\.1Deformed Lie Structures on SPD Manifolds

As shown in[Tab\.I](https://arxiv.org/html/2607.08783#S2.T1), there are three Lie groups on SPD manifolds, each with a left\-invariant metric\. These metrics include\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\},\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}, and LCM\. For clarity, we denote the group operations w\.r\.t\.\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\},\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}and LCM as⊙AI\\odot^\{\\mathrm\{AI\}\},⊙LE\\odot^\{\\mathrm\{LE\}\}and⊙LC\\odot^\{\\mathrm\{LC\}\}, respectively\.

Recently, Thanwerdas and Pennec\[[69](https://arxiv.org/html/2607.08783#bib.bib26)\]further extended\(α,β\)\(\\alpha,\\beta\)\-AIM into three\-parameter families of metrics by the pullback of matrix power functionPθ​\(⋅\)\\mathrm\{P\}\_\{\\theta\}\(\\cdot\)and scaled by1θ2\\frac\{1\}\{\\theta^\{2\}\}, denoted as\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}\. The matrix power serves as a deformation, wherein\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}encompasses\(α,β\)\(\\alpha,\\beta\)\-AIM withθ=1\\theta=1, and becomes\(α,β\)\(\\alpha,\\beta\)\-LEM asθ\\thetaapproaches 0\[[70](https://arxiv.org/html/2607.08783#bib.bib25)\]\. Inspired by the deforming utility of the power function, we define the power\-deformed metrics of\(α,β\)\(\\alpha,\\beta\)\-LEM and LCM as the pullback metrics byPθ\\operatorname\{P\}\_\{\\theta\}and scaled by1θ2\\frac\{1\}\{\\theta^\{2\}\}\. We denote these two metrics as\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}andθ​\-LCM\\theta\\text\{\-LCM\}, respectively\. We have the following results w\.r\.t\. the deformation\.

\{mymath\}

###### Proposition 5\.1\(Deformation\)\.

\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}is equal to\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}\.θ\\theta\-LCM interpolates betweeng~\\tilde\{g\}\-LEM \(θ=0\\theta=0\) and LCM \(θ=1\\theta=1\)\. Here, given anyP∈𝒮\+\+nP\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}and tangent vectorsV,W∈TP​𝒮\+\+nV,W\\in T\_\{P\}\\mathcal\{S\}^\{n\}\_\{\+\+\},g~\\tilde\{g\}\-LEM is defined as

⟨V,W⟩P=g~​\(log∗,P⁡\(V\),log∗,P⁡\(W\)\),\\langle V,W\\rangle\_\{P\}=\\tilde\{g\}\(\\operatorname\{log\}\_\{\*,P\}\(V\),\\operatorname\{log\}\_\{\*,P\}\(W\)\),\(18\)whereg~​\(V1,V2\)=12​⟨V1,V2⟩−14​⟨𝔻​\(V1\),𝔻​\(V2\)⟩\\tilde\{g\}\(V\_\{1\},V\_\{2\}\)=\\frac\{1\}\{2\}\\langle V\_\{1\},V\_\{2\}\\rangle\-\\frac\{1\}\{4\}\\langle\\mathbb\{D\}\(V\_\{1\}\),\\mathbb\{D\}\(V\_\{2\}\)\\rangle,𝔻​\(Vi\)\\mathbb\{D\}\(V\_\{i\}\)is a diagonal matrix consisting of the diagonal elements ofViV\_\{i\}, andlog∗,P\\operatorname\{log\}\_\{\*,P\}is the differential map atPP\.

###### Proof\.

The proof is presented in[Sec\.H\.6](https://arxiv.org/html/2607.08783#A8.SS6)\. ∎

As\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}is equal to\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}, we focus on\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\},\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}, andθ​\-LCM\\theta\\text\{\-LCM\}in the following\. As a diffeomorphism,Pθ\\operatorname\{P\}\_\{\\theta\}also can pull back the group operation⊙AI\\odot^\{\\mathrm\{AI\}\}and⊙LC\\odot^\{\\mathrm\{LC\}\}, denoted as⊙θ​\-AI\\odot^\{\\theta\\text\{\-AI\}\}and⊙θ​\-LC\\odot^\{\\theta\\text\{\-LC\}\}\. We have the following proposition on the invariance\.

\{mymath\}

###### Proposition 5\.2\(Invariance\)\.

\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}is left\-invariant w\.r\.t\.⊙θ​\-AI\\odot^\{\\theta\\text\{\-AI\}\}, whileθ​\-LCM\\theta\\text\{\-LCM\}is bi\-invariant w\.r\.t\.⊙θ​\-LC\\odot^\{\\theta\\text\{\-LC\}\}\.

###### Proof\.

The proof is presented in[Sec\.H\.7](https://arxiv.org/html/2607.08783#A8.SS7)\. ∎

#### 5\.1\.2SPD Right\-invariant Metrics

AIM is left\-invariant w\.r\.t\.⊙AI\\odot^\{\\mathrm\{AI\}\}\. We also can define right\-invariant metric w\.r\.t⊙AI\\odot^\{\\mathrm\{AI\}\}by definition\[[32](https://arxiv.org/html/2607.08783#bib.bib143), Ch\. 1\.2\]:

gPCRI​\(V,W\)=⟨RP⊙AI−1⁣∗,P⁡\(V\),RP⊙AI−1⁣∗,P⁡\(W\)⟩Ig^\{\\mathrm\{CRI\}\}\_\{P\}\(V,W\)=\\left\\langle\\operatorname\{R\}\_\{P\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\}\*,P\}\(V\),\\operatorname\{R\}\_\{P\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\}\*,P\}\(W\)\\right\\rangle\_\{I\}\(19\)whereR\(⋅\)\\operatorname\{R\}\_\{\(\\cdot\)\}denotes Lie group right translation,P⊙AI−1P\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\}is the inverse ofPPunder⊙AI\\odot^\{\\mathrm\{AI\}\}, and⟨⋅,⋅⟩I\\left\\langle\\cdot,\\cdot\\right\\rangle\_\{I\}denotes an arbitrary inner product onTI​𝒮\+\+nT\_\{I\}\\mathcal\{S\}^\{n\}\_\{\+\+\}\. We set⟨⋅,⋅⟩I\\left\\langle\\cdot,\\cdot\\right\\rangle\_\{I\}the same as the AIM atII,*i\.e\.*,⟨⋅,⋅⟩\(α,β\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\beta\)\}\. We call this metric the Cholesky Right Invariant Metric \(CRIM\), as the group operation is defined by the matrix product of Cholesky factors\[[72](https://arxiv.org/html/2607.08783#bib.bib79), Sec\. 3\.2\]\.

\{mymath\}

###### Theorem 5\.3\.

Given any SPD matricesP,QP,Qand tangent vectorV∈TP​𝒮\+\+nV\\in T\_\{P\}\\mathcal\{S\}^\{n\}\_\{\+\+\}, the Riemannian operators on\{𝒮\+\+n,gCRI\}\\\{\\mathcal\{S\}^\{n\}\_\{\+\+\},g^\{\\mathrm\{CRI\}\}\\\}are

gPCRI​\(V,V\)\\displaystyle g^\{\\mathrm\{CRI\}\}\_\{P\}\(V,V\)=\(‖\(L​\(L−1​V​L−⊤\)12​L−1\)Sym‖\(α,β\)\)2\\displaystyle=\\left\(\\left\\\|\\left\(L\(L^\{\-1\}VL^\{\-\\top\}\)\_\{\\frac\{1\}\{2\}\}L^\{\-1\}\\right\)\_\{\\mathrm\{Sym\}\}\\right\\\|^\{\(\\alpha,\\beta\)\}\\right\)^\{2\}\(20\)d⁡\(P,Q\)\\displaystyle\\operatorname\{d\}\(P,Q\)=‖log⁡\(Q~−12​P~​Q~−12\)‖\(α,β\),\\displaystyle=\\left\\\|\\operatorname\{log\}\\left\(\\widetilde\{Q\}^\{\-\\frac\{1\}\{2\}\}\\widetilde\{P\}\\widetilde\{Q\}^\{\-\\frac\{1\}\{2\}\}\\right\)\\right\\\|^\{\(\\alpha,\\beta\)\},\(21\)ExpP⁡\(V\)\\displaystyle\\operatorname\{Exp\}\_\{P\}\(V\)=\(ExpP~AI⁡\(−V¯\)\)⊙AI−1,\\displaystyle=\\left\(\\operatorname\{Exp\}\_\{\\widetilde\{P\}\}^\{\\mathrm\{AI\}\}\\left\(\-\\bar\{V\}\\right\)\\right\)\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\},\(22\)LogP⁡\(Q\)\\displaystyle\\operatorname\{Log\}\_\{P\}\(Q\)=−\(L​L⊤​\(L​V~​L⊤\)12⊤\)Sym,\\displaystyle=\-\\left\(LL^\{\\top\}\\left\(L\\widetilde\{V\}L^\{\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}^\{\\top\}\\right\)\_\{\\mathrm\{Sym\}\},\(23\)whereLLis the Cholesky factor ofP=L​L⊤P=LL^\{\\top\},\(⋅\)⊙AI−1\(\\cdot\)\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\}is the group inverse,Q~\\widetilde\{Q\}andP~\\widetilde\{P\}are the group inverses ofPPandQQ,V¯=\(\(L−1​V​L−⊤\)12​L−1​L−⊤\)Sym\\bar\{V\}=\\left\(\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}L^\{\-1\}L^\{\-\\top\}\\right\)\_\{\\mathrm\{Sym\}\}, andV~=LogP~AI⁡\(Q~\)\\widetilde\{V\}=\\operatorname\{Log\}^\{\\mathrm\{AI\}\}\_\{\\widetilde\{P\}\}\\left\(\\widetilde\{Q\}\\right\)\. Here,\(X\)Sym=X\+X⊤,∀X∈ℝn×n\\left\(X\\right\)\_\{\\mathrm\{Sym\}\}=X\+X^\{\\top\},\\forall X\\in\\mathbb\{R\}^\{n\\times n\}denotes symmetrization, and\(⋅\)12\(\\cdot\)\_\{\\frac\{1\}\{2\}\}as its inverse map, namely\(X\)12=⌊X⌋\+12​𝕏\(X\)\_\{\\frac\{1\}\{2\}\}=\\lfloor X\\rfloor\+\\frac\{1\}\{2\}\\mathbb\{X\}\.

###### Proof\.

The proof is presented in[Sec\.H\.8](https://arxiv.org/html/2607.08783#A8.SS8)\. ∎

\{mymath\}

###### Corollary 5\.4\.

CRIM is geodesically complete, and the associated geodesic connecting SPD matricesPPandQQis

γ\(P,Q\)​\(t\)\\displaystyle\\gamma\_\{\(P,Q\)\}\(t\)=\{γAI​\(t;P~,Q~\)\}⊙AI−1\\displaystyle=\\left\\\{\\gamma^\{\\mathrm\{AI\}\}\(t;\\widetilde\{P\},\\widetilde\{Q\}\)\\right\\\}\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\}\(24\)=\{P~12​\(P~−12​Q~​P~−12\)t​P~12\}⊙AI−1,\\displaystyle=\\left\\\{\\widetilde\{P\}^\{\\frac\{1\}\{2\}\}\\left\(\\widetilde\{P\}^\{\-\\frac\{1\}\{2\}\}\\widetilde\{Q\}\\widetilde\{P\}^\{\-\\frac\{1\}\{2\}\}\\right\)^\{t\}\\widetilde\{P\}^\{\\frac\{1\}\{2\}\}\\right\\\}\_\{\\odot^\{\\mathrm\{AI\}\}\}^\{\-1\},whereP~=P⊙AI−1\\widetilde\{P\}=P^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}andQ~=Q⊙AI−1\\widetilde\{Q\}=Q^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}are group inverses, withγAI\\gamma^\{\\mathrm\{AI\}\}as the geodesic under AIM\.

###### Proof\.

The proof is presented in[Sec\.H\.9](https://arxiv.org/html/2607.08783#A8.SS9)\. ∎

Similar to the discussion in[Sec\.5\.1\.1](https://arxiv.org/html/2607.08783#S5.SS1.SSS1), we defineθ\\theta\-CRIM as the deformed metric of CRIM by the pullback of matrix power functionPθ​\(⋅\)\\mathrm\{P\}\_\{\\theta\}\(\\cdot\)and scaled by1θ2\\frac\{1\}\{\\theta^\{2\}\}\. As the pullback of CRIM,θ\\theta\-CRIM is right\-invariant w\.r\.t\.⊙θ​\-AI\\odot^\{\\theta\\text\{\-AI\}\}by definition\.

###### Proposition 5\.5\.

θ\\theta\-CRIM is right\-invariant w\.r\.t\.⊙θ​\-AI\\odot^\{\\theta\\text\{\-AI\}\}\.

TABLE IV:Key operators in calculating LieBN on SPD manifolds\. The notations follow[Secs\.B\.3](https://arxiv.org/html/2607.08783#A2.SS3)and[5\.3](https://arxiv.org/html/2607.08783#S5.Thmtheorem3)\.Metric\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}θ\\theta\-LCMθ\\theta\-CRIMInvarianceLeft\-invarianceBi\-invarianceRight\-invarianceLieBN TypeLieBN\-LeftLieBN\-Left = LieBN\-RightLieBN\-RightPullback MapPθ\\operatorname\{P\}\_\{\\theta\}log\\operatorname\{log\}Pθ∘ψLC\\operatorname\{P\}\_\{\\theta\}\\circ\\psi\_\{\\mathrm\{LC\}\}Pθ\\operatorname\{P\}\_\{\\theta\}Codomain\{𝒮\+\+n,⊙AI,1θ2​g\(α,β\)​\-AI\}\\\{\\mathcal\{S\}^\{n\}\_\{\+\+\},\\odot^\{\\mathrm\{AI\}\},\\frac\{1\}\{\\theta^\{2\}\}g^\{\(\\alpha,\\beta\)\\text\{\-AI\}\}\\\}\{𝒮n,⟨⋅,⋅⟩\(α,β\)\}\\\{\\mathcal\{S\}^\{n\},\\langle\\cdot,\\cdot\\rangle^\{\(\\alpha,\\beta\)\}\\\}\{LTn\\\{\\mathrm\{LT\}^\{n\},1θ2⟨⋅,⋅⟩\}\\frac\{1\}\{\\theta^\{2\}\}\\langle\\cdot,\\cdot\\rangle\\\}\{𝒮\+\+n,⊙AI,1θ2​gCRI\}\\\{\\mathcal\{S\}^\{n\}\_\{\+\+\},\\odot^\{\\mathrm\{AI\}\},\\frac\{1\}\{\\theta^\{2\}\}g^\{\\mathrm\{CRI\}\}\\\}Riemannian and Lie group operators in the codomainLQ⁡\(P\)\\operatorname\{L\}\_\{Q\}\(P\)orRQ⁡\(P\)\\operatorname\{R\}\_\{Q\}\(P\)K​P​K⊤KPK^\{\\top\}P\+QP\+QP\+QP\+QL​Q​L⊤LQL^\{\\top\}LQ⊙−1⁡\(P\)\\operatorname\{L\}\_\{Q\_\{\\odot\}^\{\-1\}\}\(P\)orRQ⊙−1⁡\(P\)\\operatorname\{R\}\_\{Q\_\{\\odot\}^\{\-1\}\}\(P\)K−1​P​K−⊤K^\{\-1\}PK^\{\-\\top\}P−QP\-QP−QP\-QL−1​Q​L−⊤L^\{\-1\}QL^\{\-\\top\}ExpE⁡\[s​LogE⁡\(P\)\]\\operatorname\{Exp\}\_\{E\}\\left\[s\\operatorname\{Log\}\_\{E\}\(P\)\\right\]PsP^\{s\}s​PsPs​PsP\(\(P⊙AI−1\)s\)⊙AI−1\\left\(\\left\(P^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}\\right\)^\{s\}\\right\)^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}FMKarcher FlowArithmeticaverageArithmeticaverageKarcher FlowWFM⁡\(\{1−γ,γ\},\{P1,P2\}\)\\operatorname\{WFM\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{P\_\{1\},P\_\{2\}\\\}\)P112​\(P1−12​P2​P1−12\)γ​P112P\_\{1\}^\{\\frac\{1\}\{2\}\}\\left\(P\_\{1\}^\{\-\\frac\{1\}\{2\}\}P\_\{2\}P\_\{1\}^\{\-\\frac\{1\}\{2\}\}\\right\)^\{\\gamma\}P\_\{1\}^\{\\frac\{1\}\{2\}\}Arithmeticweighted averageArithmeticweighted average\(P~112​\(P~1−12​P~2​P~1−12\)γ​P~112\)⊙AI−1\\left\(\\widetilde\{P\}\_\{1\}^\{\\frac\{1\}\{2\}\}\\left\(\\widetilde\{P\}\_\{1\}^\{\-\\frac\{1\}\{2\}\}\\widetilde\{P\}\_\{2\}\\widetilde\{P\}\_\{1\}^\{\-\\frac\{1\}\{2\}\}\\right\)^\{\\gamma\}\\widetilde\{P\}\_\{1\}^\{\\frac\{1\}\{2\}\}\\right\)^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}

TABLE V:Key operators in calculating LieBN on the rotation matrices\. The notations follow[Sec\.B\.4](https://arxiv.org/html/2607.08783#A2.SS4)\.InvarianceLieBN TypeR⊙−1R^\{\-1\}\_\{\\odot\}LR⁡\(S\)\\operatorname\{L\}\_\{R\}\(S\)RR⁡\(S\)\\operatorname\{R\}\_\{R\}\(S\)ExpI⁡\[s​LogI⁡\(R\)\]\\operatorname\{Exp\}\_\{I\}\\left\[s\\operatorname\{Log\}\_\{I\}\(R\)\\right\]FMWFM⁡\(\{1−γ,γ\},\{R,S\}\)\\operatorname\{WFM\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{R,S\\\}\)Bi\-invarianceLieBN\-Left & LieBN\-RightR−1R^\{\-1\}R​SRSS​RSRexp⁡\(t​log⁡\(R\)\)\\operatorname\{exp\}\\left\(t\\operatorname\{log\}\\left\(R\\right\)\\right\)\[[59](https://arxiv.org/html/2607.08783#bib.bib18), Alg\. 1\]R​exp⁡\(γ​log⁡\(R⊤​S\)\)R\\operatorname\{exp\}\(\\gamma\\operatorname\{log\}\(R^\{\\top\}S\)\)

#### 5\.1\.3Manifestations on SPD Manifolds

As discussed in[Secs\.5\.1\.1](https://arxiv.org/html/2607.08783#S5.SS1.SSS1)and[5\.1\.2](https://arxiv.org/html/2607.08783#S5.SS1.SSS2), there are four families of invariant metrics on the SPD Lie groups: \(1\) left\-invariant\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}w\.r\.t\.⊙AI\\odot^\{\\mathrm\{AI\}\}; \(2\) bi\-invariant\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}w\.r\.t\.⊙LE\\odot^\{\\mathrm\{LE\}\}andθ\\theta\-LCM w\.r\.t\.⊙LC\\odot^\{\\mathrm\{LC\}\}; \(3\) right\-invariantθ\\theta\-CRIM w\.r\.t\.⊙AI\\odot^\{\\mathrm\{AI\}\}\. Since all the above metrics are pullback metrics, the LieBN based on these metrics can be simplified and calculated in the co\-domain\. We first show a general result on LieBN under the pullback metric\. We denote[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)on the Lie groupℳ\\mathcal\{M\}as

LieBN⁡\(Pi;B,s,ϵ,γ\),∀Pi∈\{P1​…​N∈ℳ\}\.\\operatorname\{LieBN\}\(P\_\{i\};B,s,\\epsilon,\\gamma\),\\forall P\_\{i\}\\in\\\{P\_\{1\\ldots N\}\\in\\mathcal\{M\}\\\}\.\(25\)Then we can obtain the following theorem\.

\{mymath\}

###### Theorem 5\.6\.

Given a Lie groupℳ1\\mathcal\{M\}\_\{1\}, a Lie groupℳ2\\mathcal\{M\}\_\{2\}with an invariant metricg2g^\{2\}, and a diffeomorphismf:ℳ1→ℳ2f:\\mathcal\{M\}\_\{1\}\\rightarrow\\mathcal\{M\}\_\{2\}, thenffinduces an invariant metricg1g^\{1\}onℳ1\\mathcal\{M\}\_\{1\}, denoted asg1=f∗​g2g^\{1\}=f^\{\*\}g^\{2\}\. For a batch of activation\{P1​…​N\}\\\{P\_\{1\\ldots N\}\\\}inℳ1\\mathcal\{M\}\_\{1\},LieBN1⁡\(Pi;B,s,ϵ,γ\)\\operatorname\{LieBN\}^\{1\}\(P\_\{i\};B,s,\\epsilon,\\gamma\)inℳ1\\mathcal\{M\}\_\{1\}can be calculated inℳ2\\mathcal\{M\}\_\{2\}by the following process:

Mapping data into​ℳ2:Pi¯=f​\(Pi\),B¯=f​\(B\),\\displaystyle\\text\{Mapping data into \}\\mathcal\{M\}\_\{2\}:\\bar\{P\_\{i\}\}=f\(P\_\{i\}\),\\bar\{B\}=f\(B\),\(26\)Performing LieBN in​ℳ2:Pi^=LieBN2⁡\(Pi¯;B¯,s,ϵ,γ\),\\displaystyle\\text\{Performing LieBN in \}\\mathcal\{M\}\_\{2\}:\\hat\{P\_\{i\}\}=\\operatorname\{LieBN\}^\{2\}\(\\bar\{P\_\{i\}\};\\bar\{B\},s,\\epsilon,\\gamma\),\(27\)Mapping the resulting data back to​ℳ1:Pi~=f−1​\(P^i\),\\displaystyle\\text\{Mapping the resulting data back to \}\\mathcal\{M\}\_\{1\}:\\tilde\{P\_\{i\}\}=f^\{\-1\}\(\\hat\{P\}\_\{i\}\),\(28\)whereLieBN2\\operatorname\{LieBN\}^\{2\}is the LieBN onℳ2\\mathcal\{M\}^\{2\}\.

###### Proof\.

The proof is presented in[Sec\.H\.10](https://arxiv.org/html/2607.08783#A8.SS10)\. ∎

Given a metricggon𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}, the power\-deformed metricg~=1θ2​Pθ∗⁡g\\tilde\{g\}=\\frac\{1\}\{\\theta^\{2\}\}\\operatorname\{P\}\_\{\\theta\}^\{\*\}gis equal toPθ∗⁡\(1θ2​g\)\\operatorname\{P\}\_\{\\theta\}^\{\*\}\(\\frac\{1\}\{\\theta^\{2\}\}g\)\.[Thm\.5\.6](https://arxiv.org/html/2607.08783#S5.Thmtheorem6)indicates that the LieBN underg~\\tilde\{g\}can be calculated by the LieBN under1θ2​g\\frac\{1\}\{\\theta^\{2\}\}g\. Besides, as the Christoffel symbols remain the same under constant scaling, the LieBNs under1θ2​g\\frac\{1\}\{\\theta^\{2\}\}gandggonly differ in the variance\. We denoteg\(α,β\)​\-AIg^\{\(\\alpha,\\beta\)\\text\{\-AI\}\}andg\(θ,α,β\)​\-AIg^\{\(\\theta,\\alpha,\\beta\)\\text\{\-AI\}\}as the metric tensors of\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}and\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}, respectively\. Based on the above discussions, the computations of the LieBN underg\(θ,α,β\)​\-AIg^\{\(\\theta,\\alpha,\\beta\)\\text\{\-AI\}\}are reduced to the LieBN under1θ2​g\(α,β\)​\-AI\\frac\{1\}\{\\theta^\{2\}\}g^\{\(\\alpha,\\beta\)\\text\{\-AI\}\}\. Similarly, denotinggCRIg^\{\\mathrm\{CRI\}\}andgθ​\-CRIg^\{\\theta\\text\{\-CRI\}\}as the metric tensors of CRIM andθ\\theta\-CRIM, then the LieBN underθ\\theta\-CRIM can be calculated by the one under1θ2​gCRI\\frac\{1\}\{\\theta^\{2\}\}g^\{\\mathrm\{CRI\}\}\. Furthermore, as shown by Chen*et al*\[[20](https://arxiv.org/html/2607.08783#bib.bib6)\],\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}is a pullback metric from the Euclidean space𝒮n\\mathcal\{S\}^\{n\}of symmetric matrices, whileθ\\theta\-LCM is a pullback metric from the Euclidean spaceLTn\\mathrm\{LT\}^\{n\}of lower triangular matrices\. As shown in[Prop\.4\.6](https://arxiv.org/html/2607.08783#S4.Thmtheorem6), the LieBN in the Euclidean space𝒮n\\mathcal\{S\}^\{n\}orLTn\\mathrm\{LT\}^\{n\}is simplified to the standard Euclidean BN\. Therefore, the LieBNs under\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}andθ\\theta\-LCM can be calculated by the Euclidean BN over𝒮n\\mathcal\{S\}^\{n\}andLTn\\mathrm\{LT\}^\{n\}, respectively\.

