Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds

# Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds
Source: [https://arxiv.org/html/2605.24390](https://arxiv.org/html/2605.24390)
\(2026\)

###### Abstract\.

The eigendecomposition of the Laplace–Beltrami Operator \(LBO\) is fundamental to geometric analysis, yet computing its low\-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large\-scale data\. To amortize this cost, we introduce the Neural Eigenspace Operator \(NEO\), a feed\-forward framework designed to predict the spectrum directly from point clouds\. Crucially, NEO circumvents the ill\-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low\-frequency subspace instead\. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh–Ritz refinement\. To handle irregular sampling, we propose a mass\-aware neural operator that incorporates per\-point area weights into attention\-based aggregation, improving robustness to non\-uniform densities and enabling zero\-shot generalization across resolutions\. Our approach achieves near\-linear runtime scaling and substantial wall\-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero\-shot transfer to high\-resolution point clouds\. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point\-wise features for downstream learning\. Code:[https://github\.com/Adversarr/NEO](https://github.com/Adversarr/NEO)\.

Spectral Geometry Processing, Eigenvalue Problem, Neural Operator

††journal:TOG††journalyear:2026††conference:Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers; July 19–23, 2026; Los Angeles, CA, USA††booktitle:Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Papers \(SIGGRAPH Conference Papers ’26\), July 19–23, 2026, Los Angeles, CA, USA††doi:10\.1145/3799902\.3811185††isbn:979\-8\-4007\-2554\-8/2026/07††ccs:Computing methodologies Shape analysis††ccs:Computing methodologies Artificial intelligence††ccs:Computing methodologies Neural networks![Refer to caption](https://arxiv.org/html/2605.24390v1/x1.png)Figure 1\.We present NEO, a neural framework that accelerates Laplace–Beltrami spectral analysis by predicting the low\-frequency eigenspace directly from raw point clouds\.Left:Example inputs with varying resolution, sampling density, and shape categories \(bottom\) and the corresponding predicted low\-frequency eigenfunctions \(top\)\.Right:The resulting spectral representation can be used in downstream geometry processing tasks such as shape matching \(functional maps\), heat\-method geodesics, and segmentation\.Teaser figure for NEO\. The left side shows several input 3D point clouds with different resolutions, sampling densities, and shape categories, together with predicted low\-frequency Laplace–Beltrami eigenfunctions visualized on the shapes\. The right side illustrates downstream uses of the predicted spectrum, including functional\-map shape matching, heat\-based geodesic computation, and segmentation\.## 1\.INTRODUCTION

Eigenmodes of the Laplace–Beltrami operator \(LBO\) provide an intrinsic spectral representation of 3D shapes\(Lévy,[2006](https://arxiv.org/html/2605.24390#bib.bib20)\)\. Much like the Fourier basis on Euclidean domains, the*low\-frequency*eigenfunctions capture global, smooth geometric structures, underpinning diverse graphics pipelines\(Vallet and Lévy,[2008](https://arxiv.org/html/2605.24390#bib.bib47); Ovsjanikov et al\.,[2012](https://arxiv.org/html/2605.24390#bib.bib33)\), from spectral mesh processing\(Lévy and Zhang,[2010](https://arxiv.org/html/2605.24390#bib.bib21); Zhang et al\.,[2007](https://arxiv.org/html/2605.24390#bib.bib54)\)and physical simulation\(Pentland and Williams,[1989](https://arxiv.org/html/2605.24390#bib.bib35)\)to geometric deep learning\(Bronstein et al\.,[2017](https://arxiv.org/html/2605.24390#bib.bib6); Sharp et al\.,[2022](https://arxiv.org/html/2605.24390#bib.bib42)\)\.

In practice, obtaining this basis reduces to solving a sparse generalized eigenvalue problem \(GEVP\) arising from an LBO discretization\. While discretization is typically local and efficient, extracting the firstkklow\-frequency modes remains a primary computational bottleneck\. Standard eigensolvers based on Krylov subspaces \(e\.g\., implicitly restarted Lanczos\) compute these modes through iterative procedures\. As a result, the computation is inherently*per\-instance*: changes in resolution, re\-sampling, or deformation often require recomputing the decomposition\. This cost is prohibitive at scale, pushing spectral analysis to offline preprocessing in many pipelines\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x2.png)Figure 2\.Ambiguities in LBO eigenspace computation\.Eigenfunctions are not uniquely defined: each mode is determined only up to a global sign, and repeated \(or nearly repeated\) eigenvalues admit arbitrary orthogonal bases within the same eigenspace\. Consequently, equally valid eigensolvers \(or the same solver under small numerical perturbations\) may return different eigenvector bases for the same shape \(A vs\. B\), including \(left\) rotations within a degenerate eigenspace, \(middle\) sign flips, and \(right\) mode mixing or re\-ordering near spectral clusters\. These ambiguities render direct eigenvector regression ill\-posed, as there is no unique basis to align with\.Comparison figure illustrating ambiguities in Laplace–Beltrami eigenvectors for the same shape\. The examples show valid solutions that differ by a rotation within a repeated eigenspace, a sign flip of an eigenfunction, or mode mixing and reordering near a spectral cluster\.Motivated by the success of feed\-forward pipelines such as RenderFormer\(Zeng et al\.,[2025](https://arxiv.org/html/2605.24390#bib.bib53)\)and VGGT\(Wang et al\.,[2025](https://arxiv.org/html/2605.24390#bib.bib48)\), we generalize this paradigm to spectral analysis by learning to solve the GEVP directly on 3D point clouds\. Our goal is to predict a high\-quality approximation of the low\-frequency LBO eigenspace in a single feed\-forward pass\. However, directly regressing the discretized eigenvectors is ill\-posed due to inherent ambiguities\(Davis and Kahan,[1970](https://arxiv.org/html/2605.24390#bib.bib12); Golub and Van Loan,[2013](https://arxiv.org/html/2605.24390#bib.bib13); Chang et al\.,[2024](https://arxiv.org/html/2605.24390#bib.bib9)\): each eigenvector is defined only up to a sign flip, and repeated eigenvalues admit arbitrary rotations within the associated eigenspace \(see Fig\.[2](https://arxiv.org/html/2605.24390#S1.F2)\)\. Consequently, regressing individual vectors forces the network to memorize the arbitrary basis choices in the training data, leading to unstable training and poor generalization\. To address this, we leverage the fact that while individual eigenfunctions are ambiguous, the low\-frequency invariant subspace they span is unique and well\-defined\. We therefore shift the learning objective from regressing individual eigenfunctions to predicting a set of functions whose span robustly captures the target eigenspace\.

To this end, we introduce the*Neural Eigenspace Operator*\(NEO\), a feed\-forward framework that predicts the low\-frequency invariant subspace from raw point clouds\. Rather than outputting exactlykkmodes, NEO predicts a modestly redundant set ofmmfunctions \(m\>km\>k\) whose span robustly captures the target eigenspace, providing slack to accommodate repeated or nearby eigenvalues\. Crucially, to handle the irregularity of point cloud sampling, we employ a*mass\-aware*attention mechanism in the neural operator\. This module explicitly incorporates point masses into feature aggregation, ensuring the neural operator approximates the continuous integral operator with respect to the underlying measure, rather than being biased by sampling density\. We train NEO with a rotation\-invariant loss that compares spans rather than individual eigenvectors, directly resolving the aforementioned eigenvector ambiguities\. When explicit eigenpairs are required, we apply a lightweight Rayleigh–Ritz refinement\. This step projects the discretized operators onto the predicted low\-dimensional basis, reducing the original large\-scale sparse GEVP to a small dense eigenproblem that is efficient to solve\.

Our experiments show that NEO achieves near\-linear inference scaling, accelerating eigensolving on point clouds ranging from thousands to over a million points\. At the same time, it preserves spectral accuracy by predicting subspaces with low span error and supporting reliable Rayleigh–Ritz eigenpair recovery\. Thanks to its mass\-aware neural operator, NEO can be trained exclusively on low\-resolution data yet exhibits strong zero\-shot transfer to dense geometry, while remaining robust under non\-uniform sampling and varied discretizations\. The recovered eigenpairs can be used in standard spectral geometry pipelines and the raw predicted functions also provide effective point\-wise features for downstream learning tasks\. We summarize our contributions as follows:

- •Neural Eigensolving Framework\.We propose a learning\-based framework that reformulates low\-frequency LBO eigensolving as invariant subspace prediction followed by Rayleigh\-Ritz refinement\. This approach enables rapid linear\-time inference after one\-time training\.
- •Geometric Learning Formulation\.We integrate a mass\-aware mechanism into the neural operator, so that it respects the surface measure and improves robustness to non\-uniform sampling\. Together with a rotation\-invariant span loss, these formulations render the ill\-posed eigenvector regression task feasible and stable\.

## 2\.RELATED WORK

### 2\.1\.LBO Eigensolvers

Classical LBO eigendecomposition relies on sparse eigensolvers for large\-scale GEVPs\. Krylov\-subspace algorithms, such as the implicitly restarted Lanczos method in ARPACK\(Lehoucq et al\.,[1998](https://arxiv.org/html/2605.24390#bib.bib19)\), and block methods such as LOBPCG\(Knyazev,[2001](https://arxiv.org/html/2605.24390#bib.bib17)\), are widely adopted due to their robustness and accuracy\. In practice, their performance can be substantially improved with preconditioning and shift\-invert schemes, especially when high\-precision low\-frequency modes are required\. To improve efficiency, geometric approximation methods\(Nasikun et al\.,[2018](https://arxiv.org/html/2605.24390#bib.bib32)\)construct explicit subspaces, but typically rely on mesh connectivity\. Nevertheless, both classical sparse eigensolvers and these approximation schemes remain per\-instance: they solve each GEVP independently and cannot exploit statistical patterns across a collection of shapes to amortize inference\.

Recent advances in physics\-informed machine learning offer optimization\-based alternatives\. Physics\-informed neural networks \(PINNs\)\(Raissi et al\.,[2019](https://arxiv.org/html/2605.24390#bib.bib39)\)and variational methods such as Deep Ritz\(Yu et al\.,[2018](https://arxiv.org/html/2605.24390#bib.bib52); Ben\-Shaul et al\.,[2023](https://arxiv.org/html/2605.24390#bib.bib5)\)optimize continuous objectives to recover eigenfunctions\. Recent work further extends these ideas to parameterized families of geometries and studies how spectra vary across a shape space\(Chang et al\.,[2024](https://arxiv.org/html/2605.24390#bib.bib9)\)\. While these methods are appealing when differentiability with respect to geometry is needed, they generally still require solving an optimization problem for each new instance\. In contrast, we target one\-shot prediction of the low\-frequency invariant subspace, which can be optionally refined by a small projected solve\.

### 2\.2\.Neural Operators

Neural operators learn mappings between function spaces\(Kovachki et al\.,[2023](https://arxiv.org/html/2605.24390#bib.bib18)\), offering a resolution\-agnostic paradigm well suited for geometric spectral problems where shapes and eigenfunctions are represented as point\-sampled fields\. Seminal works like DeepONet\(Lu et al\.,[2021](https://arxiv.org/html/2605.24390#bib.bib28)\)and Fourier Neural Operators \(FNO\)\(Li et al\.,[2020a](https://arxiv.org/html/2605.24390#bib.bib23)\)have demonstrated their efficacy on regular grids\. For irregular geometries, graph\-based approaches\(Li et al\.,[2020b](https://arxiv.org/html/2605.24390#bib.bib24); Pfaff et al\.,[2020](https://arxiv.org/html/2605.24390#bib.bib36)\)provide local message passing, yet scaling them to capture long\-range correlations over dense point sets often requires deep stacks or global mechanisms\(Alon and Yahav,[2020](https://arxiv.org/html/2605.24390#bib.bib2)\)\. To enable more efficient global interactions, methods such as Transolver\(Wu et al\.,[2024](https://arxiv.org/html/2605.24390#bib.bib49)\)and LinearNO\(Hu et al\.,[2025](https://arxiv.org/html/2605.24390#bib.bib14)\)route information using attention through a small set of latent tokens, achieving near\-linear scaling in the number of input points\. NEO adopts this scalable paradigm and adapts it by incorporating mass\-aware aggregation\. This design is reminiscent of surface attention\(Trappolini et al\.,[2021](https://arxiv.org/html/2605.24390#bib.bib46)\), introduced for shape correspondence\. In contrast, NEO injects mass information into the attention score in order to obtain a resolution\-invariant integral approximation in the neural\-operator setting\. Concurrently,Li et al\.\([2025](https://arxiv.org/html/2605.24390#bib.bib22)\)solve parametric non\-selfadjoint eigenvalue problems by learning invariant subspaces\. While their approach focuses on spectral analysis under varying physical parameters over fixed discretizations, our work emphasizes geometric variability across shapes and employs a mass\-aware mechanism to maintain resolution\-invariance on raw point clouds\.