We denote the LieBN under left and right translations as LieBN\-Left and LieBN\-Right, respectively\. Then, the LieBNs under\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}andθ\\theta\-CRIM correspond to LieBN\-Left and LieBN\-Right, respectively\. As⊙LC\\odot^\{\\mathrm\{LC\}\}and⊙LE\\odot^\{\\mathrm\{LE\}\}are commutative, the LieBN\-Left and LieBN\-Right under\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}, andθ\\theta\-LCM are equivalent\. We denoteP,Q,P1P,Q,P\_\{1\}andP2P\_\{2\}as points in the codomain,*i\.e\.*,𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}with scaled CRIM forθ\\theta\-CRIM,𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}with scaled\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}for\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\},𝒮n\\mathcal\{S\}^\{n\}for\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}, andLTn\\mathrm\{LT\}^\{n\}forθ\\theta\-LCM, respectively\. For CRIM, we denoteP~i=\(Pi\)⊙AI−1\\widetilde\{P\}\_\{i\}=\(P\_\{i\}\)^\{\-1\}\_\{\\odot^\{\\mathrm\{AI\}\}\}fori=1,2i=1,2\. We summarize all the necessary ingredients in[Tab\.IV](https://arxiv.org/html/2607.08783#S5.T4)for calculating SPD LieBN\. Note that for\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}, our scaling operation defined in[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)encompasses the scaling operation in\[[52](https://arxiv.org/html/2607.08783#bib.bib92), Eq\. \(9\)\]as a special case, when\(θ,α,β\)=\(1,1,0\)\(\\theta,\\alpha,\\beta\)=\(1,1,0\)\.

### 5\.2LieBN on Rotation Matrices

As the Riemannian metric on the rotation matrices is bi\-invariant, there are two instantiations of LieBN on this manifold,*i\.e\.*, LieBN\-Left based on the left translation and LieBN\-Right based on the right translation\. Particularly, the scaling can be further simplified:ExpI⁡\(t​LogI⁡\(R\)\)=exp⁡\(t​log⁡\(R\)\)\\operatorname\{Exp\}\_\{I\}\\left\(t\\operatorname\{Log\}\_\{I\}\\left\(R\\right\)\\right\)=\\operatorname\{exp\}\\left\(t\\operatorname\{log\}\\left\(R\\right\)\\right\)\. For the specificSO​\(3\)\\mathrm\{SO\}\(3\), the matrix exp and log can be efficiently calculated without matrix decomposition\[[39](https://arxiv.org/html/2607.08783#bib.bib70), Sec 3\.2\]\.[Tab\.V](https://arxiv.org/html/2607.08783#S5.T5)presents the expressions of the required operators in[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)\.

### 5\.3LieBN on Full\-Rank Correlation Matrices

TABLE VI:Summary of LieBN on the correlation\.⟨⋅,⋅⟩\(α,β,γ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}and⟨⋅,⋅⟩\(α,δ,ζ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}are permutation\-invariant inner products, which are discussed in[Sec\.B\.5\.2](https://arxiv.org/html/2607.08783#A2.SS5.SSS2)\.MetricECMLECMOLMLSMInvarianceBi\-invarianceLieBN TypeLieBN\-Left = LieBN\-RightPullback MapΘ\\Thetalog∘Θ\\log\\circ\\ThetaLog∘\\operatorname\{Log\}^\{\\circ\}Log⋆\\operatorname\{Log\}^\{\\star\}Codomain\{LT1n,⟨⋅,⋅⟩\}\\\{\\mathrm\{LT\}\_\{1\}^\{n\},\\left\\langle\\cdot,\\cdot\\right\\rangle\\\}\{LT0n,⟨⋅,⋅⟩\}\\\{\\mathrm\{LT\}\_\{0\}^\{n\},\\left\\langle\\cdot,\\cdot\\right\\rangle\\\}\{Holn,⟨⋅,⋅⟩\(α,β,γ\)\}\\\{\\mathrm\{Hol\}^\{n\},\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}\\\}\{Row0n,⟨⋅,⋅⟩\(α,δ,ζ\)\}\\\{\\mathrm\{Row\}\_\{0\}^\{n\},\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}\\\}

As discussed in[Sec\.2\.4](https://arxiv.org/html/2607.08783#S2.SS4), all four correlation metrics are bi\-invariant, and their associated Lie groups are commutative\. Consequently, LieBN\-Left is identical to LieBN\-Right\. Moreover, all four correlation metrics are pullback metrics from simpler Euclidean space\. Therefore, LieBN over the correlation can be implemented as[Thm\.5\.6](https://arxiv.org/html/2607.08783#S5.Thmtheorem6): \(1\) map the correlation into the prototype Euclidean space, \(2\) apply Euclidean BN, and \(3\) map back to the correlation\.

Optimization\.Lastly, let us talk about the optimization of the correlation\-valued biasing parameterB∈𝒞\+\+nB\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\. As shown by\[[72](https://arxiv.org/html/2607.08783#bib.bib79), Sec\. 4\.1\], the correlation matrix can be identified by the product of hyperbolic spaces via the Cholesky decomposition\. GivenC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}, thekk\-th row of the Cholesky factorL=Chol⁡\(C\)L=\\operatorname\{Chol\}\(C\)is\(Lk​1,…,Lk,k−1,Lk​k,0,…,0\)\\left\(L\_\{k1\},\\ldots,L\_\{k,k\-1\},L\_\{kk\},0,\\ldots,0\\right\)withLk​k\>0L\_\{kk\}\>0, which belongs to the hyperbolic space of open hemisphereHSk−1=\{x∈ℝk∣‖x‖=1,xk\>0\}\\mathrm\{HS\}^\{k\-1\}=\\left\\\{x\\in\\mathbb\{R\}^\{k\}\\mid\\\|x\\\|=1,x\_\{k\}\>0\\right\\\}\. Besides, the open hemisphereH​𝕊n\\mathrm\{H\}\\mathbb\{S\}^\{n\}is isometric to the Poincaré ballℙn=\{x∈ℝn∣‖x‖<1\}\\mathbb\{P\}^\{n\}=\\left\\\{x\\in\\mathbb\{R\}^\{n\}\\mid\\\|x\\\|<1\\right\\\}byπH​𝕊n→ℙn​\(\(x⊤,xn\+1\)⊤\)=x1\+xn\+1\\pi\_\{\\mathrm\{H\}\\mathbb\{S\}^\{n\}\\rightarrow\\mathbb\{P\}^\{n\}\}\(\(x^\{\\top\},x\_\{n\+1\}\)^\{\\top\}\)=\\frac\{x\}\{1\+x\_\{n\+1\}\}\. Therefore, each correlation can be parameterized withn−1n\-1Poincaré vectors\. Each Poincaré vector can be optimized by the well\-studied Riemannian optimization\[[6](https://arxiv.org/html/2607.08783#bib.bib42)\]\. The above process can be expressed as

C↦\(10⋯0L21L22⋯0⋮⋮⋱⋮Ln​1Ln​2⋯Ln​n\)↦\(x1∈ℙ1⋮xn−1∈ℙn−1\)\.C\{\\mapsto\}\\begin\{pmatrix\}1&0&\\cdots&0\\\\ L\_\{21\}&L\_\{22\}&\\cdots&0\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ L\_\{n1\}&L\_\{n2\}&\\cdots&L\_\{nn\}\\end\{pmatrix\}\{\\mapsto\}\\begin\{pmatrix\}x\_\{1\}\\in\\mathbb\{P\}^\{1\}\\\\ \\vdots\\\\ x\_\{n\-1\}\\in\\mathbb\{P\}^\{n\-1\}\\end\{pmatrix\}\.\(29\)

## 6Experiments

This section validates our LieBN on nine invariant metrics across the SPD, rotation, and correlation matrices\.

### 6\.1Experiments of LieBN on the SPD Manifold

Note that our LieBN layers are architecture\-agnostic and can be applied to any existing SPD neural network\. Following the previous work\[[43](https://arxiv.org/html/2607.08783#bib.bib41),[13](https://arxiv.org/html/2607.08783#bib.bib96),[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we focus on two network architectures: \(1\) SPDNet\[[43](https://arxiv.org/html/2607.08783#bib.bib41)\]for drone recognition on the Radar dataset\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\], and human action recognition on the HDM05\[[60](https://arxiv.org/html/2607.08783#bib.bib139)\]and FPHA\[[35](https://arxiv.org/html/2607.08783#bib.bib32)\]datasets; \(2\) TSMNet\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]for EEG classification on the Hinss2021 dataset\[[41](https://arxiv.org/html/2607.08783#bib.bib157)\]\. In the EEG application, TSMNet is endowed with SPD domain\-specific momentum batch normalization \(TSMNet\+SPDDSMBN\)\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], which is a domain adaptation version of\[[52](https://arxiv.org/html/2607.08783#bib.bib92)\]\. For a fair comparison, we also implement a domain\-specific momentum LieBN, referred to as DSMLieBN \(detailed in[App\.E](https://arxiv.org/html/2607.08783#A5)\)\. The backbone network architectures are represented as\{d0,d1,…,dL\}\\\{d\_\{0\},d\_\{1\},\\ldots,d\_\{L\}\\\}, where the dimension of the parameter in theii\-th BiMap layer \([App\.C](https://arxiv.org/html/2607.08783#A3)\) isdi×di−1d\_\{i\}\\times d\_\{i\-1\}\. As\(α,β\)\(\\alpha,\\beta\)only affects variance calculation throughout LieBN, we simply set\(α,β\)=\(1,0\)\(\\alpha,\\beta\)=\(1,0\)and only tune the deformation factorθ\\theta\. For each family of LieBN or DSMLieBN, we report two representatives: the standard one induced from the standard metric \(θ=1\\theta=1\), and the one induced from the deformed metric with properθ\\theta\.If the standard one is already saturated, we only report the results of the standard ones\.More details on implementation, datasets, and hyperparameter\(θ,α,β\)\(\\theta,\\alpha,\\beta\)are presented in[Sec\.G\.1](https://arxiv.org/html/2607.08783#A7.SS1)\.

TABLE VII:10\-fold average results of SPDNet with and without SPDBN or LieBN on the Radar, HDM05, and FPHA datasets\. If the LieBN under the standard metric \(θ=1\\theta=1\) is not saturated, the rightmost columns report the deformed LieBN\.\(a\)Radar dataset\.AccSPDNetSPDNetBNSPDNetLieBNθ=1\\theta=1Bestθ\\thetaAIM\-\(1\)LEM\-\(1\)LCM\-\(1\)CRIM\-\(1\)LCM\-\(\-0\.5\)Fit time \(s\)0\.981\.561\.621\.281\.111\.821\.43Mean±STD93\.25±1\.1094\.85±0\.9995\.47±0\.9094\.89±1\.0493\.52±1\.0794\.35±0\.6894\.80±0\.71Max94\.496\.1396\.2796\.895\.295\.695\.73
\(b\)HDM05 dataset\.AccSPDNetSPDNetBNSPDNetLieBNθ=1\\theta=1Bestθ\\thetaAIM\-\(1\)LEM\-\(1\)LCM\-\(1\)CRIM\-\(1\)AIM\-\(1\.5\)LCM\-\(0\.5\)CRIM\-\(0\.5\)Fit time \(s\)0\.570\.971\.140\.870\.661\.371\.461\.011\.74Mean±STD59\.13±0\.6766\.72±0\.5267\.79±0\.6565\.05±0\.6366\.68±0\.7163\.25±0\.8868\.16±0\.6870\.84±0\.9265\.76±0\.54Max60\.3467\.6668\.7566\.0568\.5264\.9469\.2572\.2766\.96
\(c\)FPHA dataset\.AccSPDNetSPDNetBNSPDNetLieBNθ=1\\theta=1Bestθ\\thetaAIM\-\(1\)LEM\-\(1\)LCM\-\(1\)CRIM\-\(1\)AIM\-\(1\.5\)LCM\-\(0\.5\)CRIM\-\(\-0\.5\)Fit time \(s\)0\.320\.620\.80\.550\.390\.921\.030\.651\.21Mean±STD85\.59±0\.7289\.33±0\.4989\.70±0\.5186\.56±0\.7977\.64±1\.0084\.65±1\.2090\.39±0\.6686\.33±0\.4386\.40±0\.57Max8690\.1790\.587\.837986\.6792\.178787\.17

Application to SPDNet:As SPDNet is the most classic SPD network, we apply our LieBN to SPDNet on the Radar, HDM05, and FPHA datasets\. Additionally, we compare our method with SPDNetBN, which applies the SPDBN in[Eqs\.6](https://arxiv.org/html/2607.08783#S3.E6)and[7](https://arxiv.org/html/2607.08783#S3.E7)to SPDNet\. Following\[[13](https://arxiv.org/html/2607.08783#bib.bib96),[25](https://arxiv.org/html/2607.08783#bib.bib7)\], we use the architectures of\{20,16,8\}\\\{20,16,8\\\},\{93,30\}\\\{93,30\\\}, and\{63,33\}\\\{63,33\\\}for the Radar, HDM05 and FPHA datasets, respectively\. The 10\-fold average results, including the average training time \(s/epoch\), are summarized in[Tab\.VII](https://arxiv.org/html/2607.08783#S6.T7)\. We have three key observations regarding the choice of metrics, deformation, and training efficiency\.

- •The choice of metrics:The metric that yields the most effective LieBN layer differs for each dataset\. Specifically, the optimal LieBN layers on these three datasets are the ones induced by AIM\-\(1\), LCM\-\(0\.5\), and AIM\-\(1\.5\), respectively,which improves the performance of SPDNet by 2\.22%, 11\.71%, and 4\.8%\. Additionally, although the LCM\-based LieBN performs worse than other LieBN variants on the Radar and FPHA datasets, it exhibits the best performance on the HDM05 dataset\. These observations highlight the advantage of the generality of our LieBN framework\.
- •The effect of deformation:Deformation patterns also vary across datasets\. Firstly, the standard AIM and CRIM are already saturated on the Radar dataset\. Secondly, the appropriate deformationθ\\thetacan further enhance the performance of LieBN\. Notably, even though the LieBNs induced by LCM\-\(1\) and CRIM\-\(1\) impede the learning of SPDNet on the FPHA datasets, they can improve the performance under an appropriate deformationθ\\theta\. These findings highlight the efficacy of the deforming geometry on the SPD manifold\.
- •Efficiency:Although our LieBN involves additional computations on variance compared with SPDNetBN, our LieBN achieves comparable or even better efficiency than SPDNetBN\. Particularly, the LieBN induced by standard LEM or LCM exhibits better efficiency than SPDNetBN\. Even with deformation, the LCM\-based LieBN is still comparable with SPDNetBN in terms of efficiency\. This phenomenon could be attributed to the fast and simple computation of LCM and LEM\.

TABLE VIII:Cross\-validation results of TSMNet with SPDDSMBN and DSMLieBN on the Hinss dataset\. If the DSMLieBN under the standard metric \(θ=1\\theta=1\) is not saturated, the bottom rows report deformed DSMLieBN\.\(a\)Inter\-session classificationMethodFit TimeMean±STDSPDDSMBN0\.1654\.12±9\.87DSMLieBNAIM\-\(1\)0\.1655\.10±7\.61LEM\-\(1\)0\.1354\.95±10\.09LCM\-\(1\)0\.1051\.54±6\.88CRIM\-\(1\)0\.2951\.86±9\.21LCM\-\(0\.5\)0\.1553\.11±5\.65
\(b\)Inter\-subject classificationMethodFit TimeMean±STDSPDDSMBN7\.7450\.10±8\.08DSMLieBNAIM\-\(1\)6\.9450\.04±8\.01LEM\-\(1\)4\.7150\.95±6\.40LCM\-\(1\)3\.5951\.86±4\.53CRIM\-\(1\)16\.3550\.71±8\.1CRIM\-\(1\.5\)19\.5151\.34±5\.82AIM\-\(\-0\.5\)8\.7153\.97±8\.78

Application to EEG classification:We apply our method to TSMNet under two scenarios, inter\-session and inter\-subject\. Following\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we adopt the architecture of\{40,20\}\\\{40,20\\\}\. Compared to SPDDSMBN, DSMLieBN\-AIM obtains the highest average scores of 55\.10% and 53\.97% in these two scenarios,outperforming SPDDSMBN by 0\.98% and 3\.87%, respectively\. In the inter\-subject scenario, the advantage of the efficiency of our LieBN over SPDDSMBN is more obvious\. Specifically, both the LEM\- and LCM\-based DSMLieBN achieve similar or better performance compared to SPDDSMBN, while requiring considerably less training time\. For example, DSMLieBN\-LCM\-\(1\) achieves better results with only half the training time of SPDDSMBN on inter\-subject tasks\. Interestingly, under the standard AIM, the sole difference between SPDDSMBN and our DSMLieBN is the way of centering and biasing\. SPDDSMBN applies the matrix inverse square root and matrix square root to fulfill centering and biasing, while AIM\-induced LieBN uses more efficient Cholesky decomposition\. As such, the DSMLieBN induced by the standard AIM is more efficient than SPDDSMBN, particularly on the inter\-subject task\. On the other hand, the CRIM\-based LieBN shows less efficiency, due to the relatively complex Riemannian computation of this metric\.

Visualization:We randomly select 50 samples and visualize the input and output of LieBN on the HDM05 dataset\. Using Riemannian t\-SNE\[[30](https://arxiv.org/html/2607.08783#bib.bib102)\], we map the30×3030\\times 30SPD matrices to2×22\\times 2low\-dimensional representations\. As shown in[Fig\.3](https://arxiv.org/html/2607.08783#S6.F3), LieBN effectively normalizes the data distribution\. Specifically, the input t\-SNE embeddings are largely scattered and their elements can reach up to 400K, whereas those of the output embeddings are mostly constrained within 20\.

![Refer to caption](https://arxiv.org/html/2607.08783v1/x2.png)Figure 3:Visualization of input and output30×3030\\times 30SPD matrices in LieBN using2×22\\times 2Riemannian t\-SNE embeddings\. The first row shows the input and output under different metrics\. Due to the significant difference in magnitude between the t\-SNE embeddings of LieBN’s input and output, the second row separately visualizes the LieBN output \(at a smaller scale\)\.
### 6\.2Experiments of LieBN on Rotation Matrices

This subsection implements our LieBN on the special orthogonal groups,*i\.e\.*,SO​\(n\)\\mathrm\{SO\}\(n\), also known as rotation matrices\. As the Riemannian metric onSO​\(n\)\\mathrm\{SO\}\(n\)is bi\-invariant, there are two instantiations of our LieBN on this group: LieBN\-Left based on the left translation and LieBN\-Right based on the right translation\. We apply our LieBN to the classic LieNet backbone\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\], where the latent space is the special orthogonal group\. Following\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\], we use three action recognition datasets, the G3D\[[10](https://arxiv.org/html/2607.08783#bib.bib43)\], HDM05\[[60](https://arxiv.org/html/2607.08783#bib.bib139)\], and NTU60\[[68](https://arxiv.org/html/2607.08783#bib.bib59)\]datasets\. We denote the LieNet models with our LieBN\-Left and LieBN\-Right as LieNetLieBN\-Left and LieNetLieBN\-Right, respectively\. More implementation details are presented in[Sec\.G\.2](https://arxiv.org/html/2607.08783#A7.SS2)\.

![Refer to caption](https://arxiv.org/html/2607.08783v1/x3.png)Figure 4:Test accuracy curves corresponding to[Tab\.IX](https://arxiv.org/html/2607.08783#S6.T9)\.TABLE IX:Results of LieNet with or without rotation LieBN\.MethodG3DHDM05NTU60Mean±STDMaxMean±STDMax2Blocks3BlocksLieNet87\.91±0\.9089\.7376\.92±1\.2779\.1162\.460\.91LieNetLieBN\-Left88\.88±1\.6290\.6778\.89±1\.0780\.8863\.5162\.62LieNetLieBN\-Right88\.12±1\.1290\.379\.39±1\.1380\.6763\.662\.72

Results:We conduct 10\-fold experiments on the G3D and HDM05 datasets under the suggested 3Blocks555Each block consists of a RotMap layer followed by a RotPooling layer\. For more details, please refer to\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\]\.and 2Blocks architectures, respectively\. On the NTU60 dataset, we validate LieBN under the 2Blocks and 3Blocks settings\. The results are presented in[Tab\.IX](https://arxiv.org/html/2607.08783#S6.T9)\. Due to differences in software, our reimplemented LieNet \(by PyTorch\) performs slightly differently from the results reported in\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\]\(by Matlab\)\. However, we still observe a clear improvement when applying our LieBN to the vanilla LieNet backbone\. Additionally, LieBN\-Right performs slightly better than LieBN\-Left\. Although the effects of left and right translations on the sample statistics under the bi\-invariant metric are identical, their transformations on each sample differ, as illustrated in[Fig\.1](https://arxiv.org/html/2607.08783#S1.F1)\. This difference could slightly affect the network performance\. The specific optimal choice of left or right translations depends on the dataset’s characteristics\.

Training Dynamics:[Fig\.4](https://arxiv.org/html/2607.08783#S6.F4)presents the test accuracy curve\. We have the following additional observations, which can be attributed to the mitigated covariate shift by our LieBN, as our LieBN can effectively normalize the sample statistics\.Accelerated convergence:LieBN significantly accelerates the convergence of LieNet\. Specifically, on the NTU60 dataset—the largest dataset involved—LieNet with LieBN converges by the 5th epoch, whereas the vanilla LieNet does not converge until the 25th epoch\. A similar phenomenon can also be observed on the HDM05 dataset\.Stabler performance:LieBN enhances the stability of network training\. Especially on the HDM05 and G3D datasets, the initial training fluctuations are greatly mitigated by our LieBN\.

### 6\.3Experiments of LieBN on Correlation Matrices

As no neural networks are specifically designed for the correlation manifold and correlation matrices are still SPD, we apply our correlation LieBN \(LieBN\-Cor\) to SPD networks\. Our experiments focus on the SPDNet backbone using the FPHA and HDM05 datasets\. LieBN\-Cor is applied before the final classification layer\. Specifically, SPD features are first activated by the power function, then mapped into correlation matrices viaCor⁡\(⋅\)\\operatorname\{Cor\}\(\\cdot\), and finally processed by LieBN\-Cor\. More details can be found in[Sec\.G\.3](https://arxiv.org/html/2607.08783#A7.SS3)\.

TABLE X:Results of SPDNet with or without correlation LieBN under different invariant metrics\.DatasetSPDNetSPDNetLieBN\-CorECMLECMOLMLSMHDM0559\.13±0\.6765\.37 ± 1\.0761\.35 ± 0\.3460\.33 ± 0\.1260\.00 ± 0\.27FPHA85\.59±0\.7287\.20 ± 0\.1287\.03 ± 0\.3286\.80 ± 0\.1286\.77 ± 0\.29

Result:The 5\-fold average results are presented in[Tab\.X](https://arxiv.org/html/2607.08783#S6.T10)\. Although LieBN\-Cor is not specifically designed for SPD networks, it still improves SPDNet’s performance, demonstrating its effectiveness\. Among the four invariant metrics, ECM achieves the best performance\. While LieBN\-SPD outperforms LieBN\-Cor when applied to SPDNet—an expected result since SPDNet is tailored for SPD matrices—this does not undermine the validity of LieBN\-Cor\. The consistent improvement over vanilla SPDNet highlights the potential of applying LieBN\-Cor to correlation manifolds\.

TABLE XI:Ablations on optimizing correlation parameter in LieBN under different metrics\.OptimMetricECMLECMOLMLSMTrivialization63\.17 ± 1\.3258\.84 ± 0\.5159\.84 ± 0\.4553\.22 ± 1\.62Riemannian65\.37 ± 1\.0761\.35 ± 0\.3460\.33 ± 0\.1260\.00 ± 0\.27

Ablations:As discussed in[Sec\.5\.3](https://arxiv.org/html/2607.08783#S5.SS3), the correlation\-valued biasing parameterB∈𝒞\+\+nB\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}is optimized via Riemannian optimization over multiple Poincaré vectors\. Alternatively, trivialization tricks\[[55](https://arxiv.org/html/2607.08783#bib.bib110)\]can be employed\. Specifically,[Thm\.5\.6](https://arxiv.org/html/2607.08783#S5.Thmtheorem6)suggests that we can instead setB¯=f​\(B\)∈V\\bar\{B\}=f\(B\)\\in Vas the parameter, whereffis the isometry andVVis the prototype space for ECM, LECM, OLM, and LSM, respectively\.[Tab\.XI](https://arxiv.org/html/2607.08783#S6.T11)presents the 5\-fold comparison, demonstrating the superiority of our Poincaré parameterization\.

## 7Conclusions

This paper presents a novel LieBN framework for batch normalization over Lie groups, leveraging natural Lie\-group\-invariant metrics\. Compared to prior approaches, LieBN provides a principled method to normalize both sample and population statistics\. Empirically, we generalize three existing Lie group structures on the SPD manifold and introduce the first non\-trivial right\-invariant SPD metric\. By employing these parameterized invariant metrics, we instantiate our framework on the SPD manifold\. Furthermore, we implement LieBN on rotation matrices using a bi\-invariant metric and on the correlation manifold using four bi\-invariant metrics\. Extensive experiments across different manifolds validate the effectiveness of our LieBN\.

## Acknowledgments

This work was supported by the MUR PNRR project FAIR \(PE00000013\) funded by the NextGenerationEU, and the EU Horizon project ELIAS \(No\. 101120237\)\. The authors are also grateful for the HPC resources from the ISCRA project\.

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![[Uncaptioned image]](https://arxiv.org/html/2607.08783v1/authors/ziheng.jpg)Ziheng Chenreceived the B\.A\. degree in logistics management from Shandong University, Jinan, China, and M\.S\. degree in computer science and technology from Jiangnan University, Wuxi, China\. He is currently working toward the Ph\.D\. degree with the Multimedia and Human Understanding Group \(MHUG\), University of Trento, Trento, Italy\. His research interests are geometric deep learning and matrix & vector manifolds\.![[Uncaptioned image]](https://arxiv.org/html/2607.08783v1/authors/yue.jpg)Yue Songreceived the B\.Sc\.*cum laude*from KU Leuven, Belgium and the joint M\.Sc\.*summa cum laude*from the University of Trento, Italy and KTH Royal Institute of Technology, Sweden, and the Ph\.D\.*summa cum laude*from the Multimedia and Human Understanding Group \(MHUG\) at the University of Trento, Italy\. Currently, he is a post\-doctoral research associate at Caltech\. His research interests are structured representation learning\.![[Uncaptioned image]](https://arxiv.org/html/2607.08783v1/authors/ruiwang.jpeg)Rui Wangreceived the M\.S\. degree in computer science from Jiangnan University, Wuxi, China, in 2018, and the Ph\.D\. degree in pattern recognition and intelligent system from Jiangnan University, Wuxi, China, in 2023\. He is currently a Lecturer with the School of Artificial Intelligence and Computer Science, Jiangnan University\. His research topics include Riemannian manifold learning, metric learning, and deep learning\.![[Uncaptioned image]](https://arxiv.org/html/2607.08783v1/authors/xiaojun.png)Xiao\-Jun Wureceived the B\.Sc\. degree in mathematics from Nanjing Normal University, Nanjing, China, in 1991\. He received the M\.S\. degree and the Ph\.D\. degree in pattern recognition and intelligent systems from Nanjing University of Science and Technology, Nanjing, China, in 1996 and 2002, respectively\. He is a distinguished professor in the School of Artificial Intelligence and Computer Science, Jiangnan University\. He is a fellow of IAPR and AAIA\.![[Uncaptioned image]](https://arxiv.org/html/2607.08783v1/authors/nicu.png)Nicu Sebeis Professor with the University of Trento, Italy, leading the research in the areas of multimedia information retrieval and human behavior understanding\. He was the General Co\-Chair of ACM Multimedia 2013, and the Program Chair of ACM Multimedia 2007 and 2011, ECCV 2016, ICCV 2017 and ICPR 2020\. He is a fellow of IAPR\.LieBN: Batch Normalization over Lie Groups \(Supplement\) Ziheng Chen, Yue Song, Rui Wang, Xiao\-Jun Wu, and Nicu Sebe

## Appendix Contents

## Appendix ANotations

For better clarity, we summarize all the notations used in this paper in[Tab\.XII](https://arxiv.org/html/2607.08783#A1.T12)\.

TABLE XII:Summary of notations\.NotationExplanation\{ℳ,⊙,g\}\\\{\\mathcal\{M\},\\odot,g\\\}or abbreviated asℳ\\mathcal\{M\}Lie group with a group operation⊙\\odotand an invariant metricgggLg^\{\\mathrm\{L\}\}andgRg^\{\\mathrm\{R\}\}Left\-invariant and right\-invariant metricsP⊙−1P\_\{\\odot\}^\{\-1\}Group inverse ofP∈ℳP\\in\\mathcal\{M\}TP​ℳT\_\{P\}\\mathcal\{M\}Tangent space atP∈ℳP\\in\\mathcal\{M\}gp​\(⋅,⋅\)g\_\{p\}\(\\cdot,\\cdot\)or⟨⋅,⋅⟩P\\langle\\cdot,\\cdot\\rangle\_\{P\}Riemannian metric atP∈ℳP\\in\\mathcal\{M\}∥⋅∥P\\\|\\cdot\\\|\_\{P\}Norm induced by⟨⋅,⋅⟩P\\langle\\cdot,\\cdot\\rangle\_\{P\}onTP​ℳT\_\{P\}\\mathcal\{M\}d⁡\(⋅,⋅\)\\operatorname\{d\}\(\\cdot,\\cdot\)Geodesic distanceFM\\operatorname\{FM\}andWFM\\operatorname\{WFM\}Fréchet mean and weighted Fréchet meanExpP\\operatorname\{Exp\}\_\{P\}andLogP\\operatorname\{Log\}\_\{P\}Riemannian exponentiation and logarithm atPPγ\(P,Q\)​\(t\)\\gamma\_\{\(P,Q\)\}\(t\)Geodesic connectingPPandQQPTP→Q\\operatorname\{PT\}\_\{P\\rightarrow Q\}Riemannian parallel transportation along the geodesic connectingPPandQQLP\\operatorname\{L\}\_\{P\}andRP\\operatorname\{R\}\_\{P\}Lie group left and right translation byP∈ℳP\\in\\mathcal\{M\}f∗,Pf\_\{\*,P\}Differential map of the smooth mapffatP∈ℳP\\in\\mathcal\{M\}f∗​gf^\{\*\}gPullback metric byfffromggℝn×n\\mathbb\{R\}^\{n\\times n\}Euclidean space ofn×nn\\times nreal matrices𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}SPD manifold ofn×nn\\times nSPD matrices𝒮n\\mathcal\{S\}^\{n\}Euclidean space ofn×nn\\times nsymmetric matricesLTn\\mathrm\{LT\}^\{n\}Euclidean space ofn×nn\\times nlower triangular matricesSO​\(n\)\\mathrm\{SO\}\(n\)Lie group ofn×nn\\times nrotation matrices𝔰​𝔬​\(n\)\\mathfrak\{so\}\(n\)Euclidean space ofn×nn\\times nskew\-symmetric matrices⟨⋅,⋅⟩\\langle\\cdot,\\cdot\\rangleand∥⋅∥F\\\|\\cdot\\\|\_\{\\mathrm\{F\}\}Standard Frobenius inner product and the induced norm⟨⋅,⋅⟩\(α,β\)\\langle\\cdot,\\cdot\\rangle^\{\(\\alpha,\\beta\)\}and∥⋅∥\(α,β\)\\\|\\cdot\\\|^\{\(\\alpha,\\beta\)\}O​\(n\)\\mathrm\{O\}\(\{n\}\)\-invariant Euclidean inner product and the induced norm𝐒𝐓\\mathbf\{ST\}𝐒𝐓=\{\(α,β\)∈ℝ2∣min⁡\(α,α\+n​β\)\>0\}\\mathbf\{ST\}=\\\{\(\\alpha,\\beta\)\\in\\mathbb\{R\}^\{2\}\\mid\\min\(\\alpha,\\alpha\+n\\beta\)\>0\\\}g\(α,β\)​\-AIg^\{\(\\alpha,\\beta\)\\text\{\-AI\}\},g\(θ,α,β\)​\-AIg^\{\(\\theta,\\alpha,\\beta\)\\text\{\-AI\}\},gCRIg^\{\\mathrm\{CRI\}\}andgθ​\-CRIg^\{\\theta\\text\{\-CRI\}\}Riemannian metric tensor of\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\},\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}, CRIM, andθ\\theta\-CRIM⊙AI\\odot^\{\\mathrm\{AI\}\},⊙LE\\odot^\{\\mathrm\{LE\}\}, and⊙LC\\odot^\{\\mathrm\{LC\}\}Group operations w\.r\.t\. AIM, LEM, and LCM\(⋅\)Sym\\left\(\\cdot\\right\)\_\{\\mathrm\{Sym\}\}\(X\)Sym=X\+X⊤,∀X∈ℝn×n\\left\(X\\right\)\_\{\\mathrm\{Sym\}\}=X\+X^\{\\top\},\\forall X\\in\\mathbb\{R\}^\{n\\times n\}\(⋅\)12\(\\cdot\)\_\{\\frac\{1\}\{2\}\}\(X\)12=⌊X⌋\+12​𝔻​\(X\),∀X∈ℝn×n\(X\)\_\{\\frac\{1\}\{2\}\}=\\lfloor X\\rfloor\+\\frac\{1\}\{2\}\\mathbb\{D\}\(X\),\\forall X\\in\\mathbb\{R\}^\{n\\times n\}𝒩​\(M,σ2\)\\mathcal\{N\}\(M,\\sigma^\{2\}\)Riemannian Gaussian distributionlog\\operatorname\{log\}andexp\\operatorname\{exp\}Matrix logarithm and exponentiationChol\\operatorname\{Chol\}Cholesky decompositionDlog\\operatorname\{Dlog\}Diagonal element\-wise logarithmψLC\\psi\_\{\\mathrm\{LC\}\}Dlog∘Chol\\operatorname\{Dlog\}\\circ\\operatorname\{Chol\}⌊⋅⌋\\lfloor\\cdot\\rfloorStrictly lower triangular part of a square matrixPθ⁡\(⋅\)\\operatorname\{P\}\_\{\\theta\}\(\\cdot\)or\(⋅\)θ\(\\cdot\)^\{\\theta\}Matrix power function𝔻​\(⋅\)\\mathbb\{D\}\(\\cdot\)Returning a diagonal matrix with diagonal elements from a square matrixdiag⁡\(⋅\)\\operatorname\{diag\}\(\\cdot\)Returns a diagonal matrix from an input vectorHoln\\mathrm\{Hol\}^\{n\}andRow0n\\mathrm\{Row\}\_\{0\}^\{n\}Subspace of𝒮n\\mathcal\{S\}^\{n\}with null diagonals and null row sumRow1n\\mathrm\{Row\}\_\{1\}^\{n\}Manifold ofn×nn\\times nSPD matrices with unit row sum\.𝒞\+\+n\\mathcal\{C\}^\{n\}\_\{\+\+\}Manifold ofn×nn\\times nfull\-rank correlation matricesLT1n\\mathrm\{LT\}\_\{1\}^\{n\}andLT0n\\mathrm\{LT\}\_\{0\}^\{n\}Euclidean subspaces ofLTn\\mathrm\{LT\}^\{n\}with unit diagonals and null diagonalsLT\+\+n\\mathrm\{LT\}\_\{\+\+\}^\{n\}Cholesky manifold ofn×nn\\times nlower triangular matrices with positive diagonals⊛\\circledastHadamard productCor\\operatorname\{Cor\}Cor:Σ∈𝒮\+\+n⟼𝔻​\(Σ\)−1/2​Σ​𝔻​\(Σ\)−1/2∈𝒞\+\+n\\operatorname\{Cor\}:\\Sigma\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}\\longmapsto\\mathbb\{D\}\(\\Sigma\)^\{\-\\nicefrac\{\{1\}\}\{\{2\}\}\}\\Sigma\\mathbb\{D\}\(\\Sigma\)^\{\-\\nicefrac\{\{1\}\}\{\{2\}\}\}\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}Θ\\ThetaΘ:C∈𝒞\+\+n⟼𝔻​\(Chol⁡\(C\)\)−1​Chol⁡\(C\)∈LT1n\\Theta:C\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longmapsto\\mathbb\{D\}\(\\operatorname\{Chol\}\(C\)\)^\{\-1\}\\operatorname\{Chol\}\(C\)\\in\\mathrm\{LT\}\_\{1\}^\{n\}Off\\mathrm\{Off\}Returns a matrix inHoln\\mathrm\{Hol\}^\{n\}consisting of off\-diagonal elementsLog∘&Exp∘\\operatorname\{Log\}^\{\\circ\}\\&\\operatorname\{Exp\}^\{\\circ\}Off\-log and its inverseLog⋆&Exp⋆\\operatorname\{Log\}^\{\\star\}\\&\\operatorname\{Exp\}^\{\\star\}Log\-scaled and its inverseIIorInI\_\{n\}&𝟎\\mathbf\{0\}Identity matrix & zero matrix⟨⋅,⋅⟩\(α,β,γ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}and⟨⋅,⋅⟩\(α,δ,ζ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}Permutation\-invariant inner products overHoln\\mathrm\{Hol\}^\{n\}andRow0n\\mathrm\{Row\}\_\{0\}^\{n\}

## Appendix BRiemannian Structures on the Involved Matrix Manifolds

### B\.1Existence and Uniqueness of the Weighted Fréchet Mean

Let\(ℳ,g\)\(\\mathcal\{M\},g\)be an orientable complete Riemannian manifold equipped with a Riemannian metricgg\. The induced distance is denoted asd⁡\(⋅,⋅\)\\operatorname\{d\}\(\\cdot,\\cdot\)\. We denote the supremum of the sectional curvatures ofℳ\\mathcal\{M\}byΔ\\Delta\. We recover the theorem on the existence and uniqueness of the Weighted Fréchet Mean \(WFM\)\[[1](https://arxiv.org/html/2607.08783#bib.bib72)\]\. We acknowledge that Chakraborty*et al*\[[14](https://arxiv.org/html/2607.08783#bib.bib87), App\. A\]has also provided a summary of the following discussions\.

###### Definition B\.1\(Geodesic Ball\)\.

LetP∈ℳP\\in\\mathcal\{M\}andr\>0r\>0\. ThenBr​\(P\)=\{Q∈ℳ∣d​\(P,Q\)<r\}B\_\{r\}\(P\)=\\\{Q\\in\\mathcal\{M\}\\mid d\(P,Q\)<r\\\}is the open geodesic ball atPPof radiusrr\.

###### Definition B\.2\(Injectivity Radius\[[59](https://arxiv.org/html/2607.08783#bib.bib18)\]\)\.

The local injectivity radius atP∈ℳP\\in\\mathcal\{M\},rinj​\(P\)r\_\{\\text\{inj\}\}\(P\), is the largest radiusrrfor whichExpP:TP​ℳ⊃Br​\(0\)→ℳ\\operatorname\{Exp\}\_\{P\}:T\_\{P\}\\mathcal\{M\}\\supset B\_\{r\}\(0\)\\to\\mathcal\{M\}is a diffeomorphism onto its image\. The injectivity radius ofℳ\\mathcal\{M\}is defined asrinj​\(ℳ\)=infP∈ℳ\{rinj​\(P\)\}r\_\{\\text\{inj\}\}\(\\mathcal\{M\}\)=\\inf\_\{P\\in\\mathcal\{M\}\}\\\{r\_\{\\text\{inj\}\}\(P\)\\\}\.

Within the local injectivity radius, the exponential map is invertible and we call the inverse map as Riemannian logarithmic map,LogP:Brinj​\(P\)​\(P\)→Brinj​\(P\)​\(0\)⊂TP​ℳ\\operatorname\{Log\}\_\{P\}:B\_\{r\_\{\\text\{inj\}\}\(P\)\}\(P\)\\rightarrow B\_\{r\_\{\\text\{inj\}\}\(P\)\}\(0\)\\subset T\_\{P\}\\mathcal\{M\}\.

###### Definition B\.3\(Regular Geodesic Ball\[[50](https://arxiv.org/html/2607.08783#bib.bib104)\]\)\.

An open geodesic ballBr​\(P\)B\_\{r\}\(P\)is a regular geodesic ball ifr<rinj​\(P\)r<r\_\{\\text\{inj\}\}\(P\)andr<π/\(2​Δ\)r<\\pi/\(2\\sqrt\{\\Delta\}\), where1/Δ1/\\sqrt\{\\Delta\}is interpreted as∞\\inftyforΔ≤0\\Delta\\leq 0\.

IfP,QP,Qin a regular geodesic ballBr​\(P\)B\_\{r\}\(P\), there exists a unique geodesicγ\(P,Q\):\[0,1\]→Br​\(P\)\\gamma\_\{\(P,Q\)\}:\[0,1\]\\rightarrow B\_\{r\}\(P\)withγ​\(0\)=P\\gamma\(0\)=Pandγ​\(1\)=Q\\gamma\(1\)=Q\[[50](https://arxiv.org/html/2607.08783#bib.bib104)\]\.

###### Definition B\.4\(Strong Convexity\[[18](https://arxiv.org/html/2607.08783#bib.bib161)\]\)\.

A subsetU⊂ℳU\\subset\\mathcal\{M\}is strongly convex if for allP,Q∈UP,Q\\in U, there exists a unique length\-minimizing geodesic segment betweenPPandQQ, and the geodesic segment lies entirely inUU\.

###### Definition B\.5\(Convexity Radius\[[37](https://arxiv.org/html/2607.08783#bib.bib103)\]\)\.

The local convexity radius atP∈ℳP\\in\\mathcal\{M\},rcvx​\(P\)r\_\{\\text\{cvx\}\}\(P\), is defined as

rcvx​\(P\)=sup\{r≤rinj​\(P\)∣Br​\(P\)​is strongly convex\}\.r\_\{\\text\{cvx\}\}\(P\)=\\sup\\\{r\\leq r\_\{\\text\{inj\}\}\(P\)\\mid B\_\{r\}\(P\)\\text\{ is strongly convex\}\\\}\.\(30\)The convexity radius ofℳ\\mathcal\{M\}is defined asrcvx​\(ℳ\)=infP∈ℳ\{rcvx​\(P\)\}r\_\{\\text\{cvx\}\}\(\\mathcal\{M\}\)=\\inf\_\{P\\in\\mathcal\{M\}\}\\\{r\_\{\\text\{cvx\}\}\(P\)\\\}\.

###### Theorem B\.6\(Existence and Uniqueness of WFM\[[1](https://arxiv.org/html/2607.08783#bib.bib72)\]\)\.

The WFM exists and is unique inside a geodesic ball of radiusrcvx​\(ℳ\)r\_\{\\text\{cvx\}\}\(\\mathcal\{M\}\)\.

In the main paper, we always assume the involvedExp\\operatorname\{Exp\},Log\\operatorname\{Log\}, and WFM are well\-defined\.

### B\.2Symmetric Matrix Functions

This subsection reviews the eigenvalue function over symmetric matrices\. For more in\-depth discussions, please refer to\[[8](https://arxiv.org/html/2607.08783#bib.bib150), Ch\. 2\.7\.13\]or\[[9](https://arxiv.org/html/2607.08783#bib.bib154), Ch\. V\.3\]\.

We denote𝒮n\\mathcal\{S\}^\{n\}as the Euclidean space ofn×nn\\times nreal symmetric matrices, and𝒮\+\+n\\mathcal\{S\}^\{n\}\_\{\+\+\}as the SPD manifold ofn×nn\\times nSPD matrices\. LetI̊\\mathring\{I\}be an open interval ofℝ\\mathbb\{R\}andf:I̊→ℝf:\\mathring\{I\}\\rightarrow\\mathbb\{R\}be a smooth function\. The smooth map induced byfffor any symmetric matrixSSwith all eigenvalues inI̊\\mathring\{I\}is defined as

f:S⟼Uf\(Σ\)U⊤∈𝒮n,withS=UΣU⊤as the eigendecomposition\.f:S\\longmapsto Uf\(\\Sigma\)U^\{\\top\}\\in\\mathcal\{S\}^\{n\},\\text\{ with \}S=U\\Sigma U^\{\\top\}\\text\{ as the eigendecomposition\.\}\(31\)Its differential is known as the Daleckĭi\-Kreĭn formula:

f∗,S​\(V\)\\displaystyle f\_\{\*,S\}\(V\)=U​\(L⊛\(U⊤​V​U\)\)​U⊤,∀V∈𝒮n,\\displaystyle=U\\left\(L\\circledast\\left\(U^\{\\top\}VU\\right\)\\right\)U^\{\\top\},\\quad\\forall V\\in\\mathcal\{S\}^\{n\},\(32\)Li,j\\displaystyle L\_\{i,j\}=\{f​\(σi\)−f​\(σj\)σi−σj,if​σi≠σjf′​\(δi\),otherwise\\displaystyle=\\begin\{cases\}\\frac\{f\(\\sigma\_\{i\}\)\-f\(\\sigma\_\{j\}\)\}\{\\sigma\_\{i\}\-\\sigma\_\{j\}\},&\\text\{ if \}\\sigma\_\{i\}\\neq\\sigma\_\{j\}\\\\ f^\{\\prime\}\(\\delta\_\{i\}\),&\\text\{ otherwise \}\\end\{cases\}\(33\)whereLLis called the Loewner matrix with the\(i,j\)\(i,j\)\-th element defined as[Eq\.33](https://arxiv.org/html/2607.08783#A2.E33), and⊙\\odotdenotes the Hadamard product\. Two special cases are the matrix logarithm:log:𝒮\+\+n→𝒮n\\log:\\mathcal\{S\}^\{n\}\_\{\+\+\}\\rightarrow\\mathcal\{S\}^\{n\}and its inverse, the matrix exponentiationexp:𝒮n→𝒮\+\+n\\exp:\\mathcal\{S\}^\{n\}\\rightarrow\\mathcal\{S\}^\{n\}\_\{\+\+\}\.

### B\.3SPD Geometries

[Tab\.XIII](https://arxiv.org/html/2607.08783#A2.T13)summarizes the Lie groups and invariant metric on the SPD manifold with the following notations\. LetP,Q∈𝒮\+\+nP,Q\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}be SPD matrices andV,W∈TP​𝒮\+\+nV,W\\in T\_\{P\}\\mathcal\{S\}^\{n\}\_\{\+\+\}be tangent vectors\. We denote the matrix logarithm, exponentiation, and Cholesky decomposition bylog⁡\(⋅\)\\operatorname\{log\}\(\\cdot\),exp⁡\(⋅\)\\operatorname\{exp\}\(\\cdot\), andChol⁡\(⋅\)\\operatorname\{Chol\}\(\\cdot\), respectively\. The differentials atPParelog∗,P\\operatorname\{log\}\_\{\*,P\}andChol∗,P\\operatorname\{Chol\}\_\{\*,P\}\. The Cholesky factors ofPPandQQare denoted asL=Chol⁡\(P\)L=\\operatorname\{Chol\}\(P\)andK=Chol⁡\(Q\)K=\\operatorname\{Chol\}\(Q\)\. The corresponding tangent vectors areX=Chol∗,P⁡\(V\)X=\\operatorname\{Chol\}\_\{\*,P\}\(V\)andY=Chol∗,P⁡\(W\)Y=\\operatorname\{Chol\}\_\{\*,P\}\(W\)for LCM\.𝕂\\mathbb\{K\},𝕃\\mathbb\{L\},𝕏\\mathbb\{X\}, and𝕐\\mathbb\{Y\}are diagonal matrices with diagonal elements fromKK,LL,XX, andYY, respectively\.⌊⋅⌋\\lfloor\\cdot\\rflooris the strictly lower part of a square matrix\. The norms induced by⟨⋅,⋅⟩\(α,β\)\\langle\\cdot,\\cdot\\rangle^\{\(\\alpha,\\beta\)\}and the standard Frobenius norm \(induced by the standard inner product⟨⋅,⋅⟩\\left\\langle\\cdot,\\cdot\\right\\rangle\) by∥⋅∥\(α,β\)\\\|\\cdot\\\|^\{\(\\alpha,\\beta\)\}and∥⋅∥F\\\|\\cdot\\\|\_\{\\mathrm\{F\}\}, respectively\.

TABLE XIII:Lie group structures and the associated Riemannian operators on SPD manifolds\.Metric\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}LCMQ⊙PQ\\odot Pexp⁡\(log⁡\(P\)\+log⁡\(Q\)\)\\operatorname\{exp\}\(\\operatorname\{log\}\(P\)\+\\operatorname\{log\}\(Q\)\)K​P​K⊤KPK^\{\\top\}Chol−1⁡\(⌊L\+K⌋\+𝕂​𝕃\)\\operatorname\{Chol\}^\{\-1\}\(\\lfloor L\+K\\rfloor\+\\mathbb\{K\}\\mathbb\{L\}\)gP​\(V,W\)g\_\{P\}\(V,W\)⟨log∗,P⁡\(V\),log∗,P⁡\(W\)⟩\(α,β\)\\langle\\operatorname\{log\}\_\{\*,P\}\(V\),\\operatorname\{log\}\_\{\*,P\}\(W\)\\rangle^\{\(\\alpha,\\beta\)\}⟨P−1​V,W​P−1⟩\(α,β\)\\langle P^\{\-1\}V,WP^\{\-1\}\\rangle^\{\(\\alpha,\\beta\)\}⟨⌊X⌋,⌊Y⌋⟩\+⟨𝕏​𝕃−1,𝕐​𝕃−1⟩\\langle\\lfloor X\\rfloor,\\lfloor Y\\rfloor\\rangle\+\\langle\\mathbb\{X\}\\mathbb\{L\}^\{\-1\},\\mathbb\{Y\}\\mathbb\{L\}^\{\-1\}\\rangled⁡\(P,Q\)\\operatorname\{d\}\(P,Q\)‖log⁡\(P\)−log⁡\(Q\)‖\(α,β\)\\left\\\|\\operatorname\{log\}\(P\)\-\\operatorname\{log\}\(Q\)\\right\\\|^\{\(\\alpha,\\beta\)\}‖log⁡\(Q−12​P​Q−12\)‖\(α,β\)\\left\\\|\\operatorname\{log\}\\left\(Q^\{\-\\frac\{1\}\{2\}\}PQ^\{\-\\frac\{1\}\{2\}\}\\right\)\\right\\\|^\{\(\\alpha,\\beta\)\}‖ψLC​\(P\)−ψLC​\(Q\)‖F\\left\\\|\\psi\_\{\\mathrm\{LC\}\}\(P\)\-\\psi\_\{\\mathrm\{LC\}\}\(Q\)\\right\\\|\_\{\\mathrm\{F\}\}FM⁡\(\{Pi\}\)\\operatorname\{FM\}\(\\\{P\_\{i\}\\\}\)exp⁡\(1n​∑ilog⁡Pi\)\\operatorname\{exp\}\\left\(\\frac\{1\}\{n\}\\sum\_\{i\}\\operatorname\{log\}\{P\_\{i\}\}\\right\)Karcher FlowψLC−1​\(1n​∑iψLC​\(Pi\)\)\\psi\_\{\\mathrm\{LC\}\}^\{\-1\}\\left\(\\frac\{1\}\{n\}\\sum\_\{i\}\\psi\_\{\\mathrm\{LC\}\}\(P\_\{i\}\)\\right\)LogP⁡Q\\operatorname\{Log\}\_\{P\}Q\(log∗,P\)−1​\[log⁡\(Q\)−log⁡\(P\)\]\(\\operatorname\{log\}\_\{\*,P\}\)^\{\-1\}\\left\[\\operatorname\{log\}\(Q\)\-\\operatorname\{log\}\(P\)\\right\]P12​log⁡\(P−12​Q​P−12\)​P12P^\{\\frac\{1\}\{2\}\}\\operatorname\{log\}\\left\(P^\{\-\\frac\{1\}\{2\}\}QP^\{\-\\frac\{1\}\{2\}\}\\right\)P^\{\\frac\{1\}\{2\}\}\(Chol−1\)∗,L​\[⌊K⌋−⌊L⌋\+𝕃​Dlog⁡\(𝕃−1​𝕂\)\]\(\\operatorname\{Chol\}^\{\-1\}\)\_\{\*,L\}\\left\[\\lfloor K\\rfloor\-\\lfloor L\\rfloor\+\\mathbb\{L\}\\operatorname\{Dlog\}\(\\mathbb\{L\}^\{\-1\}\\mathbb\{K\}\)\\right\]ExpP⁡V\\operatorname\{Exp\}\_\{P\}Vexp⁡\(log⁡\(P\)\+log∗,P⁡\(V\)\)\\operatorname\{exp\}\\left\(\\operatorname\{log\}\(P\)\+\\operatorname\{log\}\_\{\*,P\}\(V\)\\right\)P12​exp⁡\(P−12​V​P−12\)​P12P^\{\\frac\{1\}\{2\}\}\\operatorname\{exp\}\\left\(P^\{\-\\frac\{1\}\{2\}\}VP^\{\-\\frac\{1\}\{2\}\}\\right\)P^\{\\frac\{1\}\{2\}\}Chol−1⁡\(⌊L⌋\+⌊X⌋\+𝕃​exp⁡\(𝕏​𝕃−1\)\)\\operatorname\{Chol\}^\{\-1\}\\left\(\\lfloor L\\rfloor\+\\lfloor X\\rfloor\+\\mathbb\{L\}\\exp\\left\(\\mathbb\{X\}\\mathbb\{L\}^\{\-1\}\\right\)\\right\)γ\(P,Q\)​\(t\)\\gamma\_\{\(P,Q\)\}\(t\)exp⁡\[log⁡\(P\)\+t​\(log⁡\(Q\)−log⁡\(P\)\)\]\\operatorname\{exp\}\\left\[\\operatorname\{log\}\(P\)\+t\\left\(\\operatorname\{log\}\(Q\)\-\\operatorname\{log\}\(P\)\\right\)\\right\]P12​\(P−12​Q​P−12\)t​P12P^\{\\frac\{1\}\{2\}\}\\left\(P^\{\-\\frac\{1\}\{2\}\}QP^\{\-\\frac\{1\}\{2\}\}\\right\)^\{t\}P^\{\\frac\{1\}\{2\}\}Chol−1⁡\{⌊L⌋\+t​\(⌊K⌋−⌊L⌋\)\+𝕂t𝕃t−1\}\\operatorname\{Chol\}^\{\-1\}\\left\\\{\\lfloor L\\rfloor\+t\(\\lfloor K\\rfloor\-\\lfloor L\\rfloor\)\+\\frac\{\\mathbb\{K\}^\{t\}\}\{\\mathbb\{L\}^\{t\-1\}\}\\right\\\}InvarianceBi\-invarianceLeft\-invarianceBi\-invarianceReferences\[[3](https://arxiv.org/html/2607.08783#bib.bib140),[73](https://arxiv.org/html/2607.08783#bib.bib135),[25](https://arxiv.org/html/2607.08783#bib.bib7)\]\[[65](https://arxiv.org/html/2607.08783#bib.bib123),[73](https://arxiv.org/html/2607.08783#bib.bib135),[72](https://arxiv.org/html/2607.08783#bib.bib79)\]\[[56](https://arxiv.org/html/2607.08783#bib.bib126),[25](https://arxiv.org/html/2607.08783#bib.bib7),[22](https://arxiv.org/html/2607.08783#bib.bib3)\]
### B\.4Rotation Geometries

GivenR,S∈SO​\(n\)R,S\\in\\mathrm\{SO\}\(n\), and tangent vectorsA1,A2∈TR​SO​\(n\)A\_\{1\},A\_\{2\}\\in T\_\{R\}\\mathrm\{SO\}\(n\),[Tab\.XIV](https://arxiv.org/html/2607.08783#A2.T14)summarizes all the associated operators onSO​\(n\)\\mathrm\{SO\}\(n\)under the invariant metric, where⟨⋅,⋅⟩\\langle\\cdot,\\cdot\\rangledenotes the standard Frobenius inner product\.