### 2\.3\.Spectral Geometry & Geometric Learning

Spectral methods provide an intrinsic representation for geometry processing and shape analysis, rooted in manifold learning and spectral shape analysis\(Belkin and Niyogi,[2003](https://arxiv.org/html/2605.24390#bib.bib4); Coifman and Lafon,[2006](https://arxiv.org/html/2605.24390#bib.bib10); Reuter et al\.,[2006](https://arxiv.org/html/2605.24390#bib.bib41)\)\. Low\-frequency LBO eigenfunctions underpin widely used descriptors and bases, including HKS/WKS\(Sun et al\.,[2009](https://arxiv.org/html/2605.24390#bib.bib45); Aubry et al\.,[2011](https://arxiv.org/html/2605.24390#bib.bib3)\)and spectral bases for correspondence via functional maps and their refinements\(Ovsjanikov et al\.,[2012](https://arxiv.org/html/2605.24390#bib.bib33); Melzi et al\.,[2019](https://arxiv.org/html/2605.24390#bib.bib30); Ren et al\.,[2018](https://arxiv.org/html/2605.24390#bib.bib40)\)\. In geometric learning, spectral constructions have likewise informed model design, from early spectral convolutions\(Bruna et al\.,[2013](https://arxiv.org/html/2605.24390#bib.bib7); Yi et al\.,[2016](https://arxiv.org/html/2605.24390#bib.bib51)\)to recent architectures that use low\-frequency eigenvectors for feature propagation on surfaces\(Sharp et al\.,[2022](https://arxiv.org/html/2605.24390#bib.bib42); Smirnov and Solomon,[2021](https://arxiv.org/html/2605.24390#bib.bib44)\)\. In most such pipelines, eigenpairs are assumed to be precomputed\. Neural Laplacian Operator\(Pang et al\.,[2024](https://arxiv.org/html/2605.24390#bib.bib34)\)predicts a high\-quality discrete Laplacian operator directly from point clouds, enabling downstream computations via standard eigensolvers\. NeuralSound\(Jin et al\.,[2022](https://arxiv.org/html/2605.24390#bib.bib15)\)similarly uses subspace learning, but is designed for fast modal sound synthesis on voxel discretizations\. In contrast, NEO directly predicts the eigenspace on raw point clouds, providing fast access to spectral bases commonly used in geometry processing\.

## 3\.NEO INFERENCE PIPELINE

![Refer to caption](https://arxiv.org/html/2605.24390v1/x3.png)Figure 3\.Overview of the NEO Inference Pipeline\.Given a point cloudXXwith mass weightsww\(definingMM\), a mass\-aware neural operator predicts redundant raw basis functionsFFin one forward pass\. WeMM\-orthonormalizeFFto obtain a subspace basisYY, project the discrete Laplacian to a small dense matrixL^=Y⊤​L​Y\\widehat\{L\}=Y^\{\\top\}LY, solve a dense eigenproblem in the subspace, and lift the eigenvectors back asu^i=Y​vi\\widehat\{u\}\_\{i\}=Yv\_\{i\}\.Pipeline overview of NEO\. Starting from a point cloud and per\-point mass weights, the neural operator predicts a redundant set of functions\. These functions are orthonormalized with respect to the mass matrix to form a subspace basis, the Laplacian is projected into this low\-dimensional subspace, a small dense eigenproblem is solved, and the resulting eigenvectors are lifted back to the original points\.In this section, we present the inference pipeline of NEO\. We first formalize the discrete spectral problem on point clouds \([Sec\.3\.1](https://arxiv.org/html/2605.24390#S3.SS1)\)\. Then, we detail how NEO predicts a low\-frequency invariant subspace to recover explicit eigenpairs via Rayleigh–Ritz refinement, which also yields intrinsic point embeddings as a byproduct \([Sec\.3\.2](https://arxiv.org/html/2605.24390#S3.SS2)\)\. The training objectives and network architecture are detailed in[Sec\.4](https://arxiv.org/html/2605.24390#S4)\.

### 3\.1\.Problem Setup

Letℳ\\mathcal\{M\}be a \(possibly bounded\) Riemannian manifold andΔ\\Deltaits Laplace–Beltrami operator\. We consider the eigenvalue problem−Δ​ϕi=λi​ϕi\-\\Delta\\phi\_\{i\}=\\lambda\_\{i\}\\phi\_\{i\}with0=λ0≤λ1≤…0=\\lambda\_\{0\}\\leq\\lambda\_\{1\}\\leq\\dots, and assume Neumann boundary conditions for manifolds\. The eigenfunctions\{ϕi\}\\\{\\phi\_\{i\}\\\}form an orthonormal basis ofL2​\(ℳ\)L^\{2\}\(\\mathcal\{M\}\)under the inner product⟨f,g⟩ℳ=∫ℳf​\(x\)​g​\(x\)​𝑑μ​\(x\)\\langle f,g\\rangle\_\{\\mathcal\{M\}\}=\\int\_\{\\mathcal\{M\}\}f\(x\)g\(x\)\\,d\\mu\(x\)\. In this work, we focus on the invariant subspace𝒮k=span​\(\{ϕi\}i=0k−1\)\\mathcal\{S\}\_\{k\}=\\mathrm\{span\}\(\\\{\\phi\_\{i\}\\\}\_\{i=0\}^\{k\-1\}\)spanned by the firstkkeigenfunctions\.

We operate on raw point cloudsX=\{xi\}i=1N⊂ℝ3X=\\\{x\_\{i\}\\\}\_\{i=1\}^\{N\}\\subset\\mathbb\{R\}^\{3\}sampled fromℳ\\mathcal\{M\}without assuming mesh connectivity\. To define the discrete spectral problem, we assume access to a sparse, symmetric positive semi\-definite \(PSD\) Laplacian matrixL∈ℝN×NL\\in\\mathbb\{R\}^\{N\\times N\}and a*diagonal*mass matrixM=diag​\(w\)≻0M=\\mathrm\{diag\}\(w\)\\succ 0that discretize the continuous operators \(e\.g\., the intrinsic Delaunay construction\(Sharp and Crane,[2020](https://arxiv.org/html/2605.24390#bib.bib43)\)\)\. Here,wi\>0w\_\{i\}\>0approximates the local area measure atxix\_\{i\}, inducing the discrete inner product⟨u,v⟩M=u⊤​M​v\\langle u,v\\rangle\_\{M\}=u^\{\\top\}Mv\. Functions onℳ\\mathcal\{M\}are thus represented by vectorsu∈ℝNu\\in\\mathbb\{R\}^\{N\}sampled at the input points, and the eigenproblem is then discretized into the GEVP:

\(1\)L​ui=λi​M​ui,ui⊤​M​uj=δi​j,Lu\_\{i\}=\\lambda\_\{i\}Mu\_\{i\},\\qquad u\_\{i\}^\{\\top\}Mu\_\{j\}=\\delta\_\{ij\},Our goal is to replace this costly eigensolving for the firstkkmodes with a single feed\-forward evaluation\.

### 3\.2\.The NEO Pipeline

As shown in[Fig\.3](https://arxiv.org/html/2605.24390#S3.F3), to circumvent eigenvector ambiguities, NEO predicts the*invariant subspace*𝒮k\\mathcal\{S\}\_\{k\}itself, which is mathematically unique\. At inference time, the mass\-aware neural operatorℱθ\\mathcal\{F\}\_\{\\theta\}maps the input point cloud to a set ofmmbasis functions:

\(2\)F=ℱθ​\(X,w\)∈ℝN×m,F=\\mathcal\{F\}\_\{\\theta\}\(X,w\)\\in\\mathbb\{R\}^\{N\\times m\},whose span is intended to cover𝒮k\\mathcal\{S\}\_\{k\}\. We deliberately introduce redundancym\>km\>kas a relaxation: this allows the predicted subspacespan​\(F\)\\mathrm\{span\}\(F\)to robustly cover the target subspace even in the presence of ambiguities, without being forced to resolve them inside the network\. Sincem≪Nm\\ll N, this representation remains compact even for high\-resolution point clouds\. When explicit eigenpairs are needed, we refine within this subspace via Rayleigh–Ritz, as described next\.

#### Eigenpair recovery via Rayleigh–Ritz refinement\.

For applications requiring explicit eigenpairs\(λi,ui\)\(\\lambda\_\{i\},u\_\{i\}\), we apply a standard Rayleigh–Ritz procedure for the GEVP within the predicted subspace\. First, we convert the raw fieldsFFinto anMM\-orthonormal basisY∈ℝN×mY\\in\\mathbb\{R\}^\{N\\times m\}without changing the span\. SinceMMis diagonal, we can efficiently compute a weighted QR factorization: letZ=M​FZ=\\sqrt\{M\}\\,F, compute its Euclidean QR factorizationZ=Q​RZ=QR, and set

\(3\)Y=M−1/2​Q,so that​Y⊤​M​Y=Im\.Y=M^\{\-1/2\}Q,\\qquad\\text\{so that \}Y^\{\\top\}MY=I\_\{m\}\.We then project the LBO onto themm\-dimensional subspace:

\(4\)L^=Y⊤​L​Y∈ℝm×m,M^=Y⊤​M​Y=Im\.\\widehat\{L\}=Y^\{\\top\}LY\\in\\mathbb\{R\}^\{m\\times m\},\\qquad\\widehat\{M\}=Y^\{\\top\}MY=I\_\{m\}\.The original high\-dimensional GEVP is thus reduced to a small, densem×mm\\times meigen\-decomposition:

\(5\)L^​vi=λ^i​vi,u^i=Y​vi\.\\widehat\{L\}v\_\{i\}=\\widehat\{\\lambda\}\_\{i\}v\_\{i\},\\qquad\\widehat\{u\}\_\{i\}=Yv\_\{i\}\.We take thekksmallest Ritz pairs\{\(λ^i,u^i\)\}i=0k−1\\\{\(\\widehat\{\\lambda\}\_\{i\},\\widehat\{u\}\_\{i\}\)\\\}\_\{i=0\}^\{k\-1\}as the recovered low\-frequency eigenmodes\. Overall, this replaces a large sparse GEVP with: \(i\) one network forward pass, \(ii\) anMM\-orthonormalization \(𝒪​\(N​m2\)\\mathcal\{O\}\(Nm^\{2\}\)\), \(iii\) a sparse–dense projection \(𝒪​\(nnz​\(L\)​m2\)\\mathcal\{O\}\(\\mathrm\{nnz\}\(L\)\\,m^\{2\}\)\), and \(iv\) a dense eigen\-decomposition \(𝒪​\(m3\)\\mathcal\{O\}\(m^\{3\}\)\)\. Assuming the discrete Laplacian sparsity is proportional toNN\(e\.g\., constant valence\), the total inference cost scales linearly withNN\.

#### Direct embedding\.

For downstream learning tasks \(e\.g\., classification or segmentation\), explicit eigenpairs are often unnecessary\. As a byproduct of predicting a low\-frequency subspace, we can use the raw basisF∈ℝN×mF\\in\\mathbb\{R\}^\{N\\times m\}directly as point\-wise features\. This usage is*Laplacian\-free*at inference time: it requires only the point coordinates \(and optionally mass weights\), bypassing both Laplacian assembly and Rayleigh–Ritz refinement, and incurring no additional cost beyond a single forward pass\. For simplicity, in these downstream tasks we setwi=1w\_\{i\}=1when sampling is close to uniform\.

## 4\.TRAINING AND ARCHITECTURE

In this section, we open the black box of NEO and describe how the neural operatorℱθ​\(X,w\)\\mathcal\{F\}\_\{\\theta\}\(X,w\)in[Sec\.3\.2](https://arxiv.org/html/2605.24390#S3.SS2)is learned in practice\. We first define the subspace loss that enables supervised learning from ambiguous spectral data\. We then detail the architecture of the neural operator, focusing on how the mass\-aware design facilitates zero\-shot generalization to arbitrary sampling densities\.

### 4\.1\.Distance Measure between Linear Spaces

We assume access to a collection of training tuples\{\(X,w,Uk\)\}\\\{\(X,w,U\_\{k\}\)\\\}, whereUkU\_\{k\}represents the ground\-truthMM\-orthonormal eigenbasis\. Our objective is to maximize the overlap between the predicted subspacespan​\(F\)\\mathrm\{span\}\(F\)and the ground\-truth eigenspacespan​\(Uk\)\\mathrm\{span\}\(U\_\{k\}\)\.

To this end, we minimize the residual energy of the ground\-truth modes after projection onto the predicted subspace\. LetY∈ℝN×mY\\in\\mathbb\{R\}^\{N\\times m\}\(m≥km\\geq k\) be theMM\-orthonormal basis obtained from the network outputFFviaMM\-orthogonalization\. TheMM\-orthogonal projector onto the predicted subspace is given byPY=Y​Y⊤​MP\_\{Y\}=YY^\{\\top\}M\. For a specific ground\-truth eigenvectoruju\_\{j\}, the portion of energy not captured by the predicted subspace is:

\(6\)rj=‖uj−PY​uj‖M2=1−‖Y⊤​M​uj‖22,r\_\{j\}=\\\|u\_\{j\}\-P\_\{Y\}u\_\{j\}\\\|\_\{M\}^\{2\}=1\-\\\|Y^\{\\top\}Mu\_\{j\}\\\|\_\{2\}^\{2\},where the simplification holds because‖uj‖M=1\\\|u\_\{j\}\\\|\_\{M\}=1\. Intuitively, minimizingrjr\_\{j\}forces the network to align its output span with the directions of maximum spectral energy, without constraining it to match any specific rotation or sign of the basis vectors\. Aggregating over allkkmodes, we define thespan lossas:

\(7\)ℒspan=1k​∑j=1k\(1−‖Y⊤​M​uj‖22\)=1−1k​‖Y⊤​M​Uk‖F2\.\\mathcal\{L\}\_\{\\text\{span\}\}=\\frac\{1\}\{k\}\\sum\_\{j=1\}^\{k\}\\left\(1\-\\\|Y^\{\\top\}Mu\_\{j\}\\\|\_\{2\}^\{2\}\\right\)=1\-\\frac\{1\}\{k\}\\\|Y^\{\\top\}MU\_\{k\}\\\|\_\{F\}^\{2\}\.Since‖Y⊤​M​uj‖22\\\|Y^\{\\top\}Mu\_\{j\}\\\|\_\{2\}^\{2\}measures the energy ofuju\_\{j\}projected ontospan​\(Y\)\\mathrm\{span\}\(Y\), the conditionℒspan=0\\mathcal\{L\}\_\{\\text\{span\}\}=0implies that each modeuju\_\{j\}is perfectly contained within the predicted span, i\.e\.,span​\(Uk\)⊆span​\(Y\)\\mathrm\{span\}\(U\_\{k\}\)\\subseteq\\mathrm\{span\}\(Y\)\. Moreover,ℒspan\\mathcal\{L\}\_\{\\text\{span\}\}is invariant under any orthogonal change of basisUk↦Uk​RU\_\{k\}\\mapsto U\_\{k\}RforR∈O​\(k\)R\\in O\(k\), which includes sign flips and rotations within \(near\-\)degenerate eigenspaces\.

While the span loss ensures coverage, the redundancym\>km\>kcan theoretically allow the raw fieldsFFto collapse into a rank\-deficient state\. To encourage full\-rank diversity in the raw predictions, we add a weak orthogonality regularizer:

\(8\)ℒortho=‖F⊤​M​F−Im‖F2\.\\mathcal\{L\}\_\{\\text\{ortho\}\}=\\left\\\|F^\{\\top\}MF\-I\_\{m\}\\right\\\|\_\{F\}^\{2\}\.Overall, we optimizeℒtotal=ℒspan\+α​ℒortho,\\mathcal\{L\}\_\{\\text\{total\}\}=\\mathcal\{L\}\_\{\\text\{span\}\}\+\\alpha\\,\\mathcal\{L\}\_\{\\text\{ortho\}\},withα\\alphakept fixed at10−310^\{\-3\}across all experiments\.

### 4\.2\.Mass\-Aware Neural Operator

To enable resolution\-invariant inference, we instantiate the neural operatorℱθ\\mathcal\{F\}\_\{\\theta\}with a latent\-bottleneck transformer backbone, adapting the Low\-Rank Spatial Attention \(LRSA\) architecture\(Yang et al\.,[2026](https://arxiv.org/html/2605.24390#bib.bib50)\)\. Given an input point cloudX∈ℝN×3X\\in\\mathbb\{R\}^\{N\\times 3\}and its associated mass weightsw∈ℝ\>0Nw\\in\\mathbb\{R\}^\{N\}\_\{\>0\}, we first construct point\-wise embeddings by lifting coordinates into a feature space:

\(9\)hi\(0\)=MLPlift​\(\[PE​\(xi\)\]\),H\(0\)=\[h1\(0\),…,hN\(0\)\]⊤∈ℝN×d,h\_\{i\}^\{\(0\)\}=\\mathrm\{MLP\}\_\{\\mathrm\{lift\}\}\(\[\\mathrm\{PE\}\(x\_\{i\}\)\]\),\\qquad H^\{\(0\)\}=\[h\_\{1\}^\{\(0\)\},\\ldots,h\_\{N\}^\{\(0\)\}\]^\{\\top\}\\in\\mathbb\{R\}^\{N\\times d\},wherePE​\(⋅\)\\mathrm\{PE\}\(\\cdot\)denotes a sinusoidal positional encoding andMLPlift\\mathrm\{MLP\}\_\{\\mathrm\{lift\}\}is applied independently to each point\. LRSA then updates the point embeddings throughllpre\-norm transformer blocks, where the global interaction is implemented by a low\-rank latent bottleneck \(Fig\.[4](https://arxiv.org/html/2605.24390#S4.F4)top\): point embeddings are softly aggregated into a compact set of latent tokens via cross\-attention, processed in the latent space, and broadcast back to all points via a second cross\-attention\. A final point\-wise linear projection maps the updated embeddings tommoutput scalar fields,F∈ℝN×mF\\in\\mathbb\{R\}^\{N\\times m\}, which serve as raw basis functions for the subsequent operations\. However, standard cross\-attention aggregation implicitly assumes uniform sampling; we explicitly address this via a*mass injection*mechanism, as derived next\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x4.png)Figure 4\.Neural Operator with Mass Injection\.The architecture \(top\) adopts a latent\-bottleneck design to efficiently process high\-resolution geometry via global tokens\. To ensure resolution invariance, the down\-projection \(bottom\) incorporates point masses via a logarithmic bias in the cross\-attention, approximating a continuous measure\-weighted integral\.Architecture diagram of the neural operator backbone\. The top part shows a latent\-bottleneck design that maps point features to a smaller set of global latent tokens and back\. The bottom part highlights the modified cross\-attention used in down\-projection, where point masses are injected as a bias so aggregation approximates a measure\-weighted integral and remains stable under non\-uniform sampling\.#### Mass injection in cross\-attention\.

In LRSA, the down\-projection aggregates point\-wise features into a compact set of latent vectors\{zi\}i=1r\\\{z\_\{i\}\\\}\_\{i=1\}^\{r\}\(withr≪Nr\\ll N\) via cross\-attention, forming a global latent representation\. LetQ∈ℝr×dQ\\in\\mathbb\{R\}^\{r\\times d\}be a set of learnable latent queries, and letK,V∈ℝN×dK,V\\in\\mathbb\{R\}^\{N\\times d\}be the keys and values derived from the point embeddingsHH\. With a positive kernelκ​\(qi,kj\)≐exp⁡\(⟨qi,kj⟩/dh\)\\kappa\(q\_\{i\},k\_\{j\}\)\\doteq\\exp\(\\langle q\_\{i\},k\_\{j\}\\rangle/\\sqrt\{d\_\{h\}\}\), standard attention produces a normalized unweighted average:

\(10\)zi=∑j=1Nκ​\(qi,kj\)∑n=1Nκ​\(qi,kn\)⏟αi​j​vj≈∫ℳκ​\(qi,k​\(x\)\)​v​\(x\)​𝑑μ​\(x\)∫ℳκ​\(qi,k​\(x\)\)​𝑑μ​\(x\)\.z\_\{i\}=\\sum\_\{j=1\}^\{N\}\\underbrace\{\\frac\{\\kappa\(q\_\{i\},k\_\{j\}\)\}\{\\sum\_\{n=1\}^\{N\}\\kappa\(q\_\{i\},k\_\{n\}\)\}\}\_\{\\alpha\_\{ij\}\}v\_\{j\}\\;\\approx\\;\\frac\{\\int\_\{\\mathcal\{M\}\}\\kappa\(q\_\{i\},k\(x\)\)v\(x\)d\\mu\(x\)\}\{\\int\_\{\\mathcal\{M\}\}\\kappa\(q\_\{i\},k\(x\)\)d\\mu\(x\)\}\.The approximation on the right holds*only when*the discretization is uniform, so that each sample represents a comparable area\.

For geometric inputs with highly non\-uniform sampling, Eq\. \([10](https://arxiv.org/html/2605.24390#S4.E10)\) can over\-emphasize densely sampled regions and weaken generalization across resolutions\. To correct this, we inject the point masseswwdirectly into the attention logits\. By addinglog⁡w\\log wto the logits, we transform the aggregation into a consistent*quadrature rule*:

\(11\)αi​j′\\displaystyle\\alpha^\{\\prime\}\_\{ij\}=softmax​\(qi​kj⊤dh\+log⁡wj\)j=κ​\(qi,kj\)⋅wj∑n=1Nκ​\(qi,kn\)⋅wn,\\displaystyle=\\mathrm\{softmax\}\\left\(\\frac\{q\_\{i\}k\_\{j\}^\{\\top\}\}\{\\sqrt\{d\_\{h\}\}\}\+\\log w\_\{j\}\\right\)\_\{j\}=\\frac\{\\kappa\(q\_\{i\},k\_\{j\}\)\\cdot w\_\{j\}\}\{\\sum\_\{n=1\}^\{N\}\\kappa\(q\_\{i\},k\_\{n\}\)\\cdot w\_\{n\}\},zi′\\displaystyle z\_\{i\}^\{\\prime\}=∑j=1Nαi​j′​vj≈∫ℳκ​\(qi,k​\(x\)\)​v​\(x\)​𝑑μ​\(x\)∫ℳκ​\(qi,k​\(x\)\)​𝑑μ​\(x\)\.\\displaystyle=\\sum\_\{j=1\}^\{N\}\\alpha^\{\\prime\}\_\{ij\}v\_\{j\}\\;\\approx\\;\\frac\{\\int\_\{\\mathcal\{M\}\}\\kappa\(q\_\{i\},k\(x\)\)v\(x\)d\\mu\(x\)\}\{\\int\_\{\\mathcal\{M\}\}\\kappa\(q\_\{i\},k\(x\)\)d\\mu\(x\)\}\.Sinceexp⁡\(s\+log⁡wj\)=exp⁡\(s\)​wj\\exp\(s\+\\log w\_\{j\}\)=\\exp\(s\)\\,w\_\{j\}, the mass term emerges as a multiplicative quadrature weight outside the exponential kernel, making the aggregation consistent with area\-weighted integration on non\-uniform point clouds\. Importantly, this modification is invariant to the global scale ofwwand exactly recovers the standard attention of Eq\. \([10](https://arxiv.org/html/2605.24390#S4.E10)\) when the mass weights are constant\. We apply this correction only in the down\-projection, as it is the stage that aggregates point samples overℳ\\mathcal\{M\}; the up\-projection attends from points to latent tokens, and latents are not samples ofℳ\\mathcal\{M\}and therefore do not admit geometric mass weights\.

## 5\.EXPERIMENTS

In this section, we evaluate NEO as a fast spectral solver\. Complete architecture details, training hyperparameters, and evaluation statistics are provided in Appendix A\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/images/Gallery-full-v2.jpg)Figure 5\.Visualization of NEO predictions on our OOD test dataset\.The collection encompasses a wide spectrum of semantic categories, ranging from organic creatures and classic graphics models to man\-made CAD parts\. It also includes diverse topologies and structural variations, from genus\-0 solids to high\-genus shapes with holes and thin structures, stressing robustness beyond the training distribution\.Gallery of out\-of\-distribution test shapes and NEO predictions\. The figure shows many models from diverse semantic categories, including animals, classic graphics models, and CAD\-like objects, together with representative predicted low\-frequency eigenfunctions\. The collection spans different topologies, including simple solids, shapes with holes, and thin structures\.### 5\.1\.Experimental Setup

#### Data and discretization\.

To obtain a large and diverse training corpus, we adopt the ShapeNetCore\(Chang et al\.,[2015](https://arxiv.org/html/2605.24390#bib.bib8)\)dataset \(∼\\sim51k models, 9:1 train/test\) with a*mesh\-level*split to prevent data leakage from different samplings of the same geometry during training\. Unless otherwise noted, we target the firstk=96k=96modes with redundancym=192m=192, and userobust\_laplacian\(Sharp and Crane,[2020](https://arxiv.org/html/2605.24390#bib.bib43)\)to construct the point cloud Laplacian and mass matrices\. To minimize data generation costs, training is performed exclusively on low\-resolution point clouds \(Ntrain=2048N\_\{\\text\{train\}\}\\\!=\\\!2048\), where ground\-truth eigenpairs are generated using ARPACK\. We further introduce the Thingi10k dataset \(∼\\sim10k models\) to assess cross\-dataset generalization, alongside a smaller OOD dataset of standard graphics models \(shown in[Fig\.5](https://arxiv.org/html/2605.24390#S5.F5)\)\. Due to the high cost of generating exact spectra at high resolutions, evaluations involvingN≫NtrainN\\gg N\_\{\\text\{train\}\}\(runtime scaling and robustness\) are exclusively performed on this OOD dataset\.

#### Baselines, hardware, and numerical precision\.

As a classical baseline, we adopt ARPACK’s implicitly restarted Lanczos algorithm with shift\-invert\(Lehoucq et al\.,[1998](https://arxiv.org/html/2605.24390#bib.bib19)\)\. ARPACK is evaluated in double precision on CPU \(Ryzen 9 9950X, 12 threads\), while NEO is evaluated on one RTX 3090 GPU with FP32/FP16 precision backbone inference and FP32 Rayleigh–Ritz\. GPU\-based iterative solvers were evaluated but performed worse than ARPACK in runtime in our setting, motivating our choice of ARPACK as the primary baseline\. All measurements exclude\(L,M\)\(L,M\)construction unless explicitly stated\.

### 5\.2\.Accuracy

#### Metrics

LetUkU\_\{k\}be the ground\-truth eigenvectors provided by ARPACK andYYbe the predicted basis\. We evaluate performance with three metrics\. \(i\)Span Loss \(ℰspan​\(i\)=1−‖Y⊤​M​ui‖22\\mathcal\{E\}\_\{\\mathrm\{span\}\}\(i\)=1\-\\\|Y^\{\\top\}Mu\_\{i\}\\\|\_\{2\}^\{2\}\)measures the residual energy ofuiu\_\{i\}outside the predicted span\. We report the mean over modesℰ¯span\\overline\{\\mathcal\{E\}\}\_\{\\mathrm\{span\}\}\. \(ii\)Eigenvector MSE \(ℰvec​\(i\)=‖ui−u^i‖M2\\mathcal\{E\}\_\{\\mathrm\{vec\}\}\(i\)=\\\|u\_\{i\}\-\\hat\{u\}\_\{i\}\\\|\_\{M\}^\{2\}\)measures the error of recovered vectorsu^i\\hat\{u\}\_\{i\}\. \(iii\)Eigenvalue Error \(ℰval=1k−1​∑i=1k−1\|λi−λ^i\|λi\\mathcal\{E\}\_\{\\mathrm\{val\}\}=\\frac\{1\}\{k\-1\}\\sum\_\{i=1\}^\{k\-1\}\\frac\{\|\\lambda\_\{i\}\-\\hat\{\\lambda\}\_\{i\}\|\}\{\\lambda\_\{i\}\}\)is the mean relative error of eigenvalues, excluding the trivial mode\. We reportℰvec\\mathcal\{E\}\_\{\\mathrm\{vec\}\}andℰval\\mathcal\{E\}\_\{\\mathrm\{val\}\}after the Rayleigh–Ritz procedure and mode reordering; in the presence of ambiguities,ℰspan\\mathcal\{E\}\_\{\\mathrm\{span\}\}is the primary ambiguity\-free indicator\.