TABLE XIV:Lie group structures and the associated Riemannian operators on Rotation matrices\.OperatorsR⊙SR\\odot SgR​\(A1,A2\)g\_\{R\}\(A\_\{1\},A\_\{2\}\)d⁡\(R,S\)\\operatorname\{d\}\(R,S\)LogR⁡S\\operatorname\{Log\}\_\{R\}SExpR⁡\(A\)\\operatorname\{Exp\}\_\{R\}\(A\)γ\(R,S\)​\(t\)\\gamma\_\{\(R,S\)\}\(t\)FMInvarianceExpressionR​SRS⟨A1,A2⟩\\langle A\_\{1\},A\_\{2\}\\rangle‖log⁡\(R⊤​S\)‖F\\left\\\|\\operatorname\{log\}\\left\(R^\{\\top\}S\\right\)\\right\\\|\_\{\\mathrm\{F\}\}R​log⁡\(R⊤​S\)R\\operatorname\{log\}\(R^\{\\top\}S\)R​exp⁡\(R⊤​A\)R\\operatorname\{exp\}\(R^\{\\top\}A\)R​exp⁡\(t​log⁡\(R⊤​S\)\)R\\operatorname\{exp\}\(t\\operatorname\{log\}\(R^\{\\top\}S\)\)\[[59](https://arxiv.org/html/2607.08783#bib.bib18), Alg\. 1\]Bi\-invariance

### B\.5Correlation Geometries

TABLE XV:Riemannian metrics on the correlation manifold with the associated isometric prototype spaces and diffeomorphisms\.MetricPrototype spaceDiffeomorphismsPropertiesECM\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]LT1n=LT0n\+In\\mathrm\{LT\}\_\{1\}^\{n\}=\\mathrm\{LT\}\_\{0\}^\{n\}\+I\_\{n\}Θ:C∈𝒞\+\+n⟼𝔻​\(Chol⁡\(C\)\)−1​Chol⁡\(C\)∈LT1n\\Theta:C\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longmapsto\\mathbb\{D\}\(\\operatorname\{Chol\}\(C\)\)^\{\-1\}\\operatorname\{Chol\}\(C\)\\in\\mathrm\{LT\}\_\{1\}^\{n\}Θ−1=Cor∘Chol−1:LT1n⟶𝒞\+\+n\\Theta^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}:\\mathrm\{LT\}\_\{1\}^\{n\}\\longrightarrow\\mathcal\{C\}^\{n\}\_\{\+\+\}Bi\-invarianceLECM\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]LT0n\\mathrm\{LT\}\_\{0\}^\{n\}log∘Θ:𝒞\+\+n⟶LT0n\\log\\circ\\Theta:\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longrightarrow\\mathrm\{LT\}\_\{0\}^\{n\}\(log∘Θ\)−1=Cor∘Chol−1∘exp:LT0n⟶𝒞\+\+n\(\\log\\circ\\Theta\)^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}\\circ\\exp:\\mathrm\{LT\}\_\{0\}^\{n\}\\longrightarrow\\mathcal\{C\}^\{n\}\_\{\+\+\}Bi\-invarianceOLM\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\]Holn\\mathrm\{Hol\}^\{n\}Log∘:C∈𝒞\+\+n⟼Off∘log⁡\(C\)∈Holn\\operatorname\{Log\}^\{\\circ\}:C\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longmapsto\\mathrm\{Off\}\\circ\\log\(C\)\\in\\mathrm\{Hol\}^\{n\}\(Log∘\)−1=Exp∘:H∈Holn⟼exp⁡\(𝒟\+​\(H\)\+H\)∈𝒞\+\+n\(\\operatorname\{Log\}^\{\\circ\}\)^\{\-1\}=\\operatorname\{Exp\}^\{\\circ\}:H\\in\\mathrm\{Hol\}^\{n\}\\longmapsto\\exp\(\\mathcal\{D\}^\{\+\}\(H\)\+H\)\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}Bi\-invariancePermutation\-invarianceLSM\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\]Row0n\\mathrm\{Row\}\_\{0\}^\{n\}Log⋆:C∈𝒞\+\+n⟼log⁡\(𝒟⋆​\(C\)​C​𝒟⋆​\(C\)\)∈Row0n\\operatorname\{Log\}^\{\\star\}:C\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longmapsto\\log\(\\mathcal\{D\}^\{\\star\}\(C\)C\\mathcal\{D\}^\{\\star\}\(C\)\)\\in\\mathrm\{Row\}\_\{0\}^\{n\}\(Log⋆\)−1=Exp⋆:R∈Row0n⟼Cor⁡\(exp⁡\(R\)\)∈𝒞\+\+n\(\\operatorname\{Log\}^\{\\star\}\)^\{\-1\}=\\operatorname\{Exp\}^\{\\star\}:R\\in\\mathrm\{Row\}\_\{0\}^\{n\}\\longmapsto\\operatorname\{Cor\}\(\\exp\(R\)\)\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}Bi\-invariancePermutation\-invariance

The involved four geometries in[Sec\.2\.4](https://arxiv.org/html/2607.08783#S2.SS4)on the correlation matrices can be classified into two classes: \(1\) non\-permutation\-invariant metrics, including ECM and LECM; and \(2\) permutation\-invariant metrics, including OLM and LSM\.[Tab\.XV](https://arxiv.org/html/2607.08783#A2.T15)summarizes the diffeomorphisms and prototype spaces discussed in[Sec\.2\.4](https://arxiv.org/html/2607.08783#S2.SS4)\.

#### B\.5\.1None\-permutation\-invariant Metrics

The non\-permutation\-invariant metrics\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\], namely ECM and LECM, are defined by isometries:

ECM:𝒞\+\+n⇌Θ−1=Cor∘Chol−1ΘLT1n=In\+LT0n,\\displaystyle\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\\Theta^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}\]\{\\Theta\}\\mathrm\{LT\}\_\{1\}^\{n\}=I\_\{n\}\+\\mathrm\{LT\}\_\{0\}^\{n\},\(34\)LECM:𝒞\+\+n⇌\(log∘Θ\)−1=Cor∘Chol−1∘explog∘ΘLT0n,\\displaystyle\\mathcal\{C\}^\{n\}\_\{\+\+\}\\xrightleftharpoons\[\(\\log\\circ\\Theta\)^\{\-1\}=\\operatorname\{Cor\}\\circ\\operatorname\{Chol\}^\{\-1\}\\circ\\exp\]\{\\log\\circ\\Theta\}\\mathrm\{LT\}\_\{0\}^\{n\},\(35\)
ECM and LECM\.For anyC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\},V∈TC​𝒞\+\+n≅HolnV\\in T\_\{C\}\\mathcal\{C\}^\{n\}\_\{\+\+\}\\cong\\mathrm\{Hol\}^\{n\},K∈LT1nK\\in\\mathrm\{LT\}\_\{1\}^\{n\}andX,ξ∈LT0nX,\\xi\\in\\mathrm\{LT\}\_\{0\}^\{n\}, the involved maps and their differentials in ECM and LECM are

Θ​\(C\)\\displaystyle\\Theta\(C\)=𝔻​\(L\)−1​L,\\displaystyle=\\mathbb\{D\}\(L\)^\{\-1\}L,\(36\)Θ−1​\(K\)\\displaystyle\\Theta^\{\-1\}\(K\)=𝔻​\(K​K⊤\)−12​K​K⊤​𝔻​\(K​K⊤\)−12,\\displaystyle=\\mathbb\{D\}\\left\(KK^\{\\top\}\\right\)^\{\-\\frac\{1\}\{2\}\}KK^\{\\top\}\\mathbb\{D\}\\left\(KK^\{\\top\}\\right\)^\{\-\\frac\{1\}\{2\}\},\(37\)log⁡\(K\)\\displaystyle\\log\(K\)=∑k=1n−1\(−1\)k−1k​\(K−In\)k,\\displaystyle=\\sum\_\{k=1\}^\{n\-1\}\\frac\{\(\-1\)^\{k\-1\}\}\{k\}\\left\(K\-I\_\{n\}\\right\)^\{k\},\(38\)exp⁡\(ξ\)\\displaystyle\\exp\(\\xi\)=∑k=0n−11k\!​ξk,\\displaystyle=\\sum\_\{k=0\}^\{n\-1\}\\frac\{1\}\{k\!\}\\xi^\{k\},\(39\)Θ∗,C​\(V\)\\displaystyle\\Theta\_\{\*,C\}\(V\)=Θ​\(C\)​\(L−1​V​L−⊤\)12−12​𝔻​\(L−1​V​L−⊤\)​Θ​\(C\),\\displaystyle=\\Theta\(C\)\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}\-\\frac\{1\}\{2\}\\mathbb\{D\}\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\\Theta\(C\),\(40\)\(Θ∗,C\)−1​\(ξ\)\\displaystyle\\left\(\\Theta\_\{\*,C\}\\right\)^\{\-1\}\(\\xi\)=\(L​ξ⊤−C​𝔻​\(L​ξ⊤\)\)​𝔻​\(L\)\+𝔻​\(L\)​\(ξ​L⊤−𝔻​\(L​ξ⊤\)​C\),\\displaystyle=\\left\(L\\xi^\{\\top\}\-C\\mathbb\{D\}\\left\(L\\xi^\{\\top\}\\right\)\\right\)\\mathbb\{D\}\(L\)\+\\mathbb\{D\}\(L\)\\left\(\\xi L^\{\\top\}\-\\mathbb\{D\}\\left\(L\\xi^\{\\top\}\\right\)C\\right\),\(41\)log∗,K⁡\(ξ\)\\displaystyle\\log\_\{\*,K\}\(\\xi\)=∑k=1n−1\(−1\)k−1k​\[\(K−In\)k−1​ξ\+\(K−In\)k−2​ξ​\(K−In\)\+⋯\+ξ​\(K−In\)k−1\],\\displaystyle=\\sum\_\{k=1\}^\{n\-1\}\\frac\{\(\-1\)^\{k\-1\}\}\{k\}\\left\[\\left\(K\-I\_\{n\}\\right\)^\{k\-1\}\\xi\+\\left\(K\-I\_\{n\}\\right\)^\{k\-2\}\\xi\\left\(K\-I\_\{n\}\\right\)\+\\cdots\+\\xi\\left\(K\-I\_\{n\}\\right\)^\{k\-1\}\\right\],\(42\)exp∗,X⁡\(ξ\)\\displaystyle\\exp\_\{\*,X\}\(\\xi\)=∑k=1n−11k\!​\(Xk−1​ξ\+Xk−2​ξ​X\+⋯\+ξ​Xk−1\),\\displaystyle=\\sum\_\{k=1\}^\{n\-1\}\\frac\{1\}\{k\!\}\\left\(X^\{k\-1\}\\xi\+X^\{k\-2\}\\xi X\+\\cdots\+\\xi X^\{k\-1\}\\right\),\(43\)\(log∘Θ\)∗,C​\(V\)\\displaystyle\(\\log\\circ\\Theta\)\_\{\*,C\}\(V\)=log∗,Θ​\(C\)⁡\(Θ∗,C​\(V\)\),\\displaystyle=\\log\_\{\*,\\Theta\(C\)\}\\left\(\\Theta\_\{\*,C\}\(V\)\\right\),\(44\)Chol∗,C⁡\(V\)\\displaystyle\\operatorname\{Chol\}\_\{\*,C\}\(V\)=L​\(L−1​V​L−⊤\)12,\\displaystyle=L\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\},\(45\)\(Chol∗,C\)−1​\(Z\)\\displaystyle\(\\operatorname\{Chol\}\_\{\*,C\}\)^\{\-1\}\(Z\)=L​Z⊤\+Z​L⊤,∀Z∈TL​LT\+\+n≅LTn\.\\displaystyle=LZ^\{\\top\}\+ZL^\{\\top\},\\quad\\forall Z\\in T\_\{L\}\\mathrm\{LT\}\_\{\+\+\}^\{n\}\\cong\\mathrm\{LT\}^\{n\}\.\(46\)whereLLis the Cholesky factor ofCC,InI\_\{n\}is then×nn\\times nidentity matrix andLT\+\+n\\mathrm\{LT\}\_\{\+\+\}^\{n\}is the Cholesky manifold ofn×nn\\times nCholesky matrices\. Due to the nilpotency ofLT0n\\mathrm\{LT\}\_\{0\}^\{n\}, the matrix logarithm overLT1n\\mathrm\{LT\}\_\{1\}^\{n\}and exponentiation overLT0n\\mathrm\{LT\}\_\{0\}^\{n\}are free from eigendecomposition\. With the above equations,[Tab\.XVI](https://arxiv.org/html/2607.08783#A2.T16)summarizes the Riemannian operators under ECM and LECM\.

TABLE XVI:Riemannian operators under the non\-permutation\-invariant log\-Euclidean Metrics\. Here,C,C′∈𝒞\+\+nC,C^\{\\prime\}\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}are correlation matrices andV,W∈TC​𝒞\+\+n≅HolnV,W\\in T\_\{C\}\\mathcal\{C\}^\{n\}\_\{\+\+\}\\cong\\mathrm\{Hol\}^\{n\}are tangent vectors\.OperationECMLECMgC​\(V,W\)g\_\{C\}\(V,W\)⟨Θ∗,C​\(V\),Θ∗,C​\(W\)⟩\\left\\langle\\Theta\_\{\*,C\}\(V\),\\Theta\_\{\*,C\}\(W\)\\right\\rangle⟨\(log∘Θ\)∗,C​\(V\),\(log∘Θ\)∗,C​\(W\)⟩\\left\\langle\(\\log\\circ\\Theta\)\_\{\*,C\}\(V\),\(\\log\\circ\\Theta\)\_\{\*,C\}\(W\)\\right\\rangleExpC⁡\(V\)\\operatorname\{Exp\}\_\{C\}\(V\)Θ−1​\(Θ​\(C\)\+Θ∗,C​\(V\)\)\\Theta^\{\-1\}\\left\(\\Theta\\left\(C\\right\)\+\\Theta\_\{\*,C\}\\left\(V\\right\)\\right\)\(log∘Θ\)−1​\(log∘Θ​\(C\)\+\(log∘Θ\)∗,C​\(V\)\)\(\\log\\circ\\Theta\)^\{\-1\}\\left\(\\log\\circ\\Theta\\left\(C\\right\)\+\(\\log\\circ\\Theta\)\_\{\*,C\}\\left\(V\\right\)\\right\)LogC⁡\(C′\)\\operatorname\{Log\}\_\{C\}\(C^\{\\prime\}\)Θ∗,Θ​\(C\)−1​\(Θ​\(C′\)−Θ​\(C\)\)\\Theta^\{\-1\}\_\{\{\*,\\Theta\(C\)\}\}\\left\(\\Theta\\left\(C^\{\\prime\}\\right\)\-\\Theta\\left\(C\\right\)\\right\)\(log∘Θ\)∗,log∘Θ​\(C\)−1​\(log∘Θ​\(C′\)−log∘Θ​\(C\)\)\(\\log\\circ\\Theta\)^\{\-1\}\_\{\{\*,\\log\\circ\\Theta\(C\)\}\}\\left\(\\log\\circ\\Theta\\left\(C^\{\\prime\}\\right\)\-\\log\\circ\\Theta\\left\(C\\right\)\\right\)γ​\(t;C,C′\)\\gamma\(t;C,C^\{\\prime\}\)Θ−1​\(\(1−t\)​Θ​\(C\)\+t​Θ​\(C′\)\)\\Theta^\{\-1\}\\left\(\\left\(1\-t\\right\)\\Theta\\left\(C\\right\)\+t\\Theta\\left\(C^\{\\prime\}\\right\)\\right\)\(log∘Θ\)−1​\(\(1−t\)​log∘Θ​\(C\)\+t​log∘Θ​\(C′\)\)\(\\log\\circ\\Theta\)^\{\-1\}\\left\(\\left\(1\-t\\right\)\\log\\circ\\Theta\\left\(C\\right\)\+t\\log\\circ\\Theta\\left\(C^\{\\prime\}\\right\)\\right\)d⁡\(C,C′\)\\operatorname\{d\}\(C,C^\{\\prime\}\)‖Θ​\(C\)−Θ​\(C′\)‖F\\left\\\|\{\\Theta\\left\(C\\right\)\-\\Theta\\left\(C^\{\\prime\}\\right\)\}\\right\\\|\_\{\\mathrm\{F\}\}‖log∘Θ​\(C\)−log∘Θ​\(C′\)‖F\\left\\\|\{\\log\\circ\\Theta\\left\(C\\right\)\-\\log\\circ\\Theta\\left\(C^\{\\prime\}\\right\)\}\\right\\\|\_\{\\mathrm\{F\}\}Fréchet meanΘ−1​\(1k​∑i=1kΘ​\(Ci\)\)\\Theta^\{\-1\}\\left\(\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}\\Theta\\left\(C\_\{i\}\\right\)\\right\)\(log∘Θ\)−1​\(1k​∑i=1klog∘Θ​\(Ci\)\)\(\\log\\circ\\Theta\)^\{\-1\}\\left\(\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}\\log\\circ\\Theta\\left\(C\_\{i\}\\right\)\\right\)Curvature00PTC→C′⁡\(V\)\\operatorname\{PT\}\_\{C\\rightarrow C^\{\\prime\}\}\(V\)\(Θ∗,C′\)−1​\(Θ∗,C​\(V\)\)\\left\(\\Theta\_\{\*,C^\{\\prime\}\}\\right\)^\{\-1\}\\left\(\\Theta\_\{\*,C\}\\left\(V\\right\)\\right\)\(\(log∘Θ\)∗,C′\)−1​\(\(log∘Θ\)∗,C​\(V\)\)\\left\(\(\\log\\circ\\Theta\)\_\{\*,C^\{\\prime\}\}\\right\)^\{\-1\}\\left\(\(\\log\\circ\\Theta\)\_\{\*,C\}\\left\(V\\right\)\\right\)
#### B\.5\.2Permutation\-invariant Metrics

Let𝔖n\\mathfrak\{S\}^\{n\}be the group of permutation matricesPσ=\[δi,σ​\(j\)\]1⩽i,j⩽nP\_\{\\sigma\}=\\left\[\\delta\_\{i,\\sigma\(j\)\}\\right\]\_\{1\\leqslant i,j\\leqslant n\}by permutationσ\\sigma, and𝒟±​\(n\)=\{diag⁡\(\(ε1,…,εn\)\),ε∈\{−1,1\}n\}\\mathcal\{D\}^\{\\pm\}\(n\)=\\left\\\{\\operatorname\{diag\}\\left\(\\left\(\\varepsilon\_\{1\},\\ldots,\\varepsilon\_\{n\}\\right\)\\right\),\\varepsilon\\in\\\{\-1,1\\\}^\{n\}\\right\\\}be the group of diagonal matrices with coefficients in\{−1,1\}\\\{\-1,1\\\}\.\[[75](https://arxiv.org/html/2607.08783#bib.bib68), Thm\. 1\.1\]showed that the biggest congruence action on full\-rank correlation matrices is the action of signed permutation matrices:

⋆:\(A,C\)∈𝔖±\(n\)×𝒞\+\+n⟼ACA⊤∈𝒞\+\+n,\\star:\(A,C\)\\in\\mathfrak\{S\}^\{\\pm\}\(n\)\\times\\mathcal\{C\}^\{n\}\_\{\+\+\}\\longmapsto ACA^\{\\top\}\\in\\mathcal\{C\}^\{n\}\_\{\+\+\},\(47\)with𝔖±​\(n\)=𝒟±​\(n\)​𝔖n\\mathfrak\{S\}^\{\\pm\}\(n\)=\\mathcal\{D\}^\{\\pm\}\(n\)\\mathfrak\{S\}^\{n\}\. Based on this finding,\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\]proposed two permutation\-invariant metrics, namely OLM and LSM, by pulling back permutation\-invariant inner products via the following permutation\-equivariant diffeomorphisms:

𝒞\+\+n\\displaystyle\\mathcal\{C\}^\{n\}\_\{\+\+\}⇌Exp∘Log∘=Off∘logHoln,\\displaystyle\\xrightleftharpoons\[\\operatorname\{Exp\}^\{\\circ\}\]\{\\operatorname\{Log\}^\{\\circ\}=\\mathrm\{Off\}\\circ\\log\}\\mathrm\{Hol\}^\{n\},\(48\)𝒞\+\+n\\displaystyle\\mathcal\{C\}^\{n\}\_\{\+\+\}⇌Exp⋆=Cor∘expLog⋆Row0n,\\displaystyle\\xrightleftharpoons\[\\operatorname\{Exp\}^\{\\star\}=\\operatorname\{Cor\}\\circ\\exp\]\{\\operatorname\{Log\}^\{\\star\}\}\\mathrm\{Row\}\_\{0\}^\{n\},\(49\)Exp∘:Holn∋H\\displaystyle\\operatorname\{Exp\}^\{\\circ\}:\\mathrm\{Hol\}^\{n\}\\ni H⟼exp⁡\(𝒟\+​\(H\)\+H\),\\displaystyle\\longmapsto\\exp\(\\mathcal\{D\}^\{\+\}\(H\)\+H\),\(50\)Log⋆:𝒞\+\+n∋C\\displaystyle\\operatorname\{Log\}^\{\\star\}:\\mathcal\{C\}^\{n\}\_\{\+\+\}\\ni C⟼log⁡\(𝒟⋆​\(C\)​C​𝒟⋆​\(C\)\)∈Row0n\.\\displaystyle\\longmapsto\\log\(\\mathcal\{D\}^\{\\star\}\(C\)C\\mathcal\{D\}^\{\\star\}\(C\)\)\\in\\mathrm\{Row\}\_\{0\}^\{n\}\.\(51\)wherelog⁡\(⋅\)\\log\(\\cdot\)andexp⁡\(⋅\)\\exp\(\\cdot\)are symmetric matrix logarithm and exponentiation\. The involved𝒟\+\\mathcal\{D\}^\{\+\}and𝒟⋆\\mathcal\{D\}^\{\\star\}can be formally expressed as𝒟\+:Holn→𝔻n\\mathcal\{D\}^\{\+\}:\\mathrm\{Hol\}^\{n\}\\rightarrow\\mathbb\{D\}^\{n\}and𝒟⋆:𝒞\+\+n→𝔻\+n\\mathcal\{D\}^\{\\star\}:\\mathcal\{C\}^\{n\}\_\{\+\+\}\\rightarrow\\mathbb\{D\}\_\{\+\}^\{n\}, where𝔻n\\mathbb\{D\}^\{n\}denotes the Euclidean space ofn×nn\\times ndiagonal matrices, and𝔻\+n\\mathbb\{D\}\_\{\+\}^\{n\}is the submanifold of𝔻n\\mathbb\{D\}^\{n\}, consisting of positive diagonal matrices\.

The differentials ofLog∘\\operatorname\{Log\}^\{\\circ\}andLog⋆\\operatorname\{Log\}^\{\\star\}and their inverses can be calculated by the differential of symmetric matrix logarithm and exponentiation\[[75](https://arxiv.org/html/2607.08783#bib.bib68), Thms\. 2\.4 and 4\.1\]\. GivenC∈𝒞\+\+nC\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}, tangent vectorV∈TC​𝒞\+\+n≅HolnV\\in T\_\{C\}\\mathcal\{C\}^\{n\}\_\{\+\+\}\\cong\\mathrm\{Hol\}^\{n\},H,W∈HolnH,W\\in\\mathrm\{Hol\}^\{n\}, andS=H\+𝒟\+​\(H\)=U​Δ​U⊤S=H\+\\mathcal\{D\}^\{\+\}\(H\)=U\\Delta U^\{\\top\}, the differential ofLog∘\\operatorname\{Log\}^\{\\circ\}and its inverseExp∘\\operatorname\{Exp\}^\{\\circ\}are

Log∗,C∘⁡\(V\)\\displaystyle\\operatorname\{Log\}^\{\\circ\}\_\{\*,C\}\(V\)=Off​\(log∗,C⁡\(V\)\),\\displaystyle=\\mathrm\{Off\}\\left\(\\log\_\{\*,C\}\(V\)\\right\),\(52\)Exp∗,H∘⁡\(W\)\\displaystyle\\operatorname\{Exp\}^\{\\circ\}\_\{\*,H\}\(W\)=exp∗,S⁡\(W\+𝒟∗,H\+​\(W\)\),\\displaystyle=\\exp\_\{\*,S\}\\left\(W\+\\mathcal\{D\}^\{\+\}\_\{\*,H\}\(W\)\\right\),\(53\)𝒟∗,H\+​\(W\)\\displaystyle\\mathcal\{D\}^\{\+\}\_\{\*,H\}\(W\)=−diag⁡\(\(H0\)−1​𝔻​\(exp∗,S⁡\(W\)\)​𝟏\),\\displaystyle=\-\\operatorname\{diag\}\\left\(\\left\(H^\{0\}\\right\)^\{\-1\}\\mathbb\{D\}\\left\(\\exp\_\{\*,S\}\(W\)\\right\)\\boldsymbol\{1\}\\right\),\(54\)𝒮\+\+n∋Hi​l0\\displaystyle\\mathcal\{S\}^\{n\}\_\{\+\+\}\\ni H\_\{il\}^\{0\}=∑j,kPi​j​Pi​k​Pl​j​Pl​k​Lj,k,\\displaystyle=\\sum\_\{j,k\}P\_\{ij\}P\_\{ik\}P\_\{lj\}P\_\{lk\}L\_\{j,k\},\(55\)whereLLis the Loewner matrix ofexp∗,S\\exp\_\{\*,S\}, and𝟏\\boldsymbol\{1\}is the vector of all 1 entities\. Here,log∗\\log\_\{\*\}andexp∗\\exp\_\{\*\}can be calculated using the Daleckii\-Krein formula of the symmetric matrix, whilediag⁡\(⋅\):ℝn→𝔻n\\operatorname\{diag\}\(\\cdot\):\\mathbb\{R\}^\{n\}\\rightarrow\\mathbb\{D\}^\{n\}returns a diagonal matrix from an input vector\. Further denotingX,Y∈Row0⁡\(n\)X,Y\\in\\operatorname\{Row\}\_\{0\}\(n\)andΣ=𝒟⋆​\(C\)​C​𝒟⋆​\(C\)\\Sigma=\\mathcal\{D\}^\{\\star\}\(C\)C\\mathcal\{D\}^\{\\star\}\(C\), the differentials ofLog⋆\\operatorname\{Log\}^\{\\star\}and its inverseExp⋆\\operatorname\{Exp\}^\{\\star\}are

Log∗,C⋆⁡\(V\)\\displaystyle\\operatorname\{Log\}^\{\\star\}\_\{\*,C\}\(V\)=log∗,Σ⁡\(Δ​V​Δ\+12​\(V0​Σ\+Σ​V0\)\),\\displaystyle=\\log\_\{\*,\\Sigma\}\\left\(\\Delta V\\Delta\+\\frac\{1\}\{2\}\\left\(V^\{0\}\\Sigma\+\\Sigma V^\{0\}\\right\)\\right\),\(56\)Exp∗,X⋆⁡\(Y\)\\displaystyle\\operatorname\{Exp\}^\{\\star\}\_\{\*,X\}\(Y\)=Δ−1​\[exp∗,X⁡\(Y\)−12​\(Δ−2​𝔻​\(exp∗,X⁡\(Y\)\)​Σ\+Σ​𝔻​\(exp∗,X⁡\(Y\)\)​Δ−2\)\]​Δ−1\\displaystyle=\\Delta^\{\-1\}\\left\[\\exp\_\{\*,X\}\(Y\)\-\\frac\{1\}\{2\}\\left\(\\Delta^\{\-2\}\\mathbb\{D\}\\left\(\\exp\_\{\*,X\}\(Y\)\\right\)\\Sigma\+\\Sigma\\mathbb\{D\}\\left\(\\exp\_\{\*,X\}\(Y\)\\right\)\\Delta^\{\-2\}\\right\)\\right\]\\Delta^\{\-1\}\(57\)withΔ=𝔻​\(Σ\)1/2\\Delta=\\mathbb\{D\}\(\\Sigma\)^\{1/2\}andV0=−2​diag⁡\(\(In\+Σ\)−1​Δ​V​Δ​𝟏\)V^\{0\}=\-2\\operatorname\{diag\}\\left\(\\left\(I\_\{n\}\+\\Sigma\\right\)^\{\-1\}\\Delta V\\Delta\\boldsymbol\{1\}\\right\)\.

As bothLog∗⋆\\operatorname\{Log\}^\{\\star\}\_\{\*\}andLog∗∘\\operatorname\{Log\}^\{\\circ\}\_\{\*\}are permutation\-equivariant\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\], permutation\-invariant metrics over the correlation manifold can be induced by permutation\-invariant inner products overHoln\\mathrm\{Hol\}^\{n\}andRow0n\\mathrm\{Row\}\_\{0\}^\{n\}, respectively\. The following two theorems review such inner products\.

TABLE XVII:Riemannian geometries under the permutation\-invariant log\-Euclidean Metrics\.OperationOLMLSMgC​\(V,W\)g\_\{C\}\(V,W\)⟨Log∗,C∘⁡\(V\),Log∗,C∘⁡\(W\)⟩\(α,β,γ\)\\left\\langle\\operatorname\{Log\}^\{\\circ\}\_\{\*,C\}\(V\),\\operatorname\{Log\}^\{\\circ\}\_\{\*,C\}\(W\)\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}⟨Log∗,C⋆⁡\(V\),Log∗,C⋆⁡\(W\)⟩\(α,δ,ζ\)\\left\\langle\\operatorname\{Log\}^\{\\star\}\_\{\*,C\}\(V\),\\operatorname\{Log\}^\{\\star\}\_\{\*,C\}\(W\)\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}ExpC⁡\(V\)\\operatorname\{Exp\}\_\{C\}\(V\)Exp∘⁡\(Log∘⁡\(C\)\+Log∗,C∘⁡\(V\)\)\\operatorname\{Exp\}^\{\\circ\}\\left\(\\operatorname\{Log\}^\{\\circ\}\\left\(C\\right\)\+\\operatorname\{Log\}^\{\\circ\}\_\{\*,C\}\\left\(V\\right\)\\right\)Exp⋆⁡\(Log⋆⁡\(C\)\+Log∗,C⋆⁡\(V\)\)\\operatorname\{Exp\}^\{\\star\}\\left\(\\operatorname\{Log\}^\{\\star\}\\left\(C\\right\)\+\\operatorname\{Log\}^\{\\star\}\_\{\*,C\}\\left\(V\\right\)\\right\)LogC⁡\(C′\)\\operatorname\{Log\}\_\{C\}\(C^\{\\prime\}\)Exp∗,Log∘⁡\(C\)∘⁡\(Log∘⁡\(C′\)−Log∘⁡\(C\)\)\\operatorname\{Exp\}^\{\\circ\}\_\{\{\*,\\operatorname\{Log\}^\{\\circ\}\(C\)\}\}\\left\(\\operatorname\{Log\}^\{\\circ\}\\left\(C^\{\\prime\}\\right\)\-\\operatorname\{Log\}^\{\\circ\}\\left\(C\\right\)\\right\)Exp∗,Log⋆⁡\(C\)⋆⁡\(Log⋆⁡\(C′\)−Log⋆⁡\(C\)\)\\operatorname\{Exp\}^\{\\star\}\_\{\{\*,\\operatorname\{Log\}^\{\\star\}\(C\)\}\}\\left\(\\operatorname\{Log\}^\{\\star\}\\left\(C^\{\\prime\}\\right\)\-\\operatorname\{Log\}^\{\\star\}\\left\(C\\right\)\\right\)γ​\(t;C,C′\)\\gamma\(t;C,C^\{\\prime\}\)Exp∘⁡\(\(1−t\)​Log∘⁡\(C\)\+t​Log∘⁡\(C′\)\)\\operatorname\{Exp\}^\{\\circ\}\\left\(\\left\(1\-t\\right\)\\operatorname\{Log\}^\{\\circ\}\\left\(C\\right\)\+t\\operatorname\{Log\}^\{\\circ\}\\left\(C^\{\\prime\}\\right\)\\right\)Exp⋆⁡\(\(1−t\)​Log⋆⁡\(C\)\+t​Log⋆⁡\(C′\)\)\\operatorname\{Exp\}^\{\\star\}\\left\(\\left\(1\-t\\right\)\\operatorname\{Log\}^\{\\star\}\\left\(C\\right\)\+t\\operatorname\{Log\}^\{\\star\}\\left\(C^\{\\prime\}\\right\)\\right\)d⁡\(C,C′\)\\operatorname\{d\}\(C,C^\{\\prime\}\)‖Log∘⁡\(C\)−Log∘⁡\(C′\)‖\(α,β,γ\)\\left\\\|\\operatorname\{Log\}^\{\\circ\}\\left\(C\\right\)\-\\operatorname\{Log\}^\{\\circ\}\\left\(C^\{\\prime\}\\right\)\\right\\\|^\{\(\\alpha,\\beta,\\gamma\)\}‖Log⋆⁡\(C\)−Log⋆⁡\(C′\)‖\(α,δ,ζ\)\\left\\\|\\operatorname\{Log\}^\{\\star\}\\left\(C\\right\)\-\\operatorname\{Log\}^\{\\star\}\\left\(C^\{\\prime\}\\right\)\\right\\\|^\{\(\\alpha,\\delta,\\zeta\)\}Fréchet meanExp∘⁡\(1k​∑i=1kLog∘⁡\(Ci\)\)\\operatorname\{Exp\}^\{\\circ\}\\left\(\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}\\operatorname\{Log\}^\{\\circ\}\\left\(C\_\{i\}\\right\)\\right\)Exp⋆⁡\(1k​∑i=1kLog⋆⁡\(Ci\)\)\\operatorname\{Exp\}^\{\\star\}\\left\(\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}\\operatorname\{Log\}^\{\\star\}\\left\(C\_\{i\}\\right\)\\right\)Curvature00PTC→C′⁡\(V\)\\operatorname\{PT\}\_\{C\\rightarrow C^\{\\prime\}\}\(V\)\(Log∗,C′∘\)−1​\(Log∗,C∘⁡\(V\)\)\\left\(\\operatorname\{Log\}^\{\\circ\}\_\{\*,C^\{\\prime\}\}\\right\)^\{\-1\}\\left\(\\operatorname\{Log\}^\{\\circ\}\_\{\*,C\}\\left\(V\\right\)\\right\)\(Log∗,C′⋆\)−1​\(Log∗,C⋆⁡\(V\)\)\\left\(\\operatorname\{Log\}^\{\\star\}\_\{\*,C^\{\\prime\}\}\\right\)^\{\-1\}\\left\(\\operatorname\{Log\}^\{\\star\}\_\{\*,C\}\\left\(V\\right\)\\right\)InvarianceBi\-invariancePermutation\-invarianceSinged\-permutation\-invariance\(β=γ=0\)\(\\beta=\\gamma=0\)Bi\-invariancePermutation\-invariance###### Theorem B\.7\(Permutation\-Invariant Inner Products onHoln\\mathrm\{Hol\}^\{n\}\[[74](https://arxiv.org/html/2607.08783#bib.bib160)\]\)\.

Supposingn≥4n\\geq 4, permutation\-invariant inner products onHoln\\mathrm\{Hol\}^\{n\}are:

⟨X1,X2⟩\(α,β,γ\)=α​tr⁡\(X1​X2\)\+β​Sum⁡\(X1​X2\)\+γ​Sum⁡\(X1\)​Sum⁡\(X2\),∀X1,X2∈Holn,\\left\\langle X\_\{1\},X\_\{2\}\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}=\\alpha\\operatorname\{tr\}\(X\_\{1\}X\_\{2\}\)\+\\beta\\operatorname\{Sum\}\\left\(X\_\{1\}X\_\{2\}\\right\)\+\\gamma\\operatorname\{Sum\}\(X\_\{1\}\)\\operatorname\{Sum\}\(X\_\{2\}\),\\quad\\forall X\_\{1\},X\_\{2\}\\in\\mathrm\{Hol\}^\{n\},\(58\)withα\>0\\alpha\>0,2​α\+\(n−2\)​β\>02\\alpha\+\(n\-2\)\\beta\>0, andα\+\(n−1\)​\(β\+n​γ\)\>0\\alpha\+\(n\-1\)\(\\beta\+n\\gamma\)\>0\. Forn=3n=3, permutation\-invariant inner products have the same form withα=0\\alpha=0:

⟨X1,X2⟩\(α,β,γ\)=β​Sum⁡\(X1​X2\)\+γ​Sum⁡\(X1\)​Sum⁡\(X2\),with​β\>0​and​β\+3​γ\>0\.\\left\\langle X\_\{1\},X\_\{2\}\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}=\\beta\\operatorname\{Sum\}\(X\_\{1\}X\_\{2\}\)\+\\gamma\\operatorname\{Sum\}\(X\_\{1\}\)\\operatorname\{Sum\}\(X\_\{2\}\),\\quad\\text\{with \}\\beta\>0\\text\{ and \}\\beta\+3\\gamma\>0\.\(59\)Forn=2n=2, they have the same form withα=β=0\\alpha=\\beta=0:

⟨X1,X2⟩\(α,β,γ\)=γ​Sum⁡\(X1\)​Sum⁡\(X2\),with​γ\>0\.\\left\\langle X\_\{1\},X\_\{2\}\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}=\\gamma\\operatorname\{Sum\}\(X\_\{1\}\)\\operatorname\{Sum\}\(X\_\{2\}\),\\quad\\text\{with \}\\gamma\>0\.\(60\)

###### Theorem B\.8\(Permutation\-Invariant Inner Products onRow0n\\mathrm\{Row\}\_\{0\}^\{n\}\[[75](https://arxiv.org/html/2607.08783#bib.bib68)\]\)\.

Forn≥4n\\geq 4, permutation\-invariant inner products onRow0n\\mathrm\{Row\}\_\{0\}^\{n\}are

⟨Y1,Y2⟩\(α,δ,ζ\)=α​tr⁡\(Y1​Y2\)\+δ​tr⁡\(𝔻​\(Y1\)​𝔻​\(Y2\)\)\+ζ​tr⁡\(Y1\)​tr⁡\(Y2\),∀Y1,Y2∈Row0n,\\left\\langle Y\_\{1\},Y\_\{2\}\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}=\\alpha\\operatorname\{tr\}\(Y\_\{1\}Y\_\{2\}\)\+\\delta\\operatorname\{tr\}\(\\mathbb\{D\}\(Y\_\{1\}\)\\mathbb\{D\}\(Y\_\{2\}\)\)\+\\zeta\\operatorname\{tr\}\(Y\_\{1\}\)\\operatorname\{tr\}\(Y\_\{2\}\),\\quad\\forall Y\_\{1\},Y\_\{2\}\\in\\mathrm\{Row\}\_\{0\}^\{n\},\(61\)withα\>0\\alpha\>0,n​α\+\(n−2\)​δ\>0n\\alpha\+\(n\-2\)\\delta\>0, andn​α\+\(n−1\)​\(δ\+n​ζ\)\>0n\\alpha\+\(n\-1\)\(\\delta\+n\\zeta\)\>0\. Forn=3n=3, the permutation\-invariant inner products have the same form withα=0\\alpha=0\. Forn=2n=2, they have the same form withα=δ=0\\alpha=\\delta=0\.

As shown by\[[74](https://arxiv.org/html/2607.08783#bib.bib160)\], the OLM is further signed\-permutation invariant underβ=γ=0\\beta=\\gamma=0, reduced to the scaled canonical Euclidean inner product:

⟨V,W⟩\(α,β,γ\)=α​⟨V,W⟩,∀V,W∈Holn\.\\left\\langle V,W\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}=\\alpha\\left\\langle V,W\\right\\rangle,\\quad\\forall V,W\\in\\mathrm\{Hol\}^\{n\}\.\(62\)In the main paper, we assume that⟨⋅,⋅⟩\(α,β,γ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\beta,\\gamma\)\}and⟨⋅,⋅⟩\(α,δ,ζ\)\\left\\langle\\cdot,\\cdot\\right\\rangle^\{\(\\alpha,\\delta,\\zeta\)\}are the canonical Euclidean inner product\.[Tab\.XVII](https://arxiv.org/html/2607.08783#A2.T17)summarizes the Riemannian structures of OLM and LSM\.

## Appendix CBasic Layers in SPDNet and TSMNet

SPDNet\[[43](https://arxiv.org/html/2607.08783#bib.bib41)\]is the most classic SPD neural network\. SPDNet mimics the conventional densely connected feedforward network, consisting of three basic building blocks

BiMap layer:​Sk=Wk​Sk−1​Wk⊤,with​Wk​semi\-orthogonal,\\displaystyle\\text\{BiMap layer: \}S^\{k\}=W^\{k\}S^\{k\-1\}W^\{k\\top\},\\text\{ with \}W^\{k\}\\text\{ semi\-orthogonal,\}\(63\)ReEig layer:​Sk=Uk−1​max⁡\(Σk−1,ϵ​In\)​Uk−1⊤,with​Sk−1=Uk−1​Σk−1​Uk−1⊤,\\displaystyle\\text\{ReEig layer: \}S^\{k\}=U^\{k\-1\}\\max\(\\Sigma^\{k\-1\},\\epsilon I\_\{n\}\)U^\{k\-1\\top\},\\text\{ with \}S^\{k\-1\}=U^\{k\-1\}\\Sigma^\{k\-1\}U^\{k\-1\\top\},\(64\)LogEig layer:​Sk=log⁡\(Sk−1\)\.\\displaystyle\\text\{LogEig layer: \}S^\{k\}=\\log\(S^\{k\-1\}\)\.\(65\)wheremax⁡\(⋅\)\\max\(\\cdot\)is element\-wise maximization\. BiMap and ReEig mimic transformation and non\-linear activation, while LogEig maps SPD matrices into the tangent space at the identity matrix for classification\.

TSMNet\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]can be illustrate asft​c→fs​c→fB​i​M​a​p→fR​e​E​i​g→fL​o​g​E​i​gf\_\{tc\}\\rightarrow f\_\{sc\}\\rightarrow f\_\{BiMap\}\\rightarrow f\_\{ReEig\}\\rightarrow f\_\{LogEig\}, whereft​cf\_\{tc\}andfs​cf\_\{sc\}denote temporal and spatial convolution, respectively\.

## Appendix DStatistical Results of Scaling in LieBN

In this section, we will show the effect of our scaling \([Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)\) on the population\. We will see that while the resulting population variance is generally agnostic, it becomes analytic under certain circumstances, such as SPD manifolds under LEM or LCM\. As a result,[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)can normalize and transform the latent Gaussian distribution\.

To simplify, letϕs​\(P\)=ExpE⁡\[s​LogE⁡\(P\)\]\\phi\_\{s\}\(P\)=\\operatorname\{Exp\}\_\{E\}\\left\[s\\operatorname\{Log\}\_\{E\}\(P\)\\right\]\. Similar to the main paper,ℳ\\mathcal\{M\}denotes a Lie group with a left\-invariant metric\. First, we present a lemma on the resulting P\.D\.F\. of a random point transformed byϕs\\phi\_\{s\}\.

###### Lemma D\.1\.

Given a random pointXXdistributed overℳ\\mathcal\{M\}with P\.D\.F\.pXp\_\{X\}, the P\.D\.F ofY=ϕs​\(X\)Y=\\phi\_\{s\}\(X\)is given by:

pY​\(Q\)=Δ​pX​\(ϕs\(−1\)​\(Q\)\)\.p\_\{Y\}\(Q\)=\\Delta p\_\{X\}\(\\phi\_\{s\}^\{\(\-1\)\}\(Q\)\)\.\(66\)whereΔ=\|ϕs⁣∗−1\|\|Lϕs−1​\(Q\)⊙Q−1⁣∗\|\\Delta=\\frac\{\|\\phi^\{\-1\}\_\{s\*\}\|\}\{\\left\|L\_\{\\phi\_\{s\}^\{\-1\}\(Q\)\\odot Q^\{\-1\}\*\}\\right\|\}\. Here\|⋅\|\|\\cdot\|denotes the determinant, andϕs⁣∗−1\\phi^\{\-1\}\_\{s\*\}andLϕs−1​\(Q\)⊙Q−1⁣∗L\_\{\\phi\_\{s\}^\{\-1\}\(Q\)\\odot Q^\{\-1\}\*\}are the differentials\.

###### Proof\.

For the sake of simplicity, we will denoteϕs\\phi\_\{s\}asϕ\\phithroughout this proof\. The volume element w\.r\.t\. a left\-invariant metric is the Haar measure\[[64](https://arxiv.org/html/2607.08783#bib.bib107), Sec\. 3\.2\]:

dL⁡ℳ​\(P\)=d​P\|LP⁣∗,E\|,\\operatorname\{d\}\_\{L\}\\mathcal\{M\}\(P\)=\\frac\{dP\}\{\|L\_\{P\*,E\}\|\},\(67\)where\|LP⁣∗,E\|\|L\_\{P\*,E\}\|is the determinant666This should be more precisely understood as the determinant of the matrix representation ofLP⁣∗,EL\_\{P\*,E\}under a local coordinateof the differential ofLPL\_\{P\}at the neutral elementEE\. Then we have

dL⁡ℳ​\(ϕ−1​\(Q\)\)\\displaystyle\\operatorname\{d\}\_\{L\}\\mathcal\{M\}\(\\phi^\{\-1\}\(Q\)\)=d⁡ϕ−1​\(Q\)\|Lϕ−1​\(Q\)⁣∗,E\|\\displaystyle=\\frac\{\\operatorname\{d\}\\phi^\{\-1\}\(Q\)\}\{\|L\_\{\\phi^\{\-1\}\(Q\)\*,E\}\|\}\(68\)=\|\(Lϕ−1​\(Q\)⊙Q−1∘LQ\)∗,E\|−1​\|ϕ∗−1\|​d⁡Q\\displaystyle=\|\(L\_\{\\phi^\{\-1\}\(Q\)\\odot Q^\{\-1\}\}\\circ L\_\{Q\}\)\_\{\*,E\}\|^\{\-1\}\|\\phi^\{\-1\}\_\{\*\}\|\\operatorname\{d\}Q=\|ϕ∗−1\|\|Lϕ−1​\(Q\)⊙Q−1⁣∗\|​dL⁡ℳ​\(Q\)\\displaystyle=\\frac\{\|\\phi^\{\-1\}\_\{\*\}\|\}\{\|L\_\{\\phi^\{\-1\}\(Q\)\\odot Q^\{\-1\}\*\}\|\}\\operatorname\{d\}\_\{L\}\\mathcal\{M\}\(Q\)=Δ​dL⁡ℳ​\(Q\)\.\\displaystyle=\\Delta\\operatorname\{d\}\_\{L\}\\mathcal\{M\}\(Q\)\.
The probability ofQ=ϕ​\(P\)Q=\\phi\(P\)in a set𝒴⊂ℳ\\mathcal\{Y\}\\subset\\mathcal\{M\}is

F​\(ϕ​\(P\)∈𝒴\)\\displaystyle F\(\\phi\(P\)\\in\\mathcal\{Y\}\)=F​\(P∈ϕ−1​\(𝒴\)\)\\displaystyle=F\(P\\in\\phi^\{\-1\}\(\\mathcal\{Y\}\)\)\(69\)=∫ϕ−1​\(𝒴\)pX​\(P\)⋅dL​ℳ​\(P\)\\displaystyle=\\int\_\{\\phi^\{\-1\}\(\\mathcal\{Y\}\)\}p\_\{X\}\(P\)\\cdot d\_\{L\}\\mathcal\{M\}\(P\)=∫𝒴pX​\(ϕ\(−1\)​\(Q\)\)​dL​ℳ​\(ϕ\(−1\)​\(Q\)\)\\displaystyle=\\int\_\{\\mathcal\{Y\}\}p\_\{X\}\(\\phi^\{\(\-1\)\}\(Q\)\)d\_\{L\}\\mathcal\{M\}\(\\phi^\{\(\-1\)\}\(Q\)\)=∫𝒴Δ​pX​\(ϕ\(−1\)​\(Q\)\)​dL​ℳ​\(Q\)\.\\displaystyle=\\int\_\{\\mathcal\{Y\}\}\\Delta p\_\{X\}\(\\phi^\{\(\-1\)\}\(Q\)\)d\_\{L\}\\mathcal\{M\}\(Q\)\.Therefore, the density ofY=ϕ​\(X\)Y=\\phi\(X\)is

pY​\(Q\)=Δ​pX​\(ϕ\(−1\)​\(Q\)\)\.p\_\{Y\}\(Q\)=\\Delta p\_\{X\}\(\\phi^\{\(\-1\)\}\(Q\)\)\.\(70\)∎

The above lemma implies that whenΔ\\Deltais a constant,YYalso follows a Gaussian distribution\.