Table 1\.Accuracy, generalization, and numerical stability across precision\.We report means±\\pmstd below\. SNet\-32/16 and T10k\-32/16 denote the ShapeNet test split and Thingi10k evaluated in FP32/FP16, respectively\.![Refer to caption](https://arxiv.org/html/2605.24390v1/x5.png)Figure 6\.Accuracy Distribution\.Left/Middle:Per\-mode Span Lossℰspan​\(i\)\\mathcal\{E\}\_\{\\mathrm\{span\}\}\(i\)and Eigenvector MSEℰvec​\(i\)\\mathcal\{E\}\_\{\\mathrm\{vec\}\}\(i\)\. Solid lines denote median; shaded bands denote IQR \(25–75%\)\.Right:Density of mean relative eigenvalue errorℰ¯val\\overline\{\\mathcal\{E\}\}\_\{\\mathrm\{val\}\}\.Three\-panel accuracy plot\. The left panel shows per\-mode span loss across the target spectrum, the middle panel shows per\-mode eigenvector mean squared error, and the right panel shows the distribution of average relative eigenvalue error\. Solid curves indicate medians and shaded regions indicate interquartile ranges\.
#### Subspace quality and eigenpair recovery\.

We evaluate the accuracy of NEO across both the ShapeNet test set and the Thingi10k dataset\. Fig\.[6](https://arxiv.org/html/2605.24390#S5.F6)\(Left\) shows that NEO achieves low mean span lossℰspan\\mathcal\{E\}\_\{\\mathrm\{span\}\}, indicating the basis successfully captures the target low\-frequency spectral energy\. Leveraging this subspace, Rayleigh–Ritz effectively recovers eigenpairs: the eigenvectors and eigenvalues exhibit low error \(Fig\.[6](https://arxiv.org/html/2605.24390#S5.F6), Middle/Right\), demonstrating consistent spectral approximation\. While errors naturally increase with frequency as the spectral gap narrows, NEO maintains robust performance across the target spectrum\. Furthermore, Table[1](https://arxiv.org/html/2605.24390#S5.T1)shows accuracy is preserved even when using half precision or when evaluated on the Thingi10k dataset, highlighting its numerical resilience and generalizability\.

#### Impact of subspace redundancy \(m\>km\>k\)\.

We ablate the predicted subspace dimensionmmwhile keeping the target rank fixed \(k=96k=96\)\. Table[2](https://arxiv.org/html/2605.24390#S5.T2)shows a trade\-off: a strict bottleneck \(m=km=k\) is too restrictive and often fails to capture spectral clusters near the cutoff\. Introducing moderate redundancy provides the necessary slack to span these modes, thereby enhancing the reconstruction quality of the eigenpairs\. Further increasingmmyields diminishing returns and increases the inference cost\. We therefore usem=192m=192as a practical default, balancing spectral coverage with inference latency\.

Table 2\.Effect of Subspace Redundancy \(mm\) on Accuracy and Efficiency\.Targetk=96k=96is fixed, and errors are reported as medians\.

### 5\.3\.Robustness

In practice, spectral pipelines often encounter distribution shifts, including changes in resolution, sampling density, and even the choice of discrete Laplacian\. We evaluate the robustness of NEO on the OOD dataset atN=64​kN=64\\mathrm\{k\}\(32×32\\timesthe training resolution\), with additional stress tests up toN=512​kN=512\\mathrm\{k\}as shown in Table[3](https://arxiv.org/html/2605.24390#S5.T3)\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x6.png)Figure 7\.Robustness to Non\-Uniform Sampling\.We visualize recovered eigenvectors \(10th\-12th modes\) under varying sampling density biases, ranging from highly non\-uniform \(top\) to uniform \(bottom\)\. While the mass\-agnostic baseline fails under biased sampling—often overfitting to high\-density regions \(high MSE\)—our mass\-aware approach consistently matches the ground truth\. This confirms that injecting mass weights effectively decouples the learned features from sampling density\. Notably, the mass\-agnostic variant performs well on uniform data \(bottom row\), consistent with our analysis in[Sec\.4\.2](https://arxiv.org/html/2605.24390#S4.SS2)that mass injection naturally reduces to standard attention when weights are constant\.Robustness comparison under different sampling densities\. Rows show increasingly uniform point sampling, and columns compare ground\-truth eigenvectors, a mass\-agnostic baseline, and the mass\-aware method\. The mass\-agnostic variant fails under strongly non\-uniform sampling by concentrating on dense regions, while the mass\-aware method remains visually close to ground truth across all rows\.![Refer to caption](https://arxiv.org/html/2605.24390v1/x7.png)Figure 8\.Resolution scaling and discretization transfer\.We compare NEO’s predicted eigenfunctions \(modes 2, 4, 32\) against Ground Truth across varying resolutions \(2k to 1\.6M points\) and distinct Laplacian discretizations \(Mesh vs\.kk\-NN\)\. Despite being trained only on coarse 2k point clouds, NEO demonstrates strong zero\-shot generalization\. While minor deviations become visible in higher frequencies \(e\.g\., 32nd mode\), the model correctly recovers the global nodal structures and qualitative patterns across all settings\. The runtime columns highlight the efficiency gain: while ARPACK’s cost grows super\-linearly, NEO achieves over100×100\\timesspeedup at 1\.6M points with near\-linear scaling behavior\.Large comparison figure for zero\-shot transfer across resolution and discretization\. Multiple columns show ground\-truth and predicted eigenfunctions for several modes at resolutions from 2 thousand to 1\.6 million points, and for different Laplacian constructions such as mesh and k\-nearest\-neighbor graph Laplacians\. Additional columns report runtimes, showing that NEO preserves global mode structure while being much faster than ARPACK at high resolution\.#### Mass\-awareness under non\-uniform sampling\.

To isolate the effect of mass injection \(Sec\.[4\.2](https://arxiv.org/html/2605.24390#S4.SS2)\), we resample test shapes with strongly non\-uniform densities and compare against a mass\-agnostic variant where the mass term is removed\. As shown in Table[3](https://arxiv.org/html/2605.24390#S5.T3)\(middle\), removing mass injection makes the model highly sensitive to sampling density and leads to severe degradation\. In contrast, the mass\-aware design substantially improves robustness under biased sampling, consistent with the interpretation of the down\-projection attention as a measure\-weighted aggregation as shown in Fig\.[7](https://arxiv.org/html/2605.24390#S5.F7)\.

#### Zero\-shot generalization to other discretizations\.

We apply NEO to distinct discrete operators \(mesh cotangent andkk\-NN\) without fine\-tuning\.[Fig\.8](https://arxiv.org/html/2605.24390#S5.F8)and Table[3](https://arxiv.org/html/2605.24390#S5.T3)\(bottom\) show that NEO maintains strong invariance: Span Loss remains in the low10−310^\{\-3\}regime for both, indicating that the subspace retains most of the target energy\. This confirms that NEO learns the underlying geometry rather than overfitting to discrete artifacts, allowing the Rayleigh–Ritz step to recover eigenpairs regardless of the Laplacian choice\. On mesh Laplacians, NEO achieves lowerℰ¯span\\overline\{\\mathcal\{E\}\}\_\{\\mathrm\{span\}\}than FastSpectrum \(1\.08×10−21\.08\\times 10^\{\-2\}\) with better speedup; full comparison in Appendix A\.

Table 3\.Robustness Analysis\.All evaluations are performed on the OOD set\.Top:Reference performance across resolutions\.Middle:Resilience to strong non\-uniform sampling\.Bottom:Transfer to Mesh/Graph operators\.

### 5\.4\.Efficiency

#### Runtime scaling\.

We report runtimes for recovering the firstk=96k=96eigenpairs at various resolutions \(Fig\.[9](https://arxiv.org/html/2605.24390#S5.F9)left\)\. To evaluate ARPACK fairly, we measure it at both machine tolerance and at relaxed tolerance where the residual matches NEO’s\. ARPACK exhibits super\-linear empirical scaling due to its internal sparse factorization costs, while NEO achieves near\-linear scaling\. AtN=512​kN=512\\mathrm\{k\}, NEO recovers 96 eigenpairs in0\.520\.52s \(FP16\), yielding88\.2×88\.2\\timesspeedup over accurate ARPACK and a70\.2×70\.2\\timesspeedup over the relaxed variant\. We evaluated several alternative solvers for comparison: LOBPCG with multigrid preconditioning failed to converge forN\>100​kN\>100\\mathrm\{k\}, while both Spectra and GPU\-accelerated SLEPc underperform ARPACK \(Table[4](https://arxiv.org/html/2605.24390#S5.T4)\), suggesting that NEO’s speedup is primarily algorithmic rather than purely hardware\-driven\. Additional solver configurations and extended timing sweeps are reported in Appendix A\.

Table 4\.Runtime Comparison of Eigensolvers\.Wall\-clock time for recovering the firstk=96k=96low\-frequency eigenpairs at different input resolutions\.
#### Runtime breakdown\.

Fig\.[9](https://arxiv.org/html/2605.24390#S5.F9)\(right\) decomposes latency into backbone forward, weighted QR, and Rayleigh–Ritz \(projection \+ dense eigendecomposition\) atN=32​kN=32\\mathrm\{k\}and512​k512\\mathrm\{k\}\. The dominant cost remains the neural backbone forward pass, which scales efficiently on GPUs\. This profile confirms that increasing redundancymmto improve subspace robustness incurs small latency overhead\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x8.png)Figure 9\.Runtime analysis fork=96k=96eigenpairs\.Left:runtime scaling\. ”Accurate” and ”Relaxed” indicate machine precision and matched NEO precision of ARPACK solver, respectively\. Fit values showt∝N1\.16t\\propto N^\{1\.16\}for ARPACK vs near\-lineart∝Nt\\propto Nfor NEO \(R2≥0\.999R^\{2\}\\geq 0\.999\)\.Right:NEO runtime breakdown\. Forward pass remains the dominant term across resolutions\.Two\-panel runtime figure\. The left panel plots wall\-clock runtime against the number of input points for NEO and ARPACK under accurate and relaxed stopping criteria, showing near\-linear scaling for NEO and super\-linear empirical scaling for ARPACK\. The right panel breaks NEO runtime into backbone forward pass, weighted QR, and Rayleigh–Ritz components at different resolutions, with the forward pass dominating total cost\.

## 6\.Applications

### 6\.1\.Spectral Geometry Processing

We assess whether NEO’s predicted low\-frequency spectrum is accurate enough for downstream spectral geometry processing tasks\.

#### Functional maps\.

We compute correspondences using the first 30 eigenpairs from NEO\. The pipeline utilizes Heat Kernel Signatures \(HKS\)\(Sun et al\.,[2009](https://arxiv.org/html/2605.24390#bib.bib45)\)for descriptors and ZoomOut\(Melzi et al\.,[2019](https://arxiv.org/html/2605.24390#bib.bib30)\)for spectral refinement\. We compare our results against maps generated using accurate spectra from a classical solver\. As shown in Figure[10](https://arxiv.org/html/2605.24390#S6.F10), NEO yields visually comparable dense correspondences and preserves semantic part transfer\. On the FAUST dataset, using NEO’s predicted eigenpairs increases the mean geodesic error from 0\.438 to 0\.543, which remains sufficient for downstream shape matching applications\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x9.png)Figure 10\.Functional map correspondence\.Qualitative transfer results between a cat and a lion using HKS descriptors derived from exact vs\. NEO eigenpairs\. While NEO yields a slightly higher geodesic error numerically, the visual correspondence remains semantically consistent and smooth\.Functional\-map correspondence example between a cat and a lion\. The figure compares part or color transfer obtained from exact eigenpairs and from NEO\-predicted eigenpairs, showing similar semantic alignment despite slightly higher numerical error for NEO\.![Refer to caption](https://arxiv.org/html/2605.24390v1/x10.png)Figure 11\.Fast Poisson solve in the heat\-based geodesic computation\.Left:Heat geodesic distances\.Middle & Right:Convergence of the Poisson step\. Using NEO’s predicted eigenpairs as the coarse level in an additive two\-level preconditioner \(Ours\) reduces both the iteration count \(∼3×\\sim 3\\times\) and total wall\-clock time compared to the standard ICPCG baseline\.Figure for accelerated geodesic computation\. The left panel visualizes heat\-based geodesic distances on a shape\. The middle and right panels plot convergence behavior of the Poisson solve, comparing a standard incomplete\-Cholesky preconditioned conjugate\-gradient baseline with a two\-level method that uses NEO eigenpairs as a coarse space\. The NEO\-based method reaches the target tolerance in fewer iterations and less time\.
#### Accelerated geodesic distances\.