###### Corollary D\.2\.

Following the notations in[Lem\.D\.1](https://arxiv.org/html/2607.08783#A4.Thmtheorem1), ifΔ=c\\Delta=cis a constant andX∼𝒩​\(E,σ2\)X\\sim\\mathcal\{N\}\(E,\\sigma^\{2\}\), thenYYalso follows a Gaussian distribution,*i\.e\.*Y∼𝒩​\(E,s2​σ2\)Y\\sim\\mathcal\{N\}\(E,s^\{2\}\\sigma^\{2\}\)

###### Proof\.

pY​\(Q\)\\displaystyle p\_\{Y\}\(Q\)=c​k​\(σ\)​exp⁡\(−d\(ϕs−1\(Q\),E\)22​σ2\)\\displaystyle=ck\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(\\phi\_\{s\}^\{\-1\}\(Q\),E\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)\(71\)=k′​\(δ\)​exp⁡\(−d\(ExpE1/sLogE\(Q\),E\)22​σ2\)\\displaystyle=k^\{\\prime\}\(\\delta\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(\\operatorname\{Exp\}\_\{E\}\\nicefrac\{\{1\}\}\{\{s\}\}\\operatorname\{Log\}\_\{E\}\(Q\),E\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)=k′​\(δ\)​exp⁡\(−‖LogE⁡\(Q\)‖E22​s2​σ2\)\\displaystyle=k^\{\\prime\}\(\\delta\)\\exp\\left\(\-\\frac\{\\\|\\operatorname\{Log\}\_\{E\}\(Q\)\\\|\_\{E\}^\{2\}\}\{2s^\{2\}\\sigma^\{2\}\}\\right\)=k′​\(δ\)​exp⁡\(−d⁡\(Q,E\)2​s2​σ2\),\\displaystyle=k^\{\\prime\}\(\\delta\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(Q,E\)\}\{2s^\{2\}\\sigma^\{2\}\}\\right\),where∥⋅∥E\\\|\\cdot\\\|\_\{E\}is the norm of the tangent space at the neutral elementEE\. ∎

[Cor\.D\.2](https://arxiv.org/html/2607.08783#A4.Thmtheorem2)implies that whenΔ=c\\Delta=c,ϕs\\phi\_\{s\}can scale the population variance and further transform the Gaussian distribution\. Simple computations show that in the standard Euclidean spaceℝn\\mathbb\{R\}^\{n\},Δ=1/s\\Delta=\\nicefrac\{\{1\}\}\{\{s\}\}\. Therefore, it is natural to expect that the pullback ofℝn\\mathbb\{R\}^\{n\}also enjoys constantΔ\\Delta\.

###### Proposition D\.3\.

Consider annn\-dimensional Lie groupℳ\\mathcal\{M\}pulled back from the standard Euclidean spaceℝn\\mathbb\{R\}^\{n\}by the diffeomorphismψ:ℳ→ℝn\\psi:\\mathcal\{M\}\\rightarrow\\mathbb\{R\}^\{n\}\. In other words, the group operations and Riemannian metric onℳ\\mathcal\{M\}are defined byψ\\psifromℝn\\mathbb\{R\}^\{n\}\. ThenΔ\\Deltaremains constant onℳ\\mathcal\{M\}\.

###### Proof\.

To simplify notation, we denoteϕs\\phi\_\{s\}asϕ\\phi\. Under the given assumption, the group addition and Riemannian metric onℳ\\mathcal\{M\}are defined as follows:

∀P,Q∈ℳ,P⊙Q\\displaystyle\\forall P,Q\\in\\mathcal\{M\},P\\odot Q=ψ−1​\(ψ​\(P\)\+ψ​\(Q\)\)\\displaystyle=\\psi^\{\-1\}\(\\psi\(P\)\+\\psi\(Q\)\)\(72\)g\\displaystyle g=ψ∗​gE,\\displaystyle=\\psi^\{\*\}g^\{\\mathrm\{E\}\},wheregEg^\{\\mathrm\{E\}\}is the standard Euclidean metric\. Therefore,ϕ\\phican be simplified as

ϕ​\(P\)\\displaystyle\\phi\(P\)=ExpE⁡\[s​LogE⁡\(P\)\]\\displaystyle=\\operatorname\{Exp\}\_\{E\}\\left\[s\\operatorname\{Log\}\_\{E\}\(P\)\\right\]\(73\)=ψ−1​\(Exp~0​\[ψ∗,E​\(s​ψ∗,0−1​Log~0​ψ​\(P\)\)\]\)\\displaystyle=\\psi^\{\-1\}\\left\(\\tilde\{\\operatorname\{Exp\}\}\_\{0\}\\left\[\\psi\_\{\*,E\}\\left\(s\\psi^\{\-1\}\_\{\*,0\}\\tilde\{\\operatorname\{Log\}\}\_\{0\}\\psi\(P\)\\right\)\\right\]\\right\)=ψ−1​\(s​ψ​\(P\)\),\\displaystyle=\\psi^\{\-1\}\(s\\psi\(P\)\),whereExp~\\tilde\{\\operatorname\{Exp\}\}andLog~\\tilde\{\\operatorname\{Log\}\}are the Riemannian exponential and logarithmic maps inℝn\\mathbb\{R\}^\{n\}, which are reduced to vector addition and subtraction, respectively\. Therefore, the inverse ofϕ\\phiis

ϕ−1​\(P\)=ψ−1​\(1/s​ψ​\(P\)\)\.\\phi^\{\-1\}\(P\)=\\psi^\{\-1\}\\left\(\\nicefrac\{\{1\}\}\{\{s\}\}\\psi\(P\)\\right\)\.\(74\)Besides,Lϕ−1​\(Q\)⊙Q−1L\_\{\\phi^\{\-1\}\(Q\)\\odot Q^\{\-1\}\}can also be further simplified:

Lϕ−1​\(Q\)⊙Q−1​\(P\)=ψ−1​\(1/s​ψ​\(Q\)−ψ​\(Q\)\+ψ​\(P\)\)L\_\{\\phi^\{\-1\}\(Q\)\\odot Q^\{\-1\}\}\(P\)=\\psi^\{\-1\}\\left\(\\nicefrac\{\{1\}\}\{\{s\}\}\\psi\(Q\)\-\\psi\(Q\)\+\\psi\(P\)\\right\)\(75\)The differentials of[Eqs\.74](https://arxiv.org/html/2607.08783#A4.E74)and[75](https://arxiv.org/html/2607.08783#A4.E75)atQQare

ϕ∗,Q−1\\displaystyle\\phi^\{\-1\}\_\{\*,Q\}=1s​ψ∗,1/s​ψ​\(Q\)−1∘ψ∗,Q,\\displaystyle=\\frac\{1\}\{s\}\\psi^\{\-1\}\_\{\*,\\nicefrac\{\{1\}\}\{\{s\}\}\\psi\(Q\)\}\\circ\\psi\_\{\*,Q\},\(76\)Lϕ−1​\(Q\)⊙Q−1⁣∗,Q\\displaystyle L\_\{\\phi^\{\-1\}\(Q\)\\odot Q^\{\-1\}\*,Q\}=ψ∗,1/s​ψ​\(Q\)−1∘ψ∗,Q\.\\displaystyle=\\psi^\{\-1\}\_\{\*,\\nicefrac\{\{1\}\}\{\{s\}\}\\psi\(Q\)\}\\circ\\psi\_\{\*,Q\}\.\(77\)Therefore,Δ=1/s\\Delta=\\nicefrac\{\{1\}\}\{\{s\}\}for allQ∈ℳQ\\in\\mathcal\{M\}\. ∎

By[Prop\.D\.3](https://arxiv.org/html/2607.08783#A4.Thmtheorem3), we can directly obtain the following corollary\.

###### Corollary D\.4\.

Given a Lie groupℳ\\mathcal\{M\}pulled back from the Euclidean space, and a random pointX∼𝒩​\(E,σ2\)X\\sim\\mathcal\{N\}\(E,\\sigma^\{2\}\)overℳ\\mathcal\{M\},Y=ϕs​\(X\)∼𝒩​\(E,s2​σ2\)Y=\\phi\_\{s\}\(X\)\\sim\\mathcal\{N\}\(E,s^\{2\}\\sigma^\{2\}\)

In machine learning, several Lie groups are derived by the pullback from the standard Euclidean space\. As shown by Chen*et al*\[[20](https://arxiv.org/html/2607.08783#bib.bib6)\],\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}andθ​\-LCM\\theta\\text\{\-LCM\}are pullback metrics from the Euclidean metric\. Therefore, for the Lie groups of SPD manifolds w\.r\.t\.\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}andθ​\-LCM\\theta\\text\{\-LCM\},[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)can transform the Gaussian distribution\. Specifically, given a random pointX∼𝒩​\(M,σ2\)X\\sim\\mathcal\{N\}\(M,\\sigma^\{2\}\),[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11),[12](https://arxiv.org/html/2607.08783#S4.E12)and[13](https://arxiv.org/html/2607.08783#S4.E13)transform the Gaussian distribution as:

𝒩​\(M,σ2\)→𝒩​\(E,σ2\)→𝒩​\(E,s2\)→𝒩​\(B,s2\),\\mathcal\{N\}\(M,\\sigma^\{2\}\)\\rightarrow\\mathcal\{N\}\(E,\\sigma^\{2\}\)\\rightarrow\\mathcal\{N\}\(E,s^\{2\}\)\\rightarrow\\mathcal\{N\}\(B,s^\{2\}\),\(78\)whereMMandσ\\sigmaare employed to normalizeXX, andϵ\\epsilonin[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)is omitted\. The above process exactly mirrors the transformation of Gaussian distributions within the framework of standard BN\[[46](https://arxiv.org/html/2607.08783#bib.bib37)\]\.

## Appendix EDomain\-specific Momentum LieBN for EEG Classification

Input :A batch of activations

\{P1​…​N\}\\\{P\_\{1\\ldots N\}\\\}over the Lie group

\{ℳ,⊙,g\}\\\{\\mathcal\{M\},\\odot,g\\\}, and a small positive constant

ϵ\\epsilon
running mean

M¯r=E\\bar\{M\}\_\{r\}=E, running variance

v¯r2=1\\bar\{v\}^\{2\}\_\{r\}=1for training

running mean

M~r=E\\tilde\{M\}\_\{r\}=E, running variance

v~r2=1\\tilde\{v\}^\{2\}\_\{r\}=1for testing

biasing parameter

B∈ℳB\\in\\mathcal\{M\}, scaling parameter

s∈ℝ/\{0\}s\\in\\mathbb\{R\}/\\\{0\\\},

momentum for training and testing

γt​r​a​i​n,γ∈\[0,1\]\\gamma\_\{train\},\\gamma\\in\[0,1\]
Output :Normalized activations

\{P~1​…​N\}\\\{\\tilde\{P\}\_\{1\\ldots N\}\\\}
1exif*training*then

Compute batch mean

MbM\_\{b\}and variance

vb2v\_\{b\}^\{2\}of

\{P1​…​N\}\\\{P\_\{1\\ldots N\}\\\};

M¯r←WFM⁡\(\{1−γt​r​a​i​n,γt​r​a​i​n\},\{M¯r,Mb\}\)\\bar\{M\}\_\{r\}\\leftarrow\\operatorname\{WFM\}\(\\\{1\-\\gamma\_\{train\},\\gamma\_\{train\}\\\},\\\{\\bar\{M\}\_\{r\},M\_\{b\}\\\}\);

v¯r2←\(1−γt​r​a​i​n\)​v¯r2\+γt​r​a​i​n​vb2\\bar\{v\}^\{2\}\_\{r\}\\leftarrow\(1\-\\gamma\_\{train\}\)\\bar\{v\}^\{2\}\_\{r\}\+\\gamma\_\{train\}v^\{2\}\_\{b\};

M~r←WFM⁡\(\{1−γ,γ\},\{M~r,Mb\}\)\\tilde\{M\}\_\{r\}\\leftarrow\\operatorname\{WFM\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{\\tilde\{M\}\_\{r\},M\_\{b\}\\\}\);

v~r2←\(1−γ\)​v~r2\+γ​vb2\\tilde\{v\}^\{2\}\_\{r\}\\leftarrow\(1\-\\gamma\)\\tilde\{v\}^\{2\}\_\{r\}\+\\gamma v^\{2\}\_\{b\};

end if

if*training*then

M←M¯r,v2←v¯r2M\\leftarrow\\bar\{M\}\_\{r\},v^\{2\}\\leftarrow\\bar\{v\}^\{2\}\_\{r\};

else

M←M~r,v2←v~r2M\\leftarrow\\tilde\{M\}\_\{r\},v^\{2\}\\leftarrow\\tilde\{v\}^\{2\}\_\{r\};

for*i←1i\\leftarrow 1toNN*do

Centering to the neutral element

EE:

if*ggis left\-invariant*then

P¯i←LM⊙−1⁡\(Pi\)\\bar\{P\}\_\{i\}\\leftarrow\\operatorname\{L\}\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\);

else

P¯i←RM⊙−1⁡\(Pi\)\\bar\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{M\_\{\\odot\}^\{\-1\}\}\(P\_\{i\}\);

Scaling the dispersion:

P^i←ExpE⁡\[sv2\+ϵ​LogE⁡\(P¯i\)\]\\hat\{P\}\_\{i\}\\leftarrow\\operatorname\{Exp\}\_\{E\}\\left\[\\frac\{s\}\{\\sqrt\{v^\{2\}\+\\epsilon\}\}\\operatorname\{Log\}\_\{E\}\(\\bar\{P\}\_\{i\}\)\\right\]
Biasing towards parameter

BB:

if*ggis left\-invariant*then

P~i←LB⁡\(P^i\)\\tilde\{P\}\_\{i\}\\leftarrow\\operatorname\{L\}\_\{B\}\(\\hat\{P\}\_\{i\}\);

else

P~i←RB⁡\(P^i\)\\tilde\{P\}\_\{i\}\\leftarrow\\operatorname\{R\}\_\{B\}\(\\hat\{P\}\_\{i\}\);

end for

Algorithm 2Momentum LieBN \(MLieBN\) AlgorithmKobler*et al*\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]proposed SPD domain\-specific momentum batch normalization \(SPDDSMBN\) as a domain adaptation approach for EEG classification\. SPDDSMBN, based on[Eq\.9](https://arxiv.org/html/2607.08783#S3.E9), performed normalization of mean and variance on SPD manifolds under the specific AIM\. Additionally, SPDDSMBN utilized separate momentums for updating training and testing running statistics, inspired by the work of\[[90](https://arxiv.org/html/2607.08783#bib.bib53)\]\. Following\[[51](https://arxiv.org/html/2607.08783#bib.bib61), Alg\. 1\], we also present a momentum LieBN \(MLieBN\) in[Algorithm2](https://arxiv.org/html/2607.08783#algorithm2)\. Hereγ\\gammais fixed andγt​r​a​i​n\\gamma\_\{train\}is defined as

γt​r​a​i​n=1−ρ1K−1​max⁡\(K−k,0\)\+ρ,where​ρ=1d​o​m​a​i​n​s​\_​p​e​r​\_​b​a​t​c​h\\gamma\_\{train\}=1\-\\rho^\{\\frac\{1\}\{K\-1\}\\max\(K\-k,0\)\}\+\\rho,\\ \\text\{where\}\\ \\rho=\\frac\{1\}\{domains\\\_per\\\_batch\}\(79\)
Furthermore, in line with\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we adopt multi\-channel mechanisms for domain\-specific MLieBN \(DSMLieBN\), where each domain has its own MLieBN layer\. Similar to\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we set the biasing parameter equal to the neutral element, and the scaling factor is shared across all domains\. We denote[Algorithm2](https://arxiv.org/html/2607.08783#algorithm2)asMLieBN⁡\(Pj\|M,s,ϵ,γ,γt​r​a​i​n\)\\operatorname\{MLieBN\}\(P\_\{j\}\|M,s,\\epsilon,\\gamma,\\gamma\_\{train\}\)\. Then our DSMLieBN follows

DSMLieBN⁡\(Pj,i\)=MLieBNi⁡\(Pj\|E,s,ϵ,γ,γt​r​a​i​n\),∀Pj∈\{P1​…​N\},\\operatorname\{DSMLieBN\}\(P\_\{j\},i\)=\\operatorname\{MLieBN\}\_\{i\}\(P\_\{j\}\|E,s,\\epsilon,\\gamma,\\gamma\_\{train\}\),\\forall P\_\{j\}\\in\\\{P\_\{1\\ldots N\}\\\},\(80\)whereiiis the index of the domain\. We follow the official code of SPDDSMBN777https://github\.com/rkobler/TSMNetto implement our DSMLieBN\. In a word, the only difference between DSMLieBN and SPDDSMBN is the different way of normalization\.

Analogous to[Thm\.5\.6](https://arxiv.org/html/2607.08783#S5.Thmtheorem6), computations for DSMLieBN under pullback metrics can also be performed by mapping, calculating, and then remapping\.

## Appendix FBackpropagation of Matrix Functions

Our implementation of LieBN on SPD and correlation manifolds involves several matrix functions\. Thus, we employ matrix backpropagation \(BP\)\[[47](https://arxiv.org/html/2607.08783#bib.bib34)\]for gradient computation\. These matrix operations can be divided into Cholesky decomposition and the functions based on Eigendecomposition\.

The differentiation of the Cholesky decomposition can be found in\[[61](https://arxiv.org/html/2607.08783#bib.bib132), Eq\. 8\]or\[[56](https://arxiv.org/html/2607.08783#bib.bib126), Props\. 4\]\. Besides, our homemade BP of the Cholesky decomposition yields a similar gradient to the one generated by autograd oftorch\.linalg\.cholesky\. Therefore, during the experiments, we usetorch\.linalg\.cholesky\.

The second type of matrix functions is based on Eigendecomposition, such as matrix exponential, logarithm, and power\. Although torch\[[63](https://arxiv.org/html/2607.08783#bib.bib95)\]supports autograd of Eigendecomposition, it requires the computation of1δi−δj\\frac\{1\}\{\\delta\_\{i\}\-\\delta\_\{j\}\}\[[47](https://arxiv.org/html/2607.08783#bib.bib34), Props\. 1\], whereδi\\delta\_\{i\}andδj\\delta\_\{j\}denote eigenvalues\. This might trigger numerical instability whenδi\\delta\_\{i\}approximatesδj\\delta\_\{j\}\. Following\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\], we use the Daleckĭi\-Kreĭn formula\[[9](https://arxiv.org/html/2607.08783#bib.bib154), Thm\. V\.3\.3\]to calculate the BP of Eigen\-based matrix functions\. In detail, for a matrix function defined asX=f​\(S\)=U​f​\(Σ\)​U⊤X=f\(S\)=Uf\(\\Sigma\)U^\{\\top\}, withS=U​Σ​U⊤S=U\\Sigma U^\{\\top\}as the eigendecomposition of an SPD matrix, its BP is expressed as

∇SL\\displaystyle\\nabla\_\{S\}L=U​\[K⊙\(UT​\(∇XL\)​U\)\]​UT\.\\displaystyle=U\[K\\odot\(U^\{T\}\(\\nabla\_\{X\}L\)U\)\]U^\{T\}\.\(81\)where∇XL\\nabla\_\{X\}Lis the Euclidean gradient of the loss functionLLw\.r\.t\.XX\. MatrixKKis defined as

Ki​j=\{f​\(σi\)−f​\(σj\)σi−σjif​σi≠σjf′​\(σi\)otherwiseK\_\{ij\}=\\begin\{cases\}\\frac\{f\\left\(\\sigma\_\{i\}\\right\)\-f\\left\(\\sigma\_\{j\}\\right\)\}\{\\sigma\_\{i\}\-\\sigma\_\{j\}\}&\\text\{ if \}\\sigma\_\{i\}\\neq\\sigma\_\{j\}\\\\ f^\{\\prime\}\\left\(\\sigma\_\{i\}\\right\)&\\text\{ otherwise \}\\end\{cases\}\(82\)whereΣ=diag\(σ1,σ2,⋯,σd\\Sigma=\\operatorname\{diag\}\(\\sigma\_\{1\},\\sigma\_\{2\},\\cdots,\\sigma\_\{d\}\)\.[Eq\.82](https://arxiv.org/html/2607.08783#A6.E82)demonstrates the numerical stability of Daleckĭi\-Kreĭn formula\.

## Appendix GExperimental Details and Additional Discussions

### G\.1Experimental Details and Additional Discussion on the SPD Manifold

#### G\.1\.1Datasets and Preprocessing

TheRadardataset\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\]contains 3,000 synthetic radar signals\. Following the protocol in\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\], each signal is split into windows of length 20, resulting in 3,000 covariance matrices of20×2020\\times 20equally distributed in 3 classes\.

TheHDM05dataset\[[60](https://arxiv.org/html/2607.08783#bib.bib139)\]consists of 2,273 skeleton\-based motion capture sequences executed by different actors\. Each frame consists of 3D coordinates of 31 joints, allowing the representation of each sequence as a93×9393\\times 93covariance matrix\. In line with\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\], we trim the dataset to 2086 instances scattered throughout 117 classes by removing some under\-represented clips\.

TheFPHA\[[35](https://arxiv.org/html/2607.08783#bib.bib32)\]includes 1,175 skeleton\-based first\-person hand gesture videos of 45 different categories with 600 clips for training and 575 for testing\. Following\[[85](https://arxiv.org/html/2607.08783#bib.bib83)\], we represent each sequence as a63×6363\\times 63covariance matrix\.

TheHinss2021dataset\[[41](https://arxiv.org/html/2607.08783#bib.bib157)\]is a recently released competition dataset containing EEG signals for mental workload estimation\. The dataset is employed for two tasks, inter\-session and inter\-subject, which are treated as domain adaptation problems\. Geometry\-aware methods\[[88](https://arxiv.org/html/2607.08783#bib.bib86),[51](https://arxiv.org/html/2607.08783#bib.bib61)\]have demonstrated promising performance in EEG classification\. We follow\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]for data preprocessing\. In detail, the python package MOABB\[[48](https://arxiv.org/html/2607.08783#bib.bib113)\]and MNE\[[36](https://arxiv.org/html/2607.08783#bib.bib112)\]are used to preprocess the datasets\. The applied steps include resampling the EEG signals to 250/256 Hz, applying temporal filters to extract oscillatory EEG activity in the 4 to 36 Hz range, extracting short segments \(≤3\\leq 3s\) associated with a class label, and finally obtaining40×4040\\times 40SPD covariance matrices\.

#### G\.1\.2Implementation details

We use the official code of SPDNetBN888[https://proceedings\.neurips\.cc/paper\_files/paper/2019/file/6e69ebbfad976d4637bb4b39de261bf7\-Supplemental\.zip](https://proceedings.neurips.cc/paper_files/paper/2019/file/6e69ebbfad976d4637bb4b39de261bf7-Supplemental.zip)\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\]and TSMNet999[https://github\.com/rkobler/TSMNet](https://github.com/rkobler/TSMNet)\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\]to implement our experiments on the SPDNet and TSMNet backbones\. For the SPDNet architecture, we compare our LieBN with SPDNetBN\[[13](https://arxiv.org/html/2607.08783#bib.bib96)\], which applies the SPDBN \([Eqs\.6](https://arxiv.org/html/2607.08783#S3.E6)and[7](https://arxiv.org/html/2607.08783#S3.E7)\) to SPDNet\. Similar to SPDNetBN, we apply our LieBN after each transformation layer \(BiMap layer in[App\.C](https://arxiv.org/html/2607.08783#A3)\)\. In the EEG application, one of the state\-of\-the\-art methods is TSMNet with SPD domain\-specific momentum batch normalization \(TSMNet\+SPDDSMBN\)\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], which is a domain adaptation version of\[[52](https://arxiv.org/html/2607.08783#bib.bib92)\]\. For a fair comparison, we also implement a domain\-specific momentum LieBN, referred to as DSMLieBN \(detailed in[App\.E](https://arxiv.org/html/2607.08783#A5)\)\. Following\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we apply our DSMLieBN before the LogEig layer \(detailed in[App\.C](https://arxiv.org/html/2607.08783#A3)\) in TSMNet\. We use the standard cross\-entropy loss and optimize the parameters with the Riemannian AMSGrad optimizer\[[6](https://arxiv.org/html/2607.08783#bib.bib42)\]\. The network architectures are represented as\{d0,d1,…,dL\}\\\{d\_\{0\},d\_\{1\},\\ldots,d\_\{L\}\\\}, where the dimension of the parameter in theii\-th BiMap layer isdi×di−1d\_\{i\}\\times d\_\{i\-1\}\. The experiments are conducted with a learning rate of5​e−35e^\{\-3\}, batch size of 30, and training epoch of 200 on the Radar, HDM05, and FPHA datasets\. For the Hinss2021 dataset, following\[[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we use a learning rate of1​e−31e^\{\-3\}with a weight decay of1​e−41e^\{\-4\}, a batch size of 50, and a training epoch of 50\.

In line with the previous work\[[13](https://arxiv.org/html/2607.08783#bib.bib96),[51](https://arxiv.org/html/2607.08783#bib.bib61)\], we use accuracy as the scoring metric for the Radar, HDM05, and FPHA datasets, and balanced accuracy \(*i\.e\.*, the average recall across classes\) for the Hinss2021 dataset\. Ten\-fold experiments on the Radar, HDM05, and FPHA datasets are carried out with randomized initialization and split \(split is officially fixed for the FPHA dataset\), while on the Hinss2021 dataset, models are fit and evaluated with a randomized leave 5% of the sessions \(inter\-session\) or subjects \(inter\-subject\) out cross\-validation scheme\.

#### G\.1\.3Candidate Values of Hyper\-parameters

We implement the SPD LieBN and DSMLieBN induced by four standard invariant metrics, namely AIM, LEM, LCM, and CRIM, along with their deformed metrics\. Therefore, our method has a maximum of three hyper\-parameters,*i\.e\.*,\(θ,α,β\)\(\\theta,\\alpha,\\beta\)\. As\(α,β\)\(\\alpha,\\beta\)only affects variance calculation in the LieBN framework, we set\(α,β\)=\(1,0\)\(\\alpha,\\beta\)=\(1,0\)and only tune the deformation factorθ\\thetafrom the candidate values of±0\.5\\pm 0\.5,±1\\pm 1, and±1\.5\\pm 1\.5\. We denote \[Baseline\]\+\[BN\_Type\]\+\[Metric\]\-\[θ\\theta\] as the baseline endowed with a specific LieBN, such as SPDNet\+LieBN\-AIM\-\(1\) and TSMNet\+DSMLieBN\-LCM\-\(1\)\.

#### G\.1\.4Empirical Insights on the Hyper\-parameters

Our SPD LieBN has at most three types of hyper\-parameters: Riemannian metric, deformation factorθ\\theta, andO​\(n\)\\mathrm\{O\}\(\{n\}\)\-invariance parameters\(α,β\)\(\\alpha,\\beta\)\. The general order of importance should be Riemannian metric\>\>θ\\theta\>\>\(α,β\)\(\\alpha,\\beta\)\.

The most significant parameter is the choice of Riemannian metric, as all the geometric properties are sourced from a metric\. A safe choice would start with AIM, and then decide whether to explore other metrics further\. The most important reason is the property of affine invariance of AIM, which is a natural characteristic of covariance matrices\. In our experiments, the LieBN\-AIM generally achieves the best performance\. However, AIM is not always the best metric\. As shown in[Tab\.VII](https://arxiv.org/html/2607.08783#S6.T7), the best result on the HDM05 dataset is achieved by LCM\-based LieBN, which improves the vanilla SPDNet by 11\.71%\. Therefore, when choosing Riemannian metrics on SPD manifolds, a safe choice would start with AIM and extend to other metrics\. Besides, if efficiency is an important factor, one should first consider LCM, as it is the most efficient one\.

The second one is the deformation factorθ\\theta\. As we discussed in[Sec\.5\.1\.1](https://arxiv.org/html/2607.08783#S5.SS1.SSS1),θ\\thetainterpolates between different types of metrics \(θ=1\\theta=1andθ→0\\theta\\rightarrow 0\)\. Inspired by this, we selectθ\\thetaaround its deformation boundaries \(1 and 0\)\. In this paper we roughly selectθ\\thetafrom\{±0\.5,±1,±1\.5\}\\\{\\pm 0\.5,\\pm 1,\\pm 1\.5\\\}

The less important parameters are\(α,β\)\(\\alpha,\\beta\)\. Recalling[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)\. and[Tab\.XIII](https://arxiv.org/html/2607.08783#A2.T13),\(α,β\)\(\\alpha,\\beta\)only affects the calculation of variance, which should have less effects compared with the above two parameters\. Therefore, we simply set\(α,β\)=\(1,0\)\(\\alpha,\\beta\)=\(1,0\)during experiments\.

#### G\.1\.5The Effect ofβ\\beta

Recalling[Eq\.3](https://arxiv.org/html/2607.08783#S2.E3),β\\betacontrols the relative importance of the trace part against the inner product\. Therefore, we set the candidate values ofβ\\betaas\{1,1/n,1/n2,0,−1/n\+ϵ,−1/n2\}\\\{1,\\nicefrac\{\{1\}\}\{\{n\}\},\\nicefrac\{\{1\}\}\{\{n^\{2\}\}\},0,\-\\nicefrac\{\{1\}\}\{\{n\}\}\+\\epsilon,\-\\nicefrac\{\{1\}\}\{\{n^\{2\}\}\}\\\}, wherennis the input dimension of LieBN, andϵ\\epsilonis a small positive scalar to ensureO​\(n\)\\mathrm\{O\}\(\{n\}\)\-invariance,*i\.e\.*\(α,β\)∈𝐒𝐓\(\\alpha,\\beta\)\\in\\mathbf\{ST\}\.1/n2\\nicefrac\{\{1\}\}\{\{n^\{2\}\}\}and1/n\\nicefrac\{\{1\}\}\{\{n\}\}means averaging the trace in[Eq\.3](https://arxiv.org/html/2607.08783#S2.E3), while the sign ofβ\\betadenotes suppressing \(\-\), enhancing \(\+\), or neutralizing \(0\) the trace\.

We focus on AIM\-based LieBN on the HDM05 dataset\. We setθ=1\.5\\theta=1\.5, as it is the best deformation factor under this scenario\. Other network settings remain the same as the main paper\. The 10\-fold average results are presented in[Tab\.XVIII](https://arxiv.org/html/2607.08783#A7.T18)\. Note that on this setting,n=30n=30\. As expected,β\\betahas minor effects on our LieBN\.

TABLE XVIII:The effect of differentβ\\betafor AIM\-based LieBN on the HDM05 dataset\.Beta−1/302\\nicefrac\{\{\-1\}\}\{\{30^\{2\}\}\}\-0\.031/302\\nicefrac\{\{1\}\}\{\{30^\{2\}\}\}1/30\\nicefrac\{\{1\}\}\{\{30\}\}10Mean±STD68\.18±0\.8668\.12±0\.7468\.20±0\.8568\.18±0\.8568\.16±0\.8068\.16±0\.68

### G\.2Implementation Details on the Rotation Matrix

#### G\.2\.1Datasets and Preprocessing

TheG3Ddataset\[[10](https://arxiv.org/html/2607.08783#bib.bib43)\]consists of 663 sequences of 20 different gaming actions\. Each sequence is recorded by 3D locations of 20 joints \(*i\.e\.*, 19 bones\)\. Following\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\], we use the cross\-subject setting, where half of the subjects are used for training, and the other half for testing, respectively\.

TheHDM05dataset\[[60](https://arxiv.org/html/2607.08783#bib.bib139)\]has been discussed in[Sec\.G\.1\.1](https://arxiv.org/html/2607.08783#A7.SS1.SSS1)\.

TheNTU60\[[68](https://arxiv.org/html/2607.08783#bib.bib59)\]\.It has 56,880 sequences of 3D skeleton data classified into 60 classes, where each frame contains the 3D coordinates of 25 or 50 body joints\. We focus on the cross\-view protocol setting\[[68](https://arxiv.org/html/2607.08783#bib.bib59)\]\.

Following\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\], we use the code101010[https://ravitejav\.weebly\.com/kbac\.html](https://ravitejav.weebly.com/kbac.html)of\[[79](https://arxiv.org/html/2607.08783#bib.bib27)\]to represent each skeleton sequence as a point on the Lie groupSON×T​\(3\)\\mathrm\{SO\}^\{N\\times T\}\(3\), whereNNandTTdenote spatial and temporal dimensions\. As preprocessed in\[[44](https://arxiv.org/html/2607.08783#bib.bib28)\], we setTTas 100, 64, and 16 on these three datasets, respectively\.

#### G\.2\.2LieNet

The LieNet consists of three basic layers: RotMap, RotPooling, and LogMap layers\. The RotMap mimics the convolutional layer, while the RotPooling extends the pooling layers to rotation matrices\. The LogMap layer maps the rotation matrix into the tangent space at the identity for classification\. Note that the official code of LieNet111111[https://github\.com/zhiwu\-huang/LieNet](https://github.com/zhiwu-huang/LieNet)is developed by Matlab\. We use the open\-sourced Pytorch code121212[https://github\.com/hjf1997/LieNet](https://github.com/hjf1997/LieNet)to implement our experiments\. To reproduce LieNet more faithfully, we made the following modifications to this Pytorch code\. We re\-code the LogMap and RotPooling layers to make them consistent with the official Matlab implementation\. In addition, we also extend the Riemannian computation of geoopt\[[6](https://arxiv.org/html/2607.08783#bib.bib42)\]intoSO​\(3\)\\mathrm\{SO\}\(3\)to allow for Riemannian optimizer onSO​\(3\)\\mathrm\{SO\}\(3\), which is missing in the current package\. We apply our LieBN before the LogMap layer\. Note that the dimension of input features in LieNet isB×N×T×3×3B\\times N\\times T\\times 3\\times 3\. We calculate Lie group statistics along the batch and temporal dimensions \(B×TB\\times T\)\. We denote the LieNet models with our LieBN\-Left and LieBN\-Right as LieNetLieBN\-Left and LieNetLieBN\-Right, respectively\.

#### G\.2\.3Training Details

We find that SGD is the most effective optimizer for LieNet, and thus, we adopt it for our experiments\. The learning rate is set to1​e−21e^\{\-2\}\. The batch sizes are 30, 30, and 256 for the G3D, HDM05, and NTU60 datasets, respectively\. On the NTU60 dataset, the learning rate is reduced by a factor of 10 upon model convergence, specifically at the 5th and 25th epochs for LieNetLieBN and LieNet, respectively\. For each model, we applytorch\.nn\.utils\.clip\_grad\_norm\_withmax\_norm=5to the transformation matrix in the final FC layer\.

### G\.3Implementation Details on the Correlation Matrix

We follow the same settings as the experiments on the SPD manifold with respect to the backbone architecture, batch size, training epoch, optimizer, and learning rate\. The network architecture can be denoted as BiMap\-\[Power\-Cov2Cor\-LieBN\-Cor\]\-LogEig, where Power denotes the matrix power and Cov2Cor isCor⁡\(⋅\):𝒮\+\+n→𝒞\+\+n\\operatorname\{Cor\}\(\\cdot\):\\mathcal\{S\}^\{n\}\_\{\+\+\}\\to\\mathcal\{C\}^\{n\}\_\{\+\+\}\. The matrix powers used for each dataset are presented in[Tab\.XIX](https://arxiv.org/html/2607.08783#A7.T19)\. A single iteration is sufficient to achieve saturated network performance with respect to calculating𝒟\+\\mathcal\{D\}^\{\+\}and𝒟⋆\\mathcal\{D\}^\{\\star\}in OLM and LSM, except for𝒟⋆\\mathcal\{D\}^\{\\star\}on the HDM05 dataset, which requires convergence with a maximum of 20 iterations\.

TABLE XIX:Matrix powers in LieBN\-Cor under different metrics on each dataset\.DatasetMetricECMLECMOLMLSMHDM050\.750\.50\.5\-0\.5FPHA\-0\.5\-0\.25\-0\.25\-0\.25
### G\.4Hardware

All experiments use an Intel Core i9\-7960X CPU with 32GB RAM and an NVIDIA GeForce RTX 2080 Ti GPU\.

### G\.5Lie Algebra Normalization vs\. Lie Group Normalization

This subsection conducts ablations on comparing BN via Lie algebra against our LieBN\.

#### G\.5\.1Formulation

An alternative approach to defining the Gaussian distribution is through the Euclidean Gaussian on the tangent space, such as the Lie algebra\[[86](https://arxiv.org/html/2607.08783#bib.bib109),[31](https://arxiv.org/html/2607.08783#bib.bib101)\]\. In this formulation, the associated mean and variance are reduced to their Euclidean counterparts in the Lie algebra\. Based on this, we design the Lie Algebra Batch Normalization \(LieAlgebraBN\) as:

- •Mapping data into the Lie algebra by the Riemannian logarithm at the neural elementLogE\\operatorname\{Log\}\_\{E\};
- •Applying Euclidean BN over the Lie algebra;
- •Mapping back to the Lie group by the Riemannian exponentiation at the neural elementExpE\\operatorname\{Exp\}\_\{E\}\.

Disadvantages of LieAlgebraBN:However, we argue that LieAlgebraBN might distort the geometry as they solely rely on the tangent space\. Nevertheless, we compare LieAlgebraBN and LieBN on the SPD and rotation matrices in the following for completeness\.

#### G\.5\.2Experiments on the SPD Lie Groups

Following[Sec\.G\.5\.1](https://arxiv.org/html/2607.08783#A7.SS5.SSS1), we implement LieAlgebraBN on the SPD manifolds under LEM, AIM, and LCM, respectively\. The neural elements on the SPD Lie groups w\.r\.t LEM, LCM, and AIM are the identity matrixII\. Besides, the logarithmic \(exponential\) maps atIIunder AIM and LEM are identical\. Therefore, the LieAlgebraBN under AIM and LEM are identical\.

We mainly focus on the standard AIM, LEM, and LCM, where no power deformation is applied\. However, we observe that LieAlgebraBN under LCM performs significantly worse on FPHA\. To address this, we apply a matrix power of 0\.5, as[Tab\.VII\(c\)](https://arxiv.org/html/2607.08783#S6.T7.st3), to activate the LCM geometry\. Following the main paper, we conduct experiments on the SPDNet backbone while keeping all other settings consistent\.

[Tab\.XX](https://arxiv.org/html/2607.08783#A7.T20)presents the 10\-fold average comparison on the FPHA dataset\. We can observe that our LieBN is consistently better than LieAlgebraBN across different geometries\. Notably, LCM\-based LieAlgebraBN even exhibits a negative impact on the FPHA dataset, which may be attributed to the distorted geometry of the tangent space\. These findings validate the superiority of our intrinsic LieBN over the LieAlgebraBN\.

TABLE XX:Comparison of LieAlgebraBN against LieBN under different SPD invariant metrics\.DatasetBN TypeNoneLieAlgebraBNLieBNAIM/LEMLCMAIMLEMLCMFPHA85\.59±0\.7285\.63±0\.7777\.37±2\.2889\.70±0\.5186\.56±0\.7986\.33±0\.43
#### G\.5\.3Experiments on the Rotation Lie Groups

![Refer to caption](https://arxiv.org/html/2607.08783v1/x4.png)Figure 5:Comparisons of LieBN and LieAlgebraBN on the LieNet backbone\.We compare LieAlgebraBN and LieBN under the LieNet backbone on the relatively large NTU60 dataset\. Following the main paper, we adopt two\-block and three\-block architectures\. We observe that LieAlgebraBN benefits from a decreasing learning rate upon convergence\. Therefore, we apply the same learning rate schedule as in LieBN while keeping all other settings identical\.[Fig\.5](https://arxiv.org/html/2607.08783#A7.F5)presents the testing accuracy curve, demonstrating that LieBN outperforms LieAlgebraBN under both architectures\. This may be attributed to the fact that the Lie algebra, as a local tangent space approximation, distorts the intrinsic geometry of the Lie group, leading to suboptimal normalization\.

## Appendix HProofs

### H\.1Proof of[Prop\.2\.5](https://arxiv.org/html/2607.08783#S2.Thmtheorem5)

###### Proof\.

Invariance:First, we note that all four Lie groups are commutative\. Secondly, the construction of all four metrics is similar\. It suffices to show the left\-invariance of ECM\.

We only need to show thatLQ:𝒞\+\+n→𝒞\+\+nL\_\{Q\}:\\mathcal\{C\}^\{n\}\_\{\+\+\}\\to\\mathcal\{C\}^\{n\}\_\{\+\+\}for anyQ∈𝒞\+\+nQ\\in\\mathcal\{C\}^\{n\}\_\{\+\+\}is a Riemannian isometry\.LQL\_\{Q\}can be rewritten as

LQ=Θ−1∘L~Θ​\(Q\)∘Θ,\\displaystyle L\_\{Q\}=\\Theta^\{\-1\}\\circ\\widetilde\{L\}\_\{\\Theta\(Q\)\}\\circ\\Theta,\(83\)whereL~Θ​\(Q\)\\widetilde\{L\}\_\{\\Theta\(Q\)\}is the left translation overLT1n\\mathrm\{LT\}\_\{1\}^\{n\}, which is an isometry overLT1n\\mathrm\{LT\}\_\{1\}^\{n\}\. AsΘ−1\\Theta^\{\-1\},L~Θ​\(Q\)\\widetilde\{L\}\_\{\\Theta\(Q\)\}, andΘ\\Thetaare all isometries, their composition is an isometry as well\.

WFM:The WFM in the Euclidean space is reduced to arithmetic weighted average\. By the isometry offf, one can directly obtain the results\. ∎

### H\.2Proof of[Prop\.4\.1](https://arxiv.org/html/2607.08783#S4.Thmtheorem1)

###### Proof\.

Property[1](https://arxiv.org/html/2607.08783#S4.I1.i1):The MLE ofMMis

MMLE\\displaystyle M\_\{\\mathrm\{MLE\}\}=argmax⁡log⁡\(K​\(v\)\)−∑i=1Nd\(Pi,M\)22​v2\\displaystyle=\\operatorname\{argmax\}\\log\(K\(v\)\)\-\\sum\_\{i=1\}^\{N\}\\frac\{\\operatorname\{d\}\(P\_\{i\},M\)^\{2\}\}\{2v^\{2\}\}\(84\)=argmin∑i=1Nd\(Pi,M\)2\.\\displaystyle=\\operatorname\{argmin\}\\sum\_\{i=1\}^\{N\}\\operatorname\{d\}\(P\_\{i\},M\)^\{2\}\.
Property[2](https://arxiv.org/html/2607.08783#S4.I1.i2):We denoteY=LB⁡\(X\)Y=\\operatorname\{L\}\_\{B\}\(X\), andpXp\_\{X\}andpYp\_\{Y\}as the density ofXXandYY, respectively\. The density ofYYis

pY​\(Q\)\\displaystyle p\_\{Y\}\(Q\)=\(1\)pX​\(LB⊙−1⁡\(Q\)\)\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}p\_\{X\}\(\\operatorname\{L\}\_\{B^\{\-1\}\_\{\\odot\}\}\(Q\)\)\(85\)=k​\(σ\)​exp⁡\(−d\(LB⊙−1\(Q\),M\)22​σ2\)\\displaystyle=k\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(\\operatorname\{L\}\_\{B^\{\-1\}\_\{\\odot\}\}\(Q\),M\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)=\(2\)k​\(σ\)​exp⁡\(−d\(Q,LB\(M\)\)22​σ2\)\.\\displaystyle\\stackrel\{\{\\scriptstyle\(2\)\}\}\{\{=\}\}k\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(Q,\\operatorname\{L\}\_\{B\}\(M\)\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)\.The above comes from:

1. \(1\)Thm\. 7 in\[[66](https://arxiv.org/html/2607.08783#bib.bib158)\];
2. \(2\)The isometry of the left translation\.

∎

### H\.3Proof of[Prop\.4\.2](https://arxiv.org/html/2607.08783#S4.Thmtheorem2)

###### Proof\.

The isometry ofLB\\operatorname\{L\}\_\{B\}can directly obtain the homogeneity of the sample mean\. Now let us focus on[Eq\.15](https://arxiv.org/html/2607.08783#S4.E15)\. We have the following:

∑i=1Nwi​d2⁡\(ϕs​\(Pi\),E\)\\displaystyle\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\operatorname\{d\}^\{2\}\(\\phi\_\{s\}\(P\_\{i\}\),E\)=∑i=1Nwi​‖s​LogE⁡Pi‖E\\displaystyle=\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\\|s\\operatorname\{Log\}\_\{E\}P\_\{i\}\\\|\_\{E\}\(86\)=s2​∑i=1Nwi​‖LogE⁡Pi‖E2\\displaystyle=s^\{2\}\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\\|\\operatorname\{Log\}\_\{E\}P\_\{i\}\\\|^\{2\}\_\{E\}=s2​∑i=1Nwi​d2⁡\(Pi,E\),\\displaystyle=s^\{2\}\\sum\\nolimits\_\{i=1\}^\{N\}w\_\{i\}\\operatorname\{d\}^\{2\}\(P\_\{i\},E\),where∥⋅∥E\\\|\\cdot\\\|\_\{E\}is the norm onTE​ℳT\_\{E\}\\mathcal\{M\}\. ∎

### H\.4Proof of[Prop\.4\.5](https://arxiv.org/html/2607.08783#S4.Thmtheorem5)

###### Proof\.

As the right\-invariant metric shares similar the left\-invariant one, this proof follows the similar logic in the above two proofs\.

Gaussian homogeneity:We denoteY=RB⁡\(X\)Y=\\operatorname\{R\}\_\{B\}\(X\), andpXp\_\{X\}andpYp\_\{Y\}as the density ofXXandYY, respectively\. The density ofYYis

pY​\(Q\)\\displaystyle p\_\{Y\}\(Q\)=\(1\)pX​\(RB⊙−1⁡\(Q\)\)\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}p\_\{X\}\(\\operatorname\{R\}\_\{B^\{\-1\}\_\{\\odot\}\}\(Q\)\)\(87\)=k​\(σ\)​exp⁡\(−d\(RB⊙−1\(Q\),M\)22​σ2\)\\displaystyle=k\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(\\operatorname\{R\}\_\{B^\{\-1\}\_\{\\odot\}\}\(Q\),M\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)=\(2\)k​\(σ\)​exp⁡\(−d\(Q,RB\(M\)\)22​σ2\)\.\\displaystyle\\stackrel\{\{\\scriptstyle\(2\)\}\}\{\{=\}\}k\(\\sigma\)\\exp\\left\(\-\\frac\{\\operatorname\{d\}\(Q,\\operatorname\{R\}\_\{B\}\(M\)\)^\{2\}\}\{2\\sigma^\{2\}\}\\right\)\.The above comes from:

1. \(1\)Thm\. 13 in\[[66](https://arxiv.org/html/2607.08783#bib.bib158)\];
2. \(2\)The isometry of the right translation\.

Sample mean homogeneity:This is a direct corollary of the isometry of right translation\. ∎

### H\.5Proof of[Prop\.4\.6](https://arxiv.org/html/2607.08783#S4.Thmtheorem6)

###### Proof\.

Asℝn\\mathbb\{R\}^\{n\}is an abelian group and the Euclidean inner product is bi\-invariant, we focus on left\-translation in the following\. The core of this proof lies in the fact that onℝn\\mathbb\{R\}^\{n\}, \(1\) the Fréchet mean and variance are reduced to the familiar Euclidean statistics\. \(2\) the calculation of the running mean becomes the weighted arithmetic mean\. \(3\)[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11),[12](https://arxiv.org/html/2607.08783#S4.E12)and[13](https://arxiv.org/html/2607.08783#S4.E13)become[Eq\.5](https://arxiv.org/html/2607.08783#S3.E5); We prove these three points one by one\.

As stated by Luo*et al*\[[58](https://arxiv.org/html/2607.08783#bib.bib99), Prop\. G\.1 and Cor\. G\.2\], from the view of the product manifold, the element\-wise Fréchet mean and variance onℝn\\mathbb\{R\}^\{n\}are equivalent to the vector\-valued Euclidean variance and mean\.

Besides, by similar proof as Luo*et al*\[[58](https://arxiv.org/html/2607.08783#bib.bib99), Prop\. G\.1\], the weighted Fréchet mean onℝn\\mathbb\{R\}^\{n\}is simplified as the weighted arithmetic average\. Therefore, onℝn\\mathbb\{R\}^\{n\}, the calculation of running statistics in our[Algorithm1](https://arxiv.org/html/2607.08783#algorithm1)becomes the familiar moving average\.

Thirdly, onℝn\\mathbb\{R\}^\{n\}, we know thatLx⁡\(y\)=x\+y\\operatorname\{L\}\_\{x\}\(y\)=x\+y,Expx⁡v=x\+v\\operatorname\{Exp\}\_\{x\}v=x\+v,Logx⁡y=y−x\\operatorname\{Log\}\_\{x\}y=y\-x, and the neutral element is0\. Since statistics, as well as the Euclidean BN, are calculated element\-wisely, we can safely consider a single element,*i\.e\.*ℝn=ℝ\\mathbb\{R\}^\{n\}=\\mathbb\{R\}\. For a batch of activations\{xi​…​N∈ℝ\}\\\{x\_\{i\\ldots N\}\\in\\mathbb\{R\}\\\}, where the batch mean and batch variance are denoted asμb\\mu\_\{b\}andvb2v^\{2\}\_\{b\}, then[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11),[12](https://arxiv.org/html/2607.08783#S4.E12)and[13](https://arxiv.org/html/2607.08783#S4.E13)are rewrote as:

Lβ​\(Exp0⁡\[γvb2\+ϵ​Log0⁡\(L−μb⁡\(xi\)\)\]\)=γ​xi−μbvb2\+ϵ\+β\.L\_\{\\beta\}\\left\(\\operatorname\{Exp\}\_\{0\}\\left\[\\frac\{\\gamma\}\{\\sqrt\{v^\{2\}\_\{b\}\+\\epsilon\}\}\\operatorname\{Log\}\_\{0\}\(\\operatorname\{L\}\_\{\-\\mu\_\{b\}\}\(x\_\{i\}\)\)\\right\]\\right\)=\\gamma\\frac\{x\_\{i\}\-\\mu\_\{b\}\}\{\\sqrt\{v^\{2\}\_\{b\}\+\\epsilon\}\}\+\\beta\.\(88\)The above equation is the exact core computation of the standard Euclidean BN\. ∎

### H\.6Proof of[Prop\.5\.1](https://arxiv.org/html/2607.08783#S5.Thmtheorem1)

###### Proof\.

We first prove the case of\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}, and then proceed to the case ofθ​\-LCM\\theta\\text\{\-LCM\}\.