We use NEO to accelerate heat\-based geodesic distances\(Crane et al\.,[2017](https://arxiv.org/html/2605.24390#bib.bib11)\)by speeding up the Poisson step\. We construct an*additive two\-level preconditioner*: the NEO\-recovered low\-frequency eigenvectors form a coarse space that effectively removes slow\-decaying error components, while standard incomplete Cholesky handles the fine scale\. Fig\.[11](https://arxiv.org/html/2605.24390#S6.F11)shows that this two\-level approach significantly reduces both iteration count and wall\-clock time compared to the ICPCG baseline, demonstrating that our predicted subspace is high\-quality enough to accelerate classical numerical solvers\.

### 6\.2\.Point Embeddings without Explicit Spectra

We investigate whether the raw predicted subspaceFFcan serve as a robust intrinsic embedding, bypassing the eigensolving stage\. We posit that low\-frequency invariant subspaces naturally encode global intrinsic information, serving as a powerful geometric prior to facilitate downstream geometric learning\.

#### Few\-shot classification\.

We evaluate on SHREC\-11\(Lian et al\.,[2011](https://arxiv.org/html/2605.24390#bib.bib25)\)\(official split, 30 classes\)\. Here, NEO functions as a*frozen*feature extractor: we fix the backbone and train only a lightweight PointNet\(Qi et al\.,[2017](https://arxiv.org/html/2605.24390#bib.bib37)\)head on top ofFF\. We compare against training from scratch using NeRF\-style positional encodings \(NeRF\-PE\) with the same head, and a much heavier Point Transformer\(Zhao et al\.,[2021](https://arxiv.org/html/2605.24390#bib.bib55)\)baseline\. Fig\.[13](https://arxiv.org/html/2605.24390#S6.F13)\(a\) reveals a striking advantage: frozen NEO features enable a simple PointNet to reach100% accuracy\(10\-shot\), significantly outperforming the Point Transformer\. This suggests that NEO provides an effective intrinsic representation, reducing the need for complex architectural designs\.

#### Segmentation\.

We further assess dense prediction on the human body segmentation task\(Maron et al\.,[2017](https://arxiv.org/html/2605.24390#bib.bib29)\)under a fixed training budget\. Using the same setup, we train a segmentation head on top of frozenFFversus learning from coordinates\. As shown in Fig\.[13](https://arxiv.org/html/2605.24390#S6.F13)\(b\), the spectral prior provided byFFyields faster convergence and higher mIoU under the limited training budget\. While end\-to\-end models eventually narrow the gap, NEO provides explicit global information that is otherwise hard to learn from local neighborhoods \(see Fig\.[12](https://arxiv.org/html/2605.24390#S6.F12)\), significantly reducing optimization difficulty\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x11.png)Figure 12\.Segmentation\.Visual comparison of segmentation results using our predicted basis \(FF\) versus sinusoidal positional encoding \(NeRF\-PE\)\.Qualitative segmentation comparison on human shapes\. The figure contrasts predictions obtained using NEO\-derived point features with predictions using sinusoidal positional encoding, showing that NEO better separates semantically distinct but spatially adjacent body parts\.![Refer to caption](https://arxiv.org/html/2605.24390v1/x12.png)\(a\)Few\-shot classification \(SHREC\-11\)\.Validation accuracy for 1\-shot, 3\-shot, and 10\-shot settings \(higher is better\)\.Few\-shot classification plot on SHREC\-11 showing validation accuracy for 1\-shot, 3\-shot, and 10\-shot settings\. Curves or bars compare a lightweight PointNet head using frozen NEO features against baselines such as NeRF\-style positional encoding and Point Transformer, with NEO achieving the highest accuracy, including perfect accuracy in the 10\-shot case\.
![Refer to caption](https://arxiv.org/html/2605.24390v1/x13.png)\(b\)Segmentation \(Human\)\.Mean Intersection over Union \(mIoU\) at 100 and 300 epochs \(higher is better\)\.Segmentation performance plot on the human dataset showing mean intersection over union at 100 and 300 training epochs\. The comparison indicates faster convergence and better accuracy when using frozen NEO features than when learning directly from positional encoding under the same budget\.

Figure 13\.NEO as an intrinsic point embedding\.We use the raw predicted subspaceFFdirectly as point features without explicit eigensolving, comparing it against NeRF\-style positional encoding \(NeRF\-PE\) baselines\.\(a\)For classification, frozen NEO features enable a lightweight PointNet to achieve perfect accuracy, surpassing heavier end\-to\-end baselines \(e\.g\., PointTransformer\)\.\(b\)For dense segmentation, NEO’s global spectral features significantly accelerate convergence compared to learning from spatial coordinates under the same training budget\.Two\-panel figure showing NEO as an intrinsic point embedding\. Panel \(a\) reports few\-shot classification accuracy on SHREC\-11 for several training\-shot regimes and shows that frozen NEO features outperform baseline point encodings and heavier end\-to\-end models\. Panel \(b\) reports segmentation mean intersection over union at two training checkpoints and shows that NEO features improve convergence and final accuracy\.

## 7\.Conclusion

In this work, we present NEO, a feed\-forward approach for low\-frequency LBO eigensolving on 3D point clouds\. NEO predicts the low\-frequency invariant eigenspace using a mass\-aware neural operator, enabling fast spectral analysis and zero\-shot transfer from low\-resolution training to higher\-resolution inputs in our experiments\. Across standard spectral pipelines, NEO provides consistent acceleration and exhibits near\-linear runtime scaling with respect to the number of points when recovering low\-frequency modes\.

#### Limitations and future work\.

As a learning\-based method, NEO prioritizes inference speed over machine precision and is not intended to replace exact numerical solvers\. Performance can degrade on higher modes and unseen thin structures \(see[Fig\.14](https://arxiv.org/html/2605.24390#S7.F14)\)\. Promising future directions include using the predicted subspace to precondition iterative solvers and incorporatingS​E​\(3\)SE\(3\)\-equivariance\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x14.png)Figure 14\.Failure case\.We identify two main limitations: \(1\) Frequency degradation: Accuracy drops notably for higher modes, where the spectral gap narrows and oscillations become harder to approximate\. \(2\) Unseen fine details: The model struggles with thin structures or complex topological details that differ significantly from the coarse geometry seen during training\.Failure\-case figure showing two representative limitations of NEO\. One example highlights degraded recovery for higher\-frequency modes with denser oscillations, and another highlights errors on shapes with thin or fine\-scale structures that differ from the coarse training geometry\.###### Acknowledgements\.

This work was supported by Laoshan Laboratory \(No\. LSKJ202300305\) and the National Natural Science Foundation of China \(62025207\)\. Tao Du acknowledges the support from Tsinghua University and the Shanghai Qi Zhi Institute Innovation Program\. We also thank the anonymous reviewers for their valuable feedback and suggestions\.

## References

- \(1\)
- Alon and Yahav \(2020\)Uri Alon and Eran Yahav\. 2020\.On the bottleneck of graph neural networks and its practical implications\.*arXiv preprint arXiv:2006\.05205*\(2020\)\.
- Aubry et al\.\(2011\)Mathieu Aubry, Ulrich Schlickewei, and Daniel Cremers\. 2011\.The wave kernel signature: A quantum mechanical approach to shape analysis\. In*2011 IEEE international conference on computer vision workshops \(ICCV workshops\)*\. IEEE, 1626–1633\.
- Belkin and Niyogi \(2003\)Mikhail Belkin and Partha Niyogi\. 2003\.Laplacian eigenmaps for dimensionality reduction and data representation\.*Neural computation*15, 6 \(2003\), 1373–1396\.
- Ben\-Shaul et al\.\(2023\)Ido Ben\-Shaul, Leah Bar, Dalia Fishelov, and Nir Sochen\. 2023\.Deep learning solution of the eigenvalue problem for differential operators\.*Neural Computation*35, 6 \(2023\), 1100–1134\.
- Bronstein et al\.\(2017\)Michael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst\. 2017\.Geometric deep learning: going beyond euclidean data\.*IEEE Signal Processing Magazine*34, 4 \(2017\), 18–42\.
- Bruna et al\.\(2013\)Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun\. 2013\.Spectral Networks and Locally Connected Networks on Graphs\.*CoRR*abs/1312\.6203 \(2013\)\.[https://api\.semanticscholar\.org/CorpusID:17682909](https://api.semanticscholar.org/CorpusID:17682909)
- Chang et al\.\(2015\)Angel X\. Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, and Fisher Yu\. 2015\.*ShapeNet: An Information\-Rich 3D Model Repository*\.Technical Report arXiv:1512\.03012 \[cs\.GR\]\. Stanford University — Princeton University — Toyota Technological Institute at Chicago\.
- Chang et al\.\(2024\)Yue Chang, Otman Benchekroun, Maurizio M Chiaramonte, Peter Yichen Chen, and Eitan Grinspun\. 2024\.Shape Space Spectra\.*arXiv preprint arXiv:2408\.10099*\(2024\)\.
- Coifman and Lafon \(2006\)Ronald R Coifman and Stéphane Lafon\. 2006\.Diffusion maps\.*Applied and computational harmonic analysis*21, 1 \(2006\), 5–30\.
- Crane et al\.\(2017\)Keenan Crane, Clarisse Weischedel, and Max Wardetzky\. 2017\.The heat method for distance computation\.*Commun\. ACM*60, 11 \(2017\), 90–99\.
- Davis and Kahan \(1970\)Chandler Davis and William Morton Kahan\. 1970\.The rotation of eigenvectors by a perturbation\. III\.*SIAM J\. Numer\. Anal\.*7, 1 \(1970\), 1–46\.
- Golub and Van Loan \(2013\)Gene H Golub and Charles F Van Loan\. 2013\.*Matrix computations*\.JHU press\.
- Hu et al\.\(2025\)Wenjie Hu, Sidun Liu, Peng Qiao, Zhenglun Sun, and Yong Dou\. 2025\.Transolver is a Linear Transformer: Revisiting Physics\-Attention through the Lens of Linear Attention\.*arXiv preprint arXiv:2511\.06294*\(2025\)\.
- Jin et al\.\(2022\)Xutong Jin, Sheng Li, Guoping Wang, and Dinesh Manocha\. 2022\.NeuralSound: Learning\-based Modal Sound Synthesis with Acoustic Transfer\.*ACM Transactions on Graphics \(SIGGRAPH 2022\)*\.[https://doi\.org/10\.1145/3528223\.3530184](https://doi.org/10.1145/3528223.3530184)
- Jordan et al\.\(2024\)Keller Jordan, Yuchen Jin, Vlado Boza, You Jiacheng, Franz Cesista, Laker Newhouse, and Jeremy Bernstein\. 2024\.Muon: An optimizer for hidden layers in neural networks\.[https://kellerjordan\.github\.io/posts/muon/](https://kellerjordan.github.io/posts/muon/)
- Knyazev \(2001\)Andrew V Knyazev\. 2001\.Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method\.*SIAM journal on scientific computing*23, 2 \(2001\), 517–541\.
- Kovachki et al\.\(2023\)Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar\. 2023\.Neural operator: Learning maps between function spaces with applications to pdes\.*Journal of Machine Learning Research*24, 89 \(2023\), 1–97\.
- Lehoucq et al\.\(1998\)Richard B Lehoucq, Danny C Sorensen, and Chao Yang\. 1998\.*ARPACK users’ guide: solution of large\-scale eigenvalue problems with implicitly restarted Arnoldi methods*\.SIAM\.
- Lévy \(2006\)Bruno Lévy\. 2006\.Laplace\-beltrami eigenfunctions towards an algorithm that” understands” geometry\. In*IEEE International Conference on Shape Modeling and Applications 2006 \(SMI’06\)*\. IEEE, 13–13\.
- Lévy and Zhang \(2010\)Bruno Lévy and Hao Zhang\. 2010\.Spectral mesh processing\.In*ACM SIGGRAPH 2010 Courses*\. 1–312\.
- Li et al\.\(2025\)H\. Li, J\. Sun, and Z\. Zhang\. 2025\.Deep Eigenspace Network and Its Application to Parametric Non\-selfadjoint Eigenvalue Problems\.arXiv:2512\.20058 \[math\.NA\][https://arxiv\.org/abs/2512\.20058](https://arxiv.org/abs/2512.20058)
- Li et al\.\(2020a\)Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar\. 2020a\.Fourier neural operator for parametric partial differential equations\.*arXiv preprint arXiv:2010\.08895*\(2020\)\.
- Li et al\.\(2020b\)Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Andrew Stuart, Kaushik Bhattacharya, and Anima Anandkumar\. 2020b\.Multipole graph neural operator for parametric partial differential equations\.*Advances in Neural Information Processing Systems*33 \(2020\), 6755–6766\.
- Lian et al\.\(2011\)Zhouhui Lian, Afzal Godil, Benjamin Bustos, Mohamed Daoudi, Jeroen Hermans, Shun Kawamura, Yukinori Kurita, Guillaume Lavoué, Hien Van Nguyen, Ryutarou Ohbuchi, et al\.2011\.SHREC’11 Track: Shape Retrieval on Non\-rigid 3D Watertight Meshes\.*3DOR@ eurographics*5 \(2011\)\.
- Liu et al\.\(2025\)Jingyuan Liu, Jianlin Su, Xingcheng Yao, Zhejun Jiang, Guokun Lai, Yulun Du, Yidao Qin, Weixin Xu, Enzhe Lu, Junjie Yan, Yanru Chen, Huabin Zheng, Yibo Liu, Shaowei Liu, Bohong Yin, Weiran He, Han Zhu, Yuzhi Wang, Jianzhou Wang, Mengnan Dong, Zheng Zhang, Yongsheng Kang, Hao Zhang, Xinran Xu, Yutao Zhang, Yuxin Wu, Xinyu Zhou, and Zhilin Yang\. 2025\.Muon is Scalable for LLM Training\.arXiv:2502\.16982 \[cs\.LG\][https://arxiv\.org/abs/2502\.16982](https://arxiv.org/abs/2502.16982)
- Loshchilov and Hutter \(2019\)Ilya Loshchilov and Frank Hutter\. 2019\.Decoupled Weight Decay Regularization\.arXiv:1711\.05101 \[cs\.LG\][https://arxiv\.org/abs/1711\.05101](https://arxiv.org/abs/1711.05101)
- Lu et al\.\(2021\)Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis\. 2021\.Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators\.*Nature machine intelligence*3, 3 \(2021\), 218–229\.
- Maron et al\.\(2017\)Haggai Maron, Meirav Galun, Noam Aigerman, Miri Trope, Nadav Dym, Ersin Yumer, Vladimir G Kim, and Yaron Lipman\. 2017\.Convolutional neural networks on surfaces via seamless toric covers\.*ACM Trans\. Graph\.*36, 4 \(2017\), 71–1\.
- Melzi et al\.\(2019\)Simone Melzi, Jing Ren, Emanuele Rodolà, Abhishek Sharma, Peter Wonka, and Maks Ovsjanikov\. 2019\.ZoomOut: spectral upsampling for efficient shape correspondence\.*ACM Trans\. Graph\.*38, 6, Article 155 \(Nov\. 2019\), 14 pages\.[doi:10\.1145/3355089\.3356524](https://doi.org/10.1145/3355089.3356524)
- Mildenhall et al\.\(2020\)Ben Mildenhall, Pratul P\. Srinivasan, Matthew Tancik, Jonathan T\. Barron, Ravi Ramamoorthi, and Ren Ng\. 2020\.NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis\. In*ECCV*\.
- Nasikun et al\.\(2018\)Ahmad Nasikun, Christopher Brandt, and Klaus Hildebrandt\. 2018\.Fast Approximation of Laplace\-–Beltrami Eigenproblems\.*Computer Graphics Forum*37, 5 \(2018\)\.
- Ovsjanikov et al\.\(2012\)Maks Ovsjanikov, Mirela Ben\-Chen, Justin Solomon, Adrian Butscher, and Leonidas Guibas\. 2012\.Functional maps: a flexible representation of maps between shapes\.*ACM Transactions on Graphics \(ToG\)*31, 4 \(2012\), 1–11\.
- Pang et al\.\(2024\)Bo Pang, Zhongtian Zheng, Yilong Li, Guoping Wang, and Peng\-Shuai Wang\. 2024\.Neural Laplacian Operator for 3D Point Clouds\.*ACM Transactions on Graphics \(SIGGRAPH Asia\)*\(2024\)\.
- Pentland and Williams \(1989\)Alex Pentland and John Williams\. 1989\.Good vibrations: Modal dynamics for graphics and animation\. In*Proceedings of the 16th annual conference on Computer graphics and interactive techniques*\. 215–222\.
- Pfaff et al\.\(2020\)Tobias Pfaff, Meire Fortunato, Alvaro Sanchez\-Gonzalez, and Peter Battaglia\. 2020\.Learning mesh\-based simulation with graph networks\. In*International conference on learning representations*\.
- Qi et al\.\(2017\)Charles Ruizhongtai Qi, Li Yi, Hao Su, and Leonidas J Guibas\. 2017\.Pointnet\+\+: Deep hierarchical feature learning on point sets in a metric space\.*Advances in neural information processing systems*30 \(2017\)\.
- Rahaman et al\.\(2019\)Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville\. 2019\.On the Spectral Bias of Neural Networks\. In*Proceedings of the 36th International Conference on Machine Learning**\(Proceedings of Machine Learning Research, Vol\. 97\)*, Kamalika Chaudhuri and Ruslan Salakhutdinov \(Eds\.\)\. PMLR, 5301–5310\.
- Raissi et al\.\(2019\)Maziar Raissi, Paris Perdikaris, and George E Karniadakis\. 2019\.Physics\-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations\.*Journal of Computational physics*378 \(2019\), 686–707\.
- Ren et al\.\(2018\)Jing Ren, Adrien Poulenard, Peter Wonka, and Maks Ovsjanikov\. 2018\.Continuous and orientation\-preserving correspondences via functional maps\.*ACM Transactions on Graphics \(ToG\)*37, 6 \(2018\), 1–16\.
- Reuter et al\.\(2006\)Martin Reuter, Franz\-Erich Wolter, and Niklas Peinecke\. 2006\.Laplace–Beltrami spectra as ‘Shape\-DNA’of surfaces and solids\.*Computer\-Aided Design*38, 4 \(2006\), 342–366\.
- Sharp et al\.\(2022\)Nicholas Sharp, Souhaib Attaiki, Keenan Crane, and Maks Ovsjanikov\. 2022\.Diffusionnet: Discretization agnostic learning on surfaces\.*ACM Transactions on Graphics \(TOG\)*41, 3 \(2022\), 1–16\.
- Sharp and Crane \(2020\)Nicholas Sharp and Keenan Crane\. 2020\.A laplacian for nonmanifold triangle meshes\. In*Computer Graphics Forum*, Vol\. 39\. Wiley Online Library, 69–80\.
- Smirnov and Solomon \(2021\)Dmitriy Smirnov and Justin M\. Solomon\. 2021\.HodgeNet\.*ACM Transactions on Graphics \(TOG\)*40 \(2021\), 1 – 11\.[https://api\.semanticscholar\.org/CorpusID:233407503](https://api.semanticscholar.org/CorpusID:233407503)
- Sun et al\.\(2009\)Jian Sun, Maks Ovsjanikov, and Leonidas Guibas\. 2009\.A concise and provably informative multi\-scale signature based on heat diffusion\. In*Computer graphics forum*, Vol\. 28\. Wiley Online Library, 1383–1392\.
- Trappolini et al\.\(2021\)Giovanni Trappolini, Luca Cosmo, Luca Moschella, Riccardo Marin, Simone Melzi, and Emanuele Rodolà\. 2021\.Shape Registration in the Time of Transformers\. In*Advances in Neural Information Processing Systems*, M\. Ranzato, A\. Beygelzimer, Y\. Dauphin, P\.S\. Liang, and J\. Wortman Vaughan \(Eds\.\), Vol\. 34\. Curran Associates, Inc\., 5731–5744\.[https://proceedings\.neurips\.cc/paper\_files/paper/2021/file/2d3d9d5373f378108cdbd30a3c52bd3e\-Paper\.pdf](https://proceedings.neurips.cc/paper_files/paper/2021/file/2d3d9d5373f378108cdbd30a3c52bd3e-Paper.pdf)
- Vallet and Lévy \(2008\)Bruno Vallet and Bruno Lévy\. 2008\.Spectral geometry processing with manifold harmonics\. In*Computer Graphics Forum*, Vol\. 27\. Wiley Online Library, 251–260\.
- Wang et al\.\(2025\)Jianyuan Wang, Minghao Chen, Nikita Karaev, Andrea Vedaldi, Christian Rupprecht, and David Novotny\. 2025\.VGGT: Visual Geometry Grounded Transformer\. In*Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*\.
- Wu et al\.\(2024\)Haixu Wu, Huakun Luo, Haowen Wang, Jianmin Wang, and Mingsheng Long\. 2024\.Transolver: A fast transformer solver for pdes on general geometries\.*arXiv preprint arXiv:2402\.02366*\(2024\)\.
- Yang et al\.\(2026\)Zherui Yang, Haiyang Xin, Tao Du, and Ligang Liu\. 2026\.Simple yet Effective: Low\-Rank Spatial Attention for Neural Operators\.arXiv:2604\.03582 \[cs\.LG\][https://arxiv\.org/abs/2604\.03582](https://arxiv.org/abs/2604.03582)
- Yi et al\.\(2016\)L\. Yi, Hao Su, Xingwen Guo, and Leonidas J\. Guibas\. 2016\.SyncSpecCNN: Synchronized Spectral CNN for 3D Shape Segmentation\.*2017 IEEE Conference on Computer Vision and Pattern Recognition \(CVPR\)*\(2016\), 6584–6592\.[https://api\.semanticscholar\.org/CorpusID:14487589](https://api.semanticscholar.org/CorpusID:14487589)
- Yu et al\.\(2018\)Bing Yu et al\.2018\.The deep Ritz method: a deep learning\-based numerical algorithm for solving variational problems\.*Communications in Mathematics and Statistics*6, 1 \(2018\), 1–12\.
- Zeng et al\.\(2025\)Chong Zeng, Yue Dong, Pieter Peers, Hongzhi Wu, and Xin Tong\. 2025\.RenderFormer: Transformer\-based Neural Rendering of Triangle Meshes with Global Illumination\. In*ACM SIGGRAPH 2025 Conference Papers*\.
- Zhang et al\.\(2007\)Hao Zhang, Oliver van Kaick, and Ramsay Dyer\. 2007\.Spectral methods for mesh processing and analysis\. In*Proceedings of Eurographics State\-of\-the\-art Report*, Vol\. 122\. Eurographics Association Prague, 1865–1894\.
- Zhao et al\.\(2021\)Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip HS Torr, and Vladlen Koltun\. 2021\.Point transformer\. In*Proceedings of the IEEE/CVF International Conference on Computer Vision*\. 16259–16268\.

## 8\.Full Experiment Details

In this section, we provide the complete hyperparameter settings, training dynamics, and data generation details to ensure reproducibility\.

### 8\.1\.Model Architecture and Configurations

#### Architecture Overview\.

The NEO backbone is instantiated using a Low\-Rank Spatial Attention \(LRSA\) architecture\. The pipeline begins with a coordinate lifting stage, followed by a stack of mass\-aware operator blocks, and concludes with a linear projection to the output subspace functions\.

- •Input Encoding:We map input coordinatesX∈ℝN×3X\\in\\mathbb\{R\}^\{N\\times 3\}to high\-dimensional features using Sinusoidal Positional Encoding with frequencies logarithmically spaced in the range\[2−2,26\]\[2^\{\-2\},2^\{6\}\]\. These are mapped to the model dimensionDDvia a linear layer with SiLU activation\.
- •Latent\-Bottleneck Blocks:The core network consists ofLLlayers of LRSA blocks\. Each block maintains a set ofrrlearnable latent tokens that act as a bottleneck for global information propagation\. The block operates in three phases: \(1\) Down\-Projection aggregates point features into the latent tokens via Cross\-Attention; \(2\) Latent Processing mixes information among tokens via Self\-Attention and Gated Feed\-Forward Networks \(FFN\); \(3\) Up\-Projection broadcasts the refined spectral information back to the point features via Cross\-Attention\.
- •Mass Injection: To ensure robustness to non\-uniform sampling, we explicitly modify the Down\-Projection attention mechanism as described in the paper\.

#### Model Variants\.

Table[5](https://arxiv.org/html/2605.24390#S8.T5)details the specific configurations used in our experiments\. NEO\-Base \(5\.3M parameters\) is our primary model\. NEO\-Large is a deeper and wider variant used to demonstrate scaling laws\. NEO\-Deep validates the sufficiency of our default depth\. NEO\-Deep shows that additional depth can further improve subspace coverage, but NEO\-Base offers the best accuracy–efficiency trade\-off used throughout the paper\.

Table 5\.NEO Model Configurations\.All models targetk=96k=96eigenpairs\. Training time is measured on 4×\\timesA100 GPUs\.

### 8\.2\.Training and Data Pipeline

#### Data Generation\.

Our training data is derived from the ShapeNetCore dataset\. We generate the dataset by sampling each mesh twice independently withN=2048N=2048points using surface\-area\-weighted sampling\. Ground truth spectra are computed offline using ARPACK\. This preprocessing step involves solving spectra for a large collection of shapes, consuming approximately 256 CPU\-hours distributed across 64 cores\. We employ a strict 90/10 split for training and validation, ensuring no geometry leakage\.

#### Optimization Setup\.

We optimize the model using the Muon\(Liu et al\.,[2025](https://arxiv.org/html/2605.24390#bib.bib26); Jordan et al\.,[2024](https://arxiv.org/html/2605.24390#bib.bib16)\)optimizer with momentum betas\(0\.9,0\.99\)\(0\.9,0\.99\), a weight decay of0\.040\.04,400400epochs, and a maximum learning rate of8×10−48\\times 10^\{\-4\}\. To stabilize training dynamics, we apply gradient clipping with a threshold of10\.010\.0\. For learning rate scheduling, we adopt a Warmup\-Stable\-Decay \(WSD\) strategy: \(i\) Warmup: Linearly increase LR for the first 10% of steps; \(ii\) Stable: Maintain constant LR for 70% of steps; \(iii\) Decay: Exponentially decay to a final LR of1×10−61\\times 10^\{\-6\}over the final 20%\. We use half\-precision \(float16\) for the model forward pass\. Although bfloat16 may be another common choice, we found that due to the low accuracy of bfloat16, the final training result is worse than float16\. In our experiments, AdamW\(Loshchilov and Hutter,[2019](https://arxiv.org/html/2605.24390#bib.bib27)\)or OneCycle scheduler does not work better than Muon and WSD\.

#### Normalization

Regarding data preprocessing, we normalize all input point clouds to fit within the unit cube\[−1,1\]3\[\-1,1\]^\{3\}to ensure numerical stability across varying object scales\. During training, we apply online data augmentation by performing random global rotations \(sampled uniformly fromS​O​\(3\)SO\(3\)\) on the input points\. This strategy prevents overfitting to canonical poses and encourages the neural operator to learn rotation\-robust geometric features\.