\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}:For clarity, we denote the metric tensor of\(θ,α,β\)​\-LEM\(\\theta,\\alpha,\\beta\)\\text\{\-LEM\}as

g\(θ,α,β\)​\-LE=1θ2​Pθ∗⁡g\(α,β\)​\-LE,g^\{\(\\theta,\\alpha,\\beta\)\\text\{\-LE\}\}=\\frac\{1\}\{\\theta^\{2\}\}\\operatorname\{P\}\_\{\\theta\}^\{\*\}g^\{\(\\alpha,\\beta\)\\text\{\-LE\}\},\(89\)whereg\(α,β\)​\-LEg^\{\(\\alpha,\\beta\)\\text\{\-LE\}\}is the metric tensor of\(α,β\)​\-LEM\(\\alpha,\\beta\)\\text\{\-LEM\}\. LetP∈𝒮\+\+nP\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}andV,W∈TP​𝒮\+\+nV,W\\in T\_\{P\}\\mathcal\{S\}^\{n\}\_\{\+\+\}, then we have

gP\(θ,α,β\)​\-LE​\(V,W\)\\displaystyle g^\{\(\\theta,\\alpha,\\beta\)\\text\{\-LE\}\}\_\{P\}\(V,W\)=1θ2​gPθ⁡\(P\)\(α,β\)​\-LE​\(Pθ⁣∗,P⁡\(V\),Pθ⁣∗,P⁡\(W\)\)\\displaystyle=\\frac\{1\}\{\\theta^\{2\}\}g^\{\(\\alpha,\\beta\)\\text\{\-LE\}\}\_\{\\operatorname\{P\}\_\{\\theta\}\(P\)\}\\left\(\\operatorname\{P\}\_\{\\theta\*,P\}\(V\),\\operatorname\{P\}\_\{\\theta\*,P\}\(W\)\\right\)\(90\)=1θ2​⟨\(log∘Pθ\)∗,P​\(V\),\(log∘Pθ\)∗,P​\(W\)⟩\(α,β\)\\displaystyle=\\frac\{1\}\{\\theta^\{2\}\}\\langle\\left\(\\operatorname\{log\}\\circ\\operatorname\{P\}\_\{\\theta\}\\right\)\_\{\*,P\}\(V\),\\left\(\\operatorname\{log\}\\circ\\operatorname\{P\}\_\{\\theta\}\\right\)\_\{\*,P\}\(W\)\\rangle^\{\(\\alpha,\\beta\)\}=⟨log∗,P⁡\(V\),log∗,P⁡\(W\)⟩\(α,β\)\\displaystyle=\\langle\\operatorname\{log\}\_\{\*,P\}\(V\),\\operatorname\{log\}\_\{\*,P\}\(W\)\\rangle^\{\(\\alpha,\\beta\)\}=gP\(α,β\)​\-LE​\(V,W\)\.\\displaystyle=g^\{\(\\alpha,\\beta\)\\text\{\-LE\}\}\_\{P\}\(V,W\)\.
θ​\-LCM\\theta\\text\{\-LCM\}:Let us first review a well\-known fact of deformed metrics\[[71](https://arxiv.org/html/2607.08783#bib.bib81)\]\. Letg~=1θ2​Pθ∗⁡g\\tilde\{g\}=\\frac\{1\}\{\\theta^\{2\}\}\\operatorname\{P\}\_\{\\theta\}^\{\*\}gbe the power\-deformed metric on SPD Then whenθ\\thetatends to 0, for allP∈𝒮\+\+nP\\in\\mathcal\{S\}^\{n\}\_\{\+\+\}and allV∈TP​𝒮\+\+nV\\in T\_\{P\}\\mathcal\{S\}^\{n\}\_\{\+\+\}, we have

g~P​\(V,V\)→gI​\(log∗,P⁡\(V\),log∗,P⁡\(V\)\)\.\\tilde\{g\}\_\{P\}\(V,V\)\\to g\_\{I\}\(\\log\_\{\*,P\}\(V\),\\log\_\{\*,P\}\(V\)\)\.\(91\)
By[Eq\.91](https://arxiv.org/html/2607.08783#A8.E91), we can readily obtain the results\. ∎

### H\.7Proof of[Prop\.5\.2](https://arxiv.org/html/2607.08783#S5.Thmtheorem2)

###### Proof\.

\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\}is left\-invariant\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]\. As the pullback of\(α,β\)​\-AIM\(\\alpha,\\beta\)\\text\{\-AIM\},\(θ,α,β\)​\-AIM\(\\theta,\\alpha,\\beta\)\\text\{\-AIM\}is left\-invariant as well\. Besides, Chen*et al*\[[25](https://arxiv.org/html/2607.08783#bib.bib7)\]shows that LCM is the pullback metric from the Euclidean space ofLTn\\mathrm\{LT\}^\{n\}\. Therefore,θ\\theta\-LCM is bi\-invariant\. ∎

### H\.8Proof of[Thm\.5\.3](https://arxiv.org/html/2607.08783#S5.Thmtheorem3)

###### Proof\.

In the following, we denote thedL\\operatorname\{d\}^\{\\mathrm\{L\}\},LogL\\operatorname\{Log\}^\{\\mathrm\{L\}\},ExpL\\operatorname\{Exp\}^\{\\mathrm\{L\}\}as the Riemannian operators under the left\-invariant metric,*i\.e\.*AIM\. Note that the Cholesky decomposition pulls back the group operation of matrix product from the Cholesky manifoldLT\+\+n\\mathrm\{LT\}\_\{\+\+\}^\{n\}\[[72](https://arxiv.org/html/2607.08783#bib.bib79)\]\. For simplicity, we abbreviate⊙AI\\odot^\{\\mathrm\{AI\}\}as⊙\\odot\.

Let us first review the differential map of Cholesky decomposition and its inverse\[[56](https://arxiv.org/html/2607.08783#bib.bib126), Prop\. 4\]\. Following the notations in this theorem and further denoteX∈TL​LT\+\+nX\\in T\_\{L\}\\mathrm\{LT\}\_\{\+\+\}^\{n\}, we have the following

Chol∗,P⁡\(V\)\\displaystyle\\operatorname\{Chol\}\_\{\*,P\}\(V\)=L​\(L−1​V​L−⊤\)12,\\displaystyle=L\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\},\(92\)\(Chol−1\)∗,L​\(X\)\\displaystyle\(\\operatorname\{Chol\}^\{\-1\}\)\_\{\*,L\}\(X\)=L​X⊤\+X​L⊤\.\\displaystyle=LX^\{\\top\}\+XL^\{\\top\}\.\(93\)Specifically, for the differential map atII, we have

Chol∗,I⁡\(V\)\\displaystyle\\operatorname\{Chol\}\_\{\*,I\}\(V\)=\(V\)12,∀V∈TI​𝒮\+\+n\.\\displaystyle=\\left\(V\\right\)\_\{\\frac\{1\}\{2\}\},\\forall V\\in T\_\{I\}\\mathcal\{S\}^\{n\}\_\{\+\+\}\.\(94\)\(Chol−1\)∗,I​\(X\)\\displaystyle\(\\operatorname\{Chol\}^\{\-1\}\)\_\{\*,I\}\(X\)=\(X\)Sym,∀X∈TI​LT\+\+n\.\\displaystyle=\\left\(X\\right\)\_\{\\mathrm\{Sym\}\},\\forall X\\in T\_\{I\}\\mathrm\{LT\}\_\{\+\+\}^\{n\}\.\(95\)DenotingL~\\widetilde\{\\operatorname\{L\}\}andR~\\widetilde\{\\operatorname\{R\}\}as the group translation on the Cholesky manifoldLT\+\+n\\mathrm\{LT\}\_\{\+\+\}^\{n\}, we have the following w\.r\.t\. the differential maps of left and right translation:

\(LP,∗P⊙−1\)−1\\displaystyle\\left\(\\operatorname\{L\}\_\{P,\*\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}=\(1\)\(\(Chol−1\)∗,I​L~L⁣∗,L−1∘Chol∗,P⊙−1\)−1\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}\\left\(\\left\(\\operatorname\{Chol\}^\{\-1\}\\right\)\_\{\*,I\}\\widetilde\{\\operatorname\{L\}\}\_\{L\*,L^\{\-1\}\}\\circ\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}=\(Chol∗,P⊙−1\)−1​\(L~L⁣∗,L−1\)−1∘Chol∗,I,\\displaystyle=\\left\(\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\left\(\\widetilde\{\\operatorname\{L\}\}\_\{L\*,L^\{\-1\}\}\\right\)^\{\-1\}\\circ\\operatorname\{Chol\}\_\{\*,I\},\(96\)RP⊙−1⁣∗,P\\displaystyle\\operatorname\{R\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\*,P\}=\(2\)\(Chol−1\)∗,I∘R~L−1⁣∗,L∘Chol∗,P\.\\displaystyle\\stackrel\{\{\\scriptstyle\(2\)\}\}\{\{=\}\}\\left\(\\operatorname\{Chol\}^\{\-1\}\\right\)\_\{\*,I\}\\circ\\widetilde\{\\operatorname\{R\}\}\_\{L^\{\-1\}\*,L\}\\circ\\operatorname\{Chol\}\_\{\*,P\}\.\(97\)The above derivation comes from the following:

1. \(1\)LP=Chol−1∘L~L∘Chol\\operatorname\{L\}\_\{P\}=\\operatorname\{Chol\}^\{\-1\}\\circ\\widetilde\{\\operatorname\{L\}\}\_\{L\}\\circ\\operatorname\{Chol\};
2. \(2\)RP⊙−1=Chol−1∘R~L−1∘Chol\\operatorname\{R\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}=\\operatorname\{Chol\}^\{\-1\}\\circ\\widetilde\{\\operatorname\{R\}\}\_\{L^\{\-1\}\}\\circ\\operatorname\{Chol\}\.

Riemannian metric:For the differential of right translation, we have the following

RP⊙−1⁣∗,P⁡\(V\)\\displaystyle\\operatorname\{R\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\*,P\}\(V\)=\(Chol−1\)∗,I∘R~L−1⁣∗,L∘Chol∗,P⁡\(V\)\\displaystyle=\\left\(\\operatorname\{Chol\}^\{\-1\}\\right\)\_\{\*,I\}\\circ\\widetilde\{\\operatorname\{R\}\}\_\{L^\{\-1\}\*,L\}\\circ\\operatorname\{Chol\}\_\{\*,P\}\(V\)\(98\)=\(L​\(L−1​V​L−⊤\)12​L−1\)Sym\.\\displaystyle=\\left\(L\(L^\{\-1\}VL^\{\-\\top\}\)\_\{\\frac\{1\}\{2\}\}L^\{\-1\}\\right\)\_\{\\mathrm\{Sym\}\}\.By[Eq\.98](https://arxiv.org/html/2607.08783#A8.E98), one can obtain the expression for the Riemannian metric tensor\.

Riemannian geodesic and exponential map:According to\[[91](https://arxiv.org/html/2607.08783#bib.bib71)\], we have the following for the operators between left\- and right\-invariant metrics:

ExpP⁡\(V\)\\displaystyle\\operatorname\{Exp\}\_\{P\}\(V\)=\{ExpP⊙−1L⁡\(−\(LP,∗P⊙−1\)−1∘RP⊙−1⁣∗,P⁡\(V\)\)\}⊙−1,\\displaystyle=\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{P^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\operatorname\{L\}\_\{P,\*P^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\circ\\operatorname\{R\}\_\{P^\{\-1\}\_\{\\odot\}\*,P\}\(V\)\\right\)\\right\\\}^\{\-1\}\_\{\\odot\},\(99\)d⁡\(P,Q\)\\displaystyle\\operatorname\{d\}\(P,Q\)=dL⁡\(P⊙−1,Q⊙−1\)\.\\displaystyle=\\operatorname\{d\}^\{\\mathrm\{L\}\}\\left\(\{P\}^\{\-1\}\_\{\\odot\},\{Q\}^\{\-1\}\_\{\\odot\}\\right\)\.\(100\)Putting the AIM\-based geodesic distance into the RHS of[Eq\.100](https://arxiv.org/html/2607.08783#A8.E100), one can obtain the geodesic distance under CRIM\.

Now, we simplify[Eq\.99](https://arxiv.org/html/2607.08783#A8.E99)\. Putting[Eqs\.96](https://arxiv.org/html/2607.08783#A8.E96)and[97](https://arxiv.org/html/2607.08783#A8.E97)into[Eq\.99](https://arxiv.org/html/2607.08783#A8.E99), we have the following:

ExpP⁡\(V\)\\displaystyle\\operatorname\{Exp\}\_\{P\}\(V\)=\{ExpP⊙−1L⁡\(−\(Chol∗,P⊙−1\)−1∘\(L~L⁣∗,L−1\)−1∘R~L−1⁣∗,L∘Chol∗,P⁡\(V\)\)\}⊙−1\\displaystyle=\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\circ\\left\(\\widetilde\{\\operatorname\{L\}\}\_\{L\*,L^\{\-1\}\}\\right\)^\{\-1\}\\circ\\widetilde\{\\operatorname\{R\}\}\_\{L^\{\-1\}\*,L\}\\circ\\operatorname\{Chol\}\_\{\*,P\}\(V\)\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}\(101\)=\{ExpP⊙−1L⁡\(−\(Chol∗,P⊙−1\)−1​\[L−1​Chol∗,P⁡\(V\)​L−1\]\)\}⊙−1\\displaystyle=\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\left\[L^\{\-1\}\\operatorname\{Chol\}\_\{\*,P\}\(V\)L^\{\-1\}\\right\]\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}=\(1\)\{ExpP⊙−1L⁡\(−\(Chol∗,P⊙−1\)−1​\[\(L−1​V​L−⊤\)12​L−1\]\)\}⊙−1\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\left\[\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}L^\{\-1\}\\right\]\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}=\(2\)\{ExpP⊙−1L⁡\(−\(\(L−1​V​L−⊤\)12​L−1​L−⊤\)Sym\)\}⊙−1\.\\displaystyle\\stackrel\{\{\\scriptstyle\(2\)\}\}\{\{=\}\}\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\left\(L^\{\-1\}VL^\{\-\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}L^\{\-1\}L^\{\-\\top\}\\right\)\_\{\\mathrm\{Sym\}\}\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}\.The above comes from the following:

1. \(1\)
2. \(2\)

Riemannian logarithm:From the second equality in[Eq\.101](https://arxiv.org/html/2607.08783#A8.E101), we have the following

LogP⁡\(Q\)\\displaystyle\\operatorname\{Log\}\_\{P\}\(Q\)=−Chol∗,P−1⁡\{L​Chol∗,P⊙−1⁡\(LogP⊙−1L⁡\(Q⊙−1\)\)​L\}\\displaystyle=\-\\operatorname\{Chol\}\_\{\*,P\}^\{\-1\}\\left\\\{L\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\\operatorname\{Log\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\{Q\}^\{\-1\}\_\{\\odot\}\\right\)\\right\)L\\right\\\}\(102\)=\(1\)−Chol∗,P−1⁡\{\(L​V~​L⊤\)12​L\}\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}\-\\operatorname\{Chol\}\_\{\*,P\}^\{\-1\}\\left\\\{\\left\(L\\widetilde\{V\}L^\{\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}L\\right\\\}=−\(L​L⊤​\(L​V~​L⊤\)12⊤\)Sym\\displaystyle=\-\\left\(LL^\{\\top\}\\left\(L\\widetilde\{V\}L^\{\\top\}\\right\)\_\{\\frac\{1\}\{2\}\}^\{\\top\}\\right\)\_\{\\mathrm\{Sym\}\}The above comes from the following:

1. \(1\)Chol∗,P⊙−1⁡\(V\)=L−1​\(L​V​L⊤\)12,∀V∈TP⊙−1​𝒮\+\+n\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\(V\)=L^\{\-1\}\\left\(LVL^\{\\top\}\\right\)\_\{\\frac\{1\}\{2\}\},\\forall V\\in T\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\mathcal\{S\}^\{n\}\_\{\+\+\}\.

∎

### H\.9Proof of[Cor\.5\.4](https://arxiv.org/html/2607.08783#S5.Thmtheorem4)

###### Proof\.

Completeness:[Eq\.22](https://arxiv.org/html/2607.08783#S5.E22)indicates thatExpI\\operatorname\{Exp\}\_\{I\}is defined over the wholeTI​𝒮\+\+nT\_\{I\}\\mathcal\{S\}^\{n\}\_\{\+\+\}\. Besides, the SPD manifold is connected\[[65](https://arxiv.org/html/2607.08783#bib.bib123)\]\. By Cor\. 6\.20 in\[[54](https://arxiv.org/html/2607.08783#bib.bib155)\], CRIM is complete\.

Geodesic:For simplicity, we abbreviated⊙AI\\odot^\{\\mathrm\{AI\}\}as⊙\\odot\. The geodesic connectingPPandQQcan be obtained by the following:

γ​\(t;P,Q\)\\displaystyle\\gamma\{\(t;P,Q\)\}=ExpP⁡\(t​LogP⁡\(Q\)\)\\displaystyle=\\operatorname\{Exp\}\_\{P\}\\left\(t\\operatorname\{Log\}\_\{P\}\(Q\)\\right\)\(103\)=\(1\)\{ExpP⊙−1L⁡\(−\(Chol∗,P⊙−1\)−1​\[L−1​Chol∗,P⁡\(t​LogP⁡\(Q\)\)​L−1\]\)\}⊙−1\\displaystyle\\stackrel\{\{\\scriptstyle\(1\)\}\}\{\{=\}\}\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\-\\left\(\\operatorname\{Chol\}\_\{\*,\{P\}^\{\-1\}\_\{\\odot\}\}\\right\)^\{\-1\}\\left\[L^\{\-1\}\\operatorname\{Chol\}\_\{\*,P\}\\left\(t\\operatorname\{Log\}\_\{P\}\(Q\)\\right\)L^\{\-1\}\\right\]\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}=\(2\)\{ExpP⊙−1L⁡\(t​LogP⊙−1L⁡\(Q⊙−1\)\)\}⊙−1\\displaystyle\\stackrel\{\{\\scriptstyle\(2\)\}\}\{\{=\}\}\\left\\\{\\operatorname\{Exp\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(t\\operatorname\{Log\}^\{\\mathrm\{L\}\}\_\{\{P\}^\{\-1\}\_\{\\odot\}\}\\left\(\{Q\}^\{\-1\}\_\{\\odot\}\\right\)\\right\)\\right\\\}\_\{\\odot\}^\{\-1\}=\{γAI​\(t;P~,Q~\)\}⊙−1\.\\displaystyle=\\left\\\{\\gamma^\{\\mathrm\{AI\}\}\(t;\\widetilde\{P\},\\widetilde\{Q\}\)\\right\\\}\_\{\\odot\}^\{\-1\}\.The above comes from the following:

1. \(1\)The second equality in[Eq\.101](https://arxiv.org/html/2607.08783#A8.E101);
2. \(2\)The first equality in[Eq\.102](https://arxiv.org/html/2607.08783#A8.E102)\.

∎

### H\.10Proof of[Thm\.5\.6](https://arxiv.org/html/2607.08783#S5.Thmtheorem6)

###### Proof\.

With out loss of generality, we focus on the case of the left\-invariant metric\. The results of the right\-invariant metric can be proven similarly\.

We denote[Eqs\.11](https://arxiv.org/html/2607.08783#S4.E11),[12](https://arxiv.org/html/2607.08783#S4.E12)and[13](https://arxiv.org/html/2607.08783#S4.E13)onℳi,i=1,2\\mathcal\{M\}\_\{i\},i=1,2as the mappingξi\(⋅\|M,v2,B,s\)\\xi^\{i\}\(\\cdot\|M,v^\{2\},B,s\)\. Letℬ=\{P1​…​N\}\\mathcal\{B\}=\\\{P\_\{1\\ldots N\}\\\}andf​\(ℬ\)=\{f​\(P1​…​N\)\}f\(\\mathcal\{B\}\)=\\\{f\(P\_\{1\\ldots N\}\)\\\}\.

The core of this proof lies in three points:

1. 1\.The Fréchet mean and variance ofℬ\\mathcal\{B\}inℳ1\\mathcal\{M\}\_\{1\}correspond to the counterparts off​\(ℬ\)f\(\\mathcal\{B\}\)inℳ2\\mathcal\{M\}\_\{2\}\.
2. 2\.ξ1​\(Pi\|M,v2,B,s\)\\xi^\{1\}\(P\_\{i\}\|M,v^\{2\},B,s\)inℳ1\\mathcal\{M\}\_\{1\}is equal tof−1​\(ξ2​\(f​\(Pi\)\|f​\(M\),v2,f​\(B\),s\)\)f^\{\-1\}\(\\xi^\{2\}\(f\(P\_\{i\}\)\|f\(M\),v^\{2\},f\(B\),s\)\)\.
3. 3\.The updates of running statistics inℳ1\\mathcal\{M\}\_\{1\}correspond to the counterparts inℳ2\\mathcal\{M\}\_\{2\}\.

We denoteMMas the Fréchet mean ofℬ\\mathcal\{B\}, andv2v^\{2\}as the Fréchet variance ofℬ\\mathcal\{B\}\. Then, by the isometry offf, the Fréchet mean and variance off​\(ℬ\)f\(\\mathcal\{B\}\)aref​\(M\)f\(M\)andv2v^\{2\}, respectively\.

Onℳi,i=1,2\\mathcal\{M\}\_\{i\},i=1,2, we denoteLi,⊙i,Expi,Logi\\operatorname\{L\}^\{i\},\\odot^\{i\},\\operatorname\{Exp\}^\{i\},\\operatorname\{Log\}^\{i\}as the Lie group and Riemannian operators,EiE^\{i\}as the neutral element, and[Eq\.12](https://arxiv.org/html/2607.08783#S4.E12)asϕsi​\(⋅\)\\phi^\{i\}\_\{s\}\(\\cdot\)\. With the isometry and Lie group isomorphism offf, we have the following equations:

LM⊙1−11\\displaystyle L^\{1\}\_\{M^\{\-1\}\_\{\\odot^\{1\}\}\}=f−1∘Lf​\(M\)⊙2−12∘f,\\displaystyle=f^\{\-1\}\\circ\\operatorname\{L\}^\{2\}\_\{f\(M\)^\{\-1\}\_\{\\odot^\{2\}\}\}\\circ f,\(104\)ϕs1\\displaystyle\\phi^\{1\}\_\{s\}=ExpE11⁡\[s​LogE11⁡\(⋅\)\]\\displaystyle=\\operatorname\{Exp\}^\{1\}\_\{E^\{1\}\}\\left\[s\\operatorname\{Log\}^\{1\}\_\{E^\{1\}\}\(\\cdot\)\\right\]=f−1​\(ExpE21⁡\[s​LogE22⁡\(f​\(⋅\)\)\]\)\\displaystyle=f^\{\-1\}\\left\(\\operatorname\{Exp\}^\{1\}\_\{E^\{2\}\}\\left\[s\\operatorname\{Log\}^\{2\}\_\{E^\{2\}\}\(f\(\\cdot\)\)\\right\]\\right\)=f−1∘ϕs2∘f,\\displaystyle=f^\{\-1\}\\circ\\phi^\{2\}\_\{s\}\\circ f,\(105\)LB1\\displaystyle L^\{1\}\_\{B\}=f−1∘Lf​\(B\)2∘f\.\\displaystyle=f^\{\-1\}\\circ\\operatorname\{L\}^\{2\}\_\{f\(B\)\}\\circ f\.\(106\)Then we have

ξ1​\(Pi\|M,v2,B,s\)=f−1​\(ξ2​\(f​\(Pi\)\|f​\(M\),v2,f​\(B\),s\)\)\\xi^\{1\}\(P\_\{i\}\|M,v^\{2\},B,s\)=f^\{\-1\}\(\\xi^\{2\}\(f\(P\_\{i\}\)\|f\(M\),v^\{2\},f\(B\),s\)\)\(107\)
Lastly, we show the correspondence between running statistics\. Since the Fréchet variance is the same for bothℬ\\mathcal\{B\}andf​\(ℬ\)f\(\\mathcal\{B\}\), we focus on the running mean\. LetMrM\_\{r\}andf​\(Mr\)f\(M\_\{r\}\)denote the initial values of the running means inℳ1\\mathcal\{M\}\_\{1\}andℳ2\\mathcal\{M\}\_\{2\}respectively, andWFMi\\operatorname\{WFM\}^\{i\}represent the weighted Fréchet mean inℳi\\mathcal\{M\}\_\{i\}\. Then the updated running mean inℳ1\\mathcal\{M\}\_\{1\}is

WFM1⁡\(\{1−γ,γ\},\{Mr,M\}\)=f−1​\(WFM2⁡\(\{1−γ,γ\},\{f​\(Mr\),f​\(M\)\}\)\)\\operatorname\{WFM\}^\{1\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{M\_\{r\},M\\\}\)=f^\{\-1\}\(\\operatorname\{WFM\}^\{2\}\(\\\{1\-\\gamma,\\gamma\\\},\\\{f\(M\_\{r\}\),f\(M\)\\\}\)\)\(108\)we can further simply the above equation as

WFM1=f−1∘WFM2∘f\\operatorname\{WFM\}^\{1\}=f^\{\-1\}\\circ\\operatorname\{WFM\}^\{2\}\\circ f\(109\)
DenotingLieBNi\\operatorname\{LieBN\}^\{i\}as the LieBN algorithm onℳi\\mathcal\{M\}\_\{i\},[Eq\.107](https://arxiv.org/html/2607.08783#A8.E107)and[Eq\.109](https://arxiv.org/html/2607.08783#A8.E109)imply that:

LieBN1⁡\(Pi\|B,s,ϵ,γ\)=f−1​\[LieBN2⁡\(f​\(Pi\)\|f​\(B\),s,ϵ,γ\)\]\.\\operatorname\{LieBN\}^\{1\}\(P\_\{i\}\|B,s,\\epsilon,\\gamma\)=f^\{\-1\}\\left\[\\operatorname\{LieBN\}^\{2\}\(f\(P\_\{i\}\)\|f\(B\),s,\\epsilon,\\gamma\)\\right\]\.\(110\)∎