#### Hardware and Computation\.

Models are trained on a single node equipped with 4×\\timesNVIDIA A100 \(40GB\) GPUs\. We utilize a per\-GPU batch size of 200, resulting in a global effective batch size of 800\. Under these settings, NEO\-Base converges in approximately 8\.6 hours\. Scaling up the model increases costs linearly with depth: NEO\-Deep \(L=8L=8\) requires roughly 17 hours, while the wider NEO\-Large \(L=6L=6\) requires approximately 36 hours\.

## 9\.Full Experiment Results

### 9\.1\.NEO as an Eigen Solver

To strictly isolate the impact of architectural design from model capacity, we evaluate the PointNet\+\+\(Qi et al\.,[2017](https://arxiv.org/html/2605.24390#bib.bib37)\)and Point Transformer\(Zhao et al\.,[2021](https://arxiv.org/html/2605.24390#bib.bib55)\)baselines under a rigorous iso\-parameter setting\. Specifically, we scale both baselines to match the parameter budget of NEO\-Base \(≈5\.3​M\\approx 5\.3\\mathrm\{M\}\)\. Furthermore, to ensure a fair comparison, all variants are trained under a unified protocol, employing identical supervision \(loss functions and regularization\), optimization schedules, and batch configurations\. This guarantees that any observed performance gaps are attributable solely to the architectural inductive biases\.

As detailed in Table[6](https://arxiv.org/html/2605.24390#S9.T6), while all models achieve reasonable convergence on the training resolution \(2k\), their behaviors diverge sharply under distribution shifts\. PointNet\+\+ maintains moderate robustness but suffers from high latency due to its hierarchical sampling and grouping operations\. Crucially, Point Transformer exhibits catastrophic degradation when scaling to 64k points \(ℰsub\\mathcal\{E\}\_\{\\mathrm\{sub\}\}surges to4\.92​e−14\.92\\mathrm\{e\}\{\-\}1\), indicating that standard local\-attention mechanisms struggle to generalize zero\-shot to density regimes unseen during training\. In contrast, NEO demonstrates superior stability across all settings, primarily due to its mass\-aware global attention mechanism which effectively approximates resolution\-invariant integral operators\. Furthermore, our architecture is significantly more efficient, delivering a4\.8×4\.8\\timesto8\.9×8\.9\\timesspeedup over the baselines\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x15.png)Figure 15\.Error distribution across OOD shapes ranked by Span Loss \(ℰsub\\mathcal\{E\}\_\{\\mathrm\{sub\}\}\)\.We sort the OOD test shapes \(N=128​kN=128\\mathrm\{k\}\) from highest error \(left\) to lowest \(right\)\. The results highlight a clear dependency on geometric complexity: performance degrades primarily on models with intricate high\-frequency surface details or non\-trivial topology, while remaining robust for smooth manifolds\.Table 6\.Backbone Comparison details\.Performance is evaluated using the Mean Span Lossℰsub\\mathcal\{E\}\_\{\\mathrm\{sub\}\}\. Baselines are parameter\-matched to NEO\-Base\. “Test\(2k\)” evaluates the in\-distribution test split at the training resolution; “64k” and “Non\-Uniform” evaluate the OOD set\.TsampleT\_\{\\text\{sample\}\}measures end\-to\-end backbone forward time \(FP16\) on a single RTX 3090\.#### Comparison with FastSpectrum\.

We further compare NEO with FastSpectrum\(Nasikun et al\.,[2018](https://arxiv.org/html/2605.24390#bib.bib32)\), a mesh\-based approximate eigensolver\. Using the authors’ implementation with matched subspace dimensions \(m=192m=192\) and evaluation settings on our mesh datasets, we observed that while FastSpectrum offers a14×14\\timesspeedup over ARPACK \(0\.21s vs\. 2\.93s\), NEO achieves a much more significant73×73\\timesspeedup \(0\.04s\)\. As detailed in Table[7](https://arxiv.org/html/2605.24390#S9.T7), NEO also yields a lower mean Span Loss \(8\.60​e−38\.60\\mathrm\{e\}\{\-\}3compared to FastSpectrum’s1\.08​e−21\.08\\mathrm\{e\}\{\-\}2\) and lower Eigenvector MSE \(1\.97​e−11\.97\\mathrm\{e\}\{\-\}1vs\.2\.28​e−12\.28\\mathrm\{e\}\{\-\}1\), although with slightly higher variance \(1\.17​e−11\.17\\mathrm\{e\}\{\-\}1vs\.1\.08​e−11\.08\\mathrm\{e\}\{\-\}1\)\. This confirms that despite not assuming mesh connectivity, NEO predicts a high\-quality spectral subspace and offers competitive or superior performance to specialized mesh\-based approximation techniques\.

Table 7\.Comparison with FastSpectrum\.Evaluation on mesh datasets targetingk=96k=96eigenpairs with subspace dimensionm=192m=192\. NEO yields lower span loss and eigenvector MSE but marginally higher variance\.
#### Baselines Setting and Scalability Evaluation\.

To comprehensively evaluate the computational efficiency and scalability of traditional eigensolvers across varying problem sizes, we benchmarked multiple industry\-standard libraries: ARPACK, SPECTRA, and SLEPc \(CUDA accelerated\)\. We tested these solvers on matrices corresponding to point cloud discretizations ranging fromN=8,192N=8,192to over1\.51\.5million points \(N=1,572,864N=1,572,864\)\. For SLEPc, we evaluated both the Shift\-Invert and LOBPCG strategies\. The experiments demonstrated that while traditional solvers like ARPACK and SPECTRA scale reasonably well at lower resolutions, their computational cost grows significantly for large\-scale geometries\. Notably, the SLEPc LOBPCG solver failed to converge or encountered memory/numerical issues under this specific configuration\. Table[8](https://arxiv.org/html/2605.24390#S9.T8)summarizes the wall\-clock times for each method\.

Table 8\.Eigensolver Timing Comparison \(seconds\)\.The evaluated time for ARPACK, SPECTRA, and SLEPc \(Shift\-Invert\) across different problem sizesNN\. SLEPc with LOBPCG failed to evaluate under these settings\.
#### Detailed Statistical Analysis\.

In Table[9](https://arxiv.org/html/2605.24390#S9.T9), we explicitly report the median and interquartile range \(IQR\) alongside the mean errors to analyze the error distribution\. Across all metrics, the median error is consistently lower than the mean \(e\.g\., medianℰ¯sub\\overline\{\\mathcal\{E\}\}\_\{\\mathrm\{sub\}\}is15%15\\%lower than the mean\)\. This discrepancy indicates a right\-skewed error distribution, suggesting that NEO solves the vast majority of geometries with high precision, while the mean is slightly impacted by a small tail of outliers—likely shapes with severe non\-manifold artifacts or disconnected components that challenge the definition of the ground\-truth Laplacian itself\. The tight IQR confirms the reliability of our method across the diverse shape categories in ShapeNetCore\.

Table 9\.Accuracy Metrics on ShapeNet Test Split \(k=96k=96\)\.We report the mean, median, and interquartile range \(IQR\) across the test set\. Span Loss is consistently low \(10−310^\{\-3\}\), indicating robust coverage of the target low\-frequency eigenspace\.
#### Zero\-shot Scalability and Precision Analysis\.

To validate the resolution scalability of our operator, we evaluate a single NEO model, which was trained strictly onNtrain≈2​kN\_\{\\text\{train\}\}\\approx 2\\mathrm\{k\}, across OOD test point clouds ranging fromN=512N=512to1\.5×1061\.5\\times 10^\{6\}\. As shown in Figure[16](https://arxiv.org/html/2605.24390#S9.F16), the method exhibits strong zero\-shot generalization\. Interestingly, the sweet spot for performance is observed atN=8​kN=8\\mathrm\{k\}, where both span loss and eigenvector MSE reach their minimum, outperforming even the native training resolution \(2​k2\\mathrm\{k\}\)\. This suggests that the learned operator captures the underlying continuous spectral geometry better when the discretization granularity is slightly finer than training, likely because the discrete mass approximation becomes more accurate\. Performance degrades gracefully asNNincreases beyond10510^\{5\}, primarily due to the vast domain shift from the training distribution\. Regarding numerical precision, using FP16 for the neural backbone has a negligible impact on the predicted subspace \(ℰsub\\mathcal\{E\}\_\{\\mathrm\{sub\}\}\) and eigenvectors \(ℰvec\\mathcal\{E\}\_\{\\mathrm\{vec\}\}\)\. However, the eigenvalue error \(ℰval\\mathcal\{E\}\_\{\\mathrm\{val\}\}\) is more sensitive to numerical precision at very high resolutions \(N\>128​kN\>128\\mathrm\{k\}\)\. This behavior indicates that limited precision in the backbone outputFFintroduces high\-frequency noise that is orthogonal to the target low\-frequency subspace \(keepingℰvec\\mathcal\{E\}\_\{\\mathrm\{vec\}\}low\) but slightly perturbs the energy estimates in the projected Rayleigh quotient \(inflatingℰval\\mathcal\{E\}\_\{\\mathrm\{val\}\}\), particularly as the conditioning of the Laplacian degrades with increasingNN\.

![Refer to caption](https://arxiv.org/html/2605.24390v1/x16.png)Figure 16\.Zero\-shot resolution scalability and numerical precision\.We evaluate NEO, trained exclusively on 2k\-point clouds, across a wide range of test resolutions \(N=512N=512to1\.5​M1\.5\\mathrm\{M\}\)\. Solid lines denote the median error; shaded regions correspond to the interquartile range \(25–75%\)\.Left/Middle:The subspace span loss \(ℰsub\\mathcal\{E\}\_\{\\mathrm\{sub\}\}\) and eigenvector MSE \(ℰvec\\mathcal\{E\}\_\{\\mathrm\{vec\}\}\) remain stable across resolutions, with the lowest error surprisingly occurring at8​k8\\mathrm\{k\}points rather than the training resolution\.Right:While eigenvalues \(ℰval\\mathcal\{E\}\_\{\\mathrm\{val\}\}\) are more sensitive to the precision gap at extremely high resolutions \(\>128​k\>128\\mathrm\{k\}\), both eigenvector quality and subspace coverage are largely unaffected by the use of half\-precision \(FP16\) inference\.
#### Failure Case Analysis

Figure[15](https://arxiv.org/html/2605.24390#S9.F15)visualizes the error distribution across the OOD test set at high resolution \(N=128​kN=128\\text\{k\}\), revealing a correlation between spectral accuracy and geometric complexity\. The most challenging cases correspond to models rich in intricate surface details \(e\.g\.,*Dragon*,*Lucy*\), where dense micro\-structures significantly perturb local area measures\. Although our mass\-aware mechanism mitigates global bias, capturing the aggregate spectral influence of these features remains difficult when they induce high\-frequency localized modes\. Conversely, NEO achieves near\-perfect reconstruction on piecewise smooth surfaces and simple topology \(e\.g\.,*Bunny*,*Fandisk*\), confirming that the learned operator aligns well with the global low\-frequency geometry of typical manifolds\.

#### Accelerated Poisson Solver via Deflation

We provide the algorithm implemented in Section 6\.1\. \(Accelerated geodesic distances\)\. The ICPCG baseline and ICPCG with deflation using NEO’s predicted eigenpairs as presented in Alg\.[1](https://arxiv.org/html/2605.24390#algorithm1)and Alg\.[2](https://arxiv.org/html/2605.24390#algorithm2)\.

Input:System matrix

𝐀\\mathbf\{A\}, RHS

𝐛\\mathbf\{b\}, max iterations

Km​a​xK\_\{max\}, tolerance

ϵ\\epsilon
Output:Solution

𝐱\\mathbf\{x\}
//Precompute Incomplete Cholesky Factorization

𝐋←IncompleteCholesky​\(𝐀\)\\mathbf\{L\}\\leftarrow\\text\{IncompleteCholesky\}\(\\mathbf\{A\}\);

Define preconditioner solve

ℳ−1​\(𝐯\)≡\(𝐋𝐋T\)−1​𝐯\\mathcal\{M\}^\{\-1\}\(\\mathbf\{v\}\)\\equiv\(\\mathbf\{L\}\\mathbf\{L\}^\{T\}\)^\{\-1\}\\mathbf\{v\};

1ex

𝐱0←𝟎\\mathbf\{x\}\_\{0\}\\leftarrow\\mathbf\{0\};

𝐫0←𝐛−𝐀𝐱0\\mathbf\{r\}\_\{0\}\\leftarrow\\mathbf\{b\}\-\\mathbf\{A\}\\mathbf\{x\}\_\{0\};

𝐳0←ℳ−1​\(𝐫0\)\\mathbf\{z\}\_\{0\}\\leftarrow\\mathcal\{M\}^\{\-1\}\(\\mathbf\{r\}\_\{0\}\);

//Apply ICHOL preconditioner

𝐩0←𝐳0\\mathbf\{p\}\_\{0\}\\leftarrow\\mathbf\{z\}\_\{0\};

1ex

for*k=0k=0toKm​a​xK\_\{max\}*do

αk←\(𝐫kT​𝐳k\)/\(𝐩kT​𝐀𝐩k\)\\alpha\_\{k\}\\leftarrow\(\\mathbf\{r\}\_\{k\}^\{T\}\\mathbf\{z\}\_\{k\}\)/\(\\mathbf\{p\}\_\{k\}^\{T\}\\mathbf\{A\}\\mathbf\{p\}\_\{k\}\);

𝐱k\+1←𝐱k\+αk​𝐩k\\mathbf\{x\}\_\{k\+1\}\\leftarrow\\mathbf\{x\}\_\{k\}\+\\alpha\_\{k\}\\mathbf\{p\}\_\{k\};

𝐫k\+1←𝐫k−αk​𝐀𝐩k\\mathbf\{r\}\_\{k\+1\}\\leftarrow\\mathbf\{r\}\_\{k\}\-\\alpha\_\{k\}\\mathbf\{A\}\\mathbf\{p\}\_\{k\};

if*‖𝐫k\+1‖<ϵ\\\|\\mathbf\{r\}\_\{k\+1\}\\\|<\\epsilon*then

break;

end if

𝐳k\+1←ℳ−1​\(𝐫k\+1\)\\mathbf\{z\}\_\{k\+1\}\\leftarrow\\mathcal\{M\}^\{\-1\}\(\\mathbf\{r\}\_\{k\+1\}\);

//Apply ICHOL preconditioner

βk←\(𝐫k\+1T​𝐳k\+1\)/\(𝐫kT​𝐳k\)\\beta\_\{k\}\\leftarrow\(\\mathbf\{r\}\_\{k\+1\}^\{T\}\\mathbf\{z\}\_\{k\+1\}\)/\(\\mathbf\{r\}\_\{k\}^\{T\}\\mathbf\{z\}\_\{k\}\);

𝐩k\+1←𝐳k\+1\+βk​𝐩k\\mathbf\{p\}\_\{k\+1\}\\leftarrow\\mathbf\{z\}\_\{k\+1\}\+\\beta\_\{k\}\\mathbf\{p\}\_\{k\};

end for

return

𝐱k\+1\\mathbf\{x\}\_\{k\+1\};

ALGORITHM 1Incomplete Cholesky PCG \(ICPCG\)Input:System matrix

𝐀\\mathbf\{A\}, RHS

𝐛\\mathbf\{b\}, Basis vectors

𝐘∈ℝN×k\\mathbf\{Y\}\\in\\mathbb\{R\}^\{N\\times k\}
from NEOOutput:Solution

𝐱\\mathbf\{x\}
//1\. Setup Deflation Space

𝐄←𝐘T​𝐀𝐘\\mathbf\{E\}\\leftarrow\\mathbf\{Y\}^\{T\}\\mathbf\{A\}\\mathbf\{Y\};

//Compute coarse matrixk×kk\\times k

𝐄−1←Inverse​\(𝐄\)\\mathbf\{E\}^\{\-1\}\\leftarrow\\text\{Inverse\}\(\\mathbf\{E\}\);

//Dense inverse for smallkk

Define coarse solve

𝒞​\(𝐯\)≡𝐘​\(𝐄−1​\(𝐘T​𝐯\)\)\\mathcal\{C\}\(\\mathbf\{v\}\)\\equiv\\mathbf\{Y\}\(\\mathbf\{E\}^\{\-1\}\(\\mathbf\{Y\}^\{T\}\\mathbf\{v\}\)\);

//2\. Setup Base Preconditioner \(High\-Freq\)

𝐋←IncompleteCholesky​\(𝐀\)\\mathbf\{L\}\\leftarrow\\text\{IncompleteCholesky\}\(\\mathbf\{A\}\);

Define base solve

ℳb​a​s​e−1​\(𝐯\)≡\(𝐋𝐋T\)−1​𝐯\\mathcal\{M\}\_\{base\}^\{\-1\}\(\\mathbf\{v\}\)\\equiv\(\\mathbf\{L\}\\mathbf\{L\}^\{T\}\)^\{\-1\}\\mathbf\{v\};

1ex

//3\. Warm Start using coarse solution

𝐱0←𝒞​\(𝐛\)\\mathbf\{x\}\_\{0\}\\leftarrow\\mathcal\{C\}\(\\mathbf\{b\}\);

𝐫0←𝐛−𝐀𝐱0\\mathbf\{r\}\_\{0\}\\leftarrow\\mathbf\{b\}\-\\mathbf\{A\}\\mathbf\{x\}\_\{0\};

//Two\-level additive apply:𝐳=𝐳b​a​s​e\+𝐳c​o​a​r​s​e\\mathbf\{z\}=\\mathbf\{z\}\_\{base\}\+\\mathbf\{z\}\_\{coarse\}

𝐳0←ℳb​a​s​e−1​\(𝐫0\)\+𝒞​\(𝐫0\)\\mathbf\{z\}\_\{0\}\\leftarrow\\mathcal\{M\}\_\{base\}^\{\-1\}\(\\mathbf\{r\}\_\{0\}\)\+\\mathcal\{C\}\(\\mathbf\{r\}\_\{0\}\);

𝐩0←𝐳0\\mathbf\{p\}\_\{0\}\\leftarrow\\mathbf\{z\}\_\{0\};

1ex

for*k=0k=0toKm​a​xK\_\{max\}*do

αk←\(𝐫kT​𝐳k\)/\(𝐩kT​𝐀𝐩k\)\\alpha\_\{k\}\\leftarrow\(\\mathbf\{r\}\_\{k\}^\{T\}\\mathbf\{z\}\_\{k\}\)/\(\\mathbf\{p\}\_\{k\}^\{T\}\\mathbf\{A\}\\mathbf\{p\}\_\{k\}\);

𝐱k\+1←𝐱k\+αk​𝐩k\\mathbf\{x\}\_\{k\+1\}\\leftarrow\\mathbf\{x\}\_\{k\}\+\\alpha\_\{k\}\\mathbf\{p\}\_\{k\};

𝐫k\+1←𝐫k−αk​𝐀𝐩k\\mathbf\{r\}\_\{k\+1\}\\leftarrow\\mathbf\{r\}\_\{k\}\-\\alpha\_\{k\}\\mathbf\{A\}\\mathbf\{p\}\_\{k\};

if*‖𝐫k\+1‖<ϵ\\\|\\mathbf\{r\}\_\{k\+1\}\\\|<\\epsilon*then

break;

end if

//Additive Preconditioner Step

𝐳h​i​g​h←ℳb​a​s​e−1​\(𝐫k\+1\)\\mathbf\{z\}\_\{high\}\\leftarrow\\mathcal\{M\}\_\{base\}^\{\-1\}\(\\mathbf\{r\}\_\{k\+1\}\);

𝐳l​o​w←𝒞​\(𝐫k\+1\)\\mathbf\{z\}\_\{low\}\\leftarrow\\mathcal\{C\}\(\\mathbf\{r\}\_\{k\+1\}\);

𝐳k\+1←𝐳h​i​g​h\+𝐳l​o​w\\mathbf\{z\}\_\{k\+1\}\\leftarrow\\mathbf\{z\}\_\{high\}\+\\mathbf\{z\}\_\{low\};

βk←\(𝐫k\+1T​𝐳k\+1\)/\(𝐫kT​𝐳k\)\\beta\_\{k\}\\leftarrow\(\\mathbf\{r\}\_\{k\+1\}^\{T\}\\mathbf\{z\}\_\{k\+1\}\)/\(\\mathbf\{r\}\_\{k\}^\{T\}\\mathbf\{z\}\_\{k\}\);

𝐩k\+1←𝐳k\+1\+βk​𝐩k\\mathbf\{p\}\_\{k\+1\}\\leftarrow\\mathbf\{z\}\_\{k\+1\}\+\\beta\_\{k\}\\mathbf\{p\}\_\{k\};

end for

return

𝐱k\+1\\mathbf\{x\}\_\{k\+1\};

ALGORITHM 2Spectral Deflated ICPCG \(Additive\)

### 9\.2\.NEO as a Point\-wise Embedding Provider

We investigate the utility of the raw predicted functionsF∈ℝN×mF\\in\\mathbb\{R\}^\{N\\times m\}as intrinsic point embeddings for downstream tasks\. In this “Laplacian\-free” inference mode, NEO serves as a pre\-trained geometric feature extractor\. To rigorously evaluate the quality of these embeddings, we compare them against a standard positional encoding baseline \(NeRF\-PE\) combined with either a PointNet or Point Transformer head\.

#### Data Processing and Discretization\.

Since the original datasets \(SHREC’11 and Human Body\) provide triangular meshes, we convert them to point clouds to simulate a raw scanning setup\. For classification, we uniformly sampleN=8192N=8192points from the mesh surface\. For segmentation, we perform training and inference on the sampled point cloud; to evaluate against the ground\-truth labels defined on the mesh vertices, we propagate the predicted labels from the point cloud to the original vertices via nearest\-neighbor interpolation\.

#### Implementation Details\.

To ensure that performance differences arise from the input features rather than model capacity, we employ standardized lightweight heads for all baselines\. Table[10](https://arxiv.org/html/2605.24390#S9.T10)details the specific hyperparameters for the PointNet\+\+ and Point Transformer architectures used in these experiments\. All variants share a consistent feature dimension \(d=128d=128\) and activation scheme\. For the NEO\-based method, we freeze the backbone and only train the task\-specific head \(a simple PointNet MLP\) on top of the predicted featuresFF\.

#### Positional Encoding\.

To establish a competitive coordinate\-based baseline, we employ the Positional Encoding \(PE\) mechanism popularized by NeRF\(Mildenhall et al\.,[2020](https://arxiv.org/html/2605.24390#bib.bib31)\)\. Since standard MLPs exhibit a spectral bias towards low\-frequency signals\(Rahaman et al\.,[2019](https://arxiv.org/html/2605.24390#bib.bib38)\), raw coordinates𝐱∈ℝ3\\mathbf\{x\}\\in\\mathbb\{R\}^\{3\}are insufficient for capturing fine\-grained geometric details\. NeRF\-PE lifts the input into a higher\-dimensional Fourier feature space via

γ​\(𝐱\)=\[sin⁡\(20​π​𝐱\),cos⁡\(20​π​𝐱\),…,sin⁡\(2L−1​π​𝐱\),cos⁡\(2L−1​π​𝐱\)\],\\gamma\(\\mathbf\{x\}\)=\[\\sin\(2^\{0\}\\pi\\mathbf\{x\}\),\\cos\(2^\{0\}\\pi\\mathbf\{x\}\),\\dots,\\sin\(2^\{L\-1\}\\pi\\mathbf\{x\}\),\\cos\(2^\{L\-1\}\\pi\\mathbf\{x\}\)\],allowing the network to regress high\-frequency variations effectively\. In our experiments, we setL=10L=10, resulting in a 63\-dimensional input vector \(including original coordinates\)\.

Table 10\.Hyperparameters for Downstream Task Baselines\.We detail the architectures used for the few\-shot classification and segmentation heads\. Both models share the same channel width \(d=128d=128\) and general depth \(L=4L=4or 3 stages\) where applicable\. ”SA” denotes Set Abstraction and ”FP” denotes Feature Propagation modules\.
#### Few\-shot Classification\.

Table[11](https://arxiv.org/html/2605.24390#S9.T11)reports the accuracy on the SHREC’11 few\-shot classification benchmark\. Our method \(FF\+ PointNet\) significantly outperforms coordinate\-based baselines in the low\-data regime\. Notably, in the 1\-shot setting, NEO achieves an accuracy of58\.3%58\.3\\%, surpassing the Point Transformer baseline by over16%16\\%\. This suggests that the spectral subspace encapsulates a robust geometric prior that generalizes better than raw coordinates when supervision is scarce\.

Table 11\.Few\-shot classification on SHREC’11\.We report the classification accuracy \(%\) for 1, 3, and 10\-shot settings\. The comparison includes our method \(NEO\) and coordinate\-based baselines \(NeRF\-PE\) equipped with different heads\. “Mass Free” denotes the ablation where integration weights are fixed towi=1w\_\{i\}=1\.
#### Dense Segmentation\.

Table[12](https://arxiv.org/html/2605.24390#S9.T12)evaluates dense prediction performance on the Human Body segmentation task\. We observe that using NEO features leads to faster convergence and higher final accuracy\. At 100 epochs, our method reaches near\-optimal performance \(0\.80 mIoU\), whereas coordinate\-based methods require significantly more training steps to align the spatial information\. This confirms thatFFprovides a globally consistent coordinate system that simplifies the optimization landscape for dense labeling tasks\.

Table 12\.Segmentation on Human Body\.We report the mean IoU \(mIoU\) evaluated at 100 and 300 training epochs\. All methods utilize the same frozen backbone settings and a lightweight PointNet head\. The comparison highlights the convergence speed and final performance differences between spectral and coordinate\-based features\.
#### Ablation on Mass Awareness in Downstream Tasks\.

We further analyze the contribution of the mass integration weights by including a “Mass Free” variant \(wi=1w\_\{i\}=1\) in both Table[11](https://arxiv.org/html/2605.24390#S9.T11)and Table[12](https://arxiv.org/html/2605.24390#S9.T12)\. This setting simulates a simplified deployment scenario where local area estimation is skipped, treating the integration effectively as a uniform sum\. Results indicate that while the mass\-aware formulation consistently yields the best performance, the method exhibits graceful degradation when mass is omitted\. In the classification task \(Table[11](https://arxiv.org/html/2605.24390#S9.T11)\), the gap is most pronounced in the 1\-shot setting \(58\.3% vs\. 53\.3%\), suggesting that accurate geometric integration is crucial for extracting robust priors when supervision is scarce\. However, it is notable that even the Mass Free variant significantly outperforms the NeRF\-PE baseline \(53\.3% vs\. 47\.5%\), confirming that the learned spectral subspace captures intrinsic manifold structure that raw coordinates cannot, regardless of the integration scheme\.