学会接受引导：为 Classifier-Free Guidance 塑造潜在空间几何结构

# Shaping Latent Geometry for Classifier-Free Guidance
Source: [https://arxiv.org/html/2610.22885](https://arxiv.org/html/2610.22885)
## Learning to Be Guided: Shaping Latent Geometry for Classifier\-Free Guidance

Junhao ZhuAffiliation:New York University ShanghaiAffiliation:Georgia Institute of TechnologyShaoyi ZhengAffiliation:New York UniversityAffiliation:New York University ShanghaiYuanhe GuoAffiliation:New York UniversityAffiliation:New York University ShanghaiJinnuo LiuAffiliation:New York UniversityAffiliation:New York University ShanghaiShenji WanAffiliation:New York UniversityAffiliation:New York University ShanghaiHongyi WenAffiliation:New York UniversityAffiliation:New York University Shanghai

###### Abstract

††footnotetext:∗Equal contribution\. Correspondence to:\{haoming\.liu, hongyi\.wen\}@nyu\.edu\.

Classifier\-free guidance \(CFG\) is a standard component of conditional diffusion models, and guided generation performance has become thede factomeasure of state\-of\-the\-art models\. Recent work on latent diffusion models seeks to learn latent spaces that are easier to model with diffusion\. However, under CFG, generation additionally depends on how conditions are arranged in the latent space\. We study this arrangement under discrete class conditions, termed*class geometry*, and investigate what class geometry allows a latent space to work well with CFG\. Under a Gaussian class model, we show that shifting variance from intra\- to inter\-class at fixed total variance makes classes more distinguishable under noise, and derive a noisy posterior objective whose per\-sample descent direction is proportional to the oracle CFG correction\. Based on the analysis, we propose Prototype Regularization \(PR\), a lightweight surrogate for the intractable noisy posterior objective, which applies a cosine prototype classifier to spatially pooled latents\. With a controlled training budget \(20 epochs for VAE, 80 epochs for DiT\), adding PR to an LDM baseline reduces guided gFID on ImageNet from 3\.31 to 2\.28, surpassing a DINOv3\-aligned VA\-VAE \(2\.44\) with negligible overhead\. We further demonstrate that PR generalizes to pseudo\-labels derived from clustering, continuous text\-embedding conditions, and video latents\. Together, our findings highlight class geometry as a critical factor in guided generation and take a first step toward CFG\-compatible representation learning\. Our project page is available at:[https://maps\-research\.github\.io/cfg\-geometry/](https://maps-research.github.io/cfg-geometry/)\.

## 1Introduction

Diffusion models\([Ho et al\., 2020](https://arxiv.org/html/2610.22885#bib.bib34);[Song et al\., 2021](https://arxiv.org/html/2610.22885#bib.bib36);[Lipman et al\., 2023](https://arxiv.org/html/2610.22885#bib.bib28)\)have become a predominant paradigm in generative modeling, with recent advances largely driven by conditional generation on auxiliary signals such as class labels and text\([Rombach et al\., 2022](https://arxiv.org/html/2610.22885#bib.bib1);[Esser et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib2)\)\. Classifier\-free guidance \(CFG\)\([Ho and Salimans, 2021](https://arxiv.org/html/2610.22885#bib.bib23)\)is a standard technique for strengthening the conditioning: during sampling, a single model makes conditional and unconditional predictions, and their displacement \(i\.e\., the CFG correction\) is amplified by a guidance scaleω\>1\\omega\>1\. CFG improves generation quality and has become a crucial component of state\-of\-the\-art models: headline results are generally reported under CFG, usually with a large margin over unguided sampling\. Consequently, guided performance has become thede factomeasure of a conditional diffusion model\.

In parallel, latent diffusion models \(LDMs\)\([Rombach et al\., 2022](https://arxiv.org/html/2610.22885#bib.bib1)\)have become the prevailing approach for generating high\-dimensional data: an autoencoder first compresses data into a compact latent space, and a diffusion model is then trained in this latent space\. A growing line of research designs latent spaces that are easier to model, either by aligning latents with pretrained representations\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6);[Leng et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib8)\)or by adopting pretrained representations directly as the latent space\([Zheng et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib11);[Shi et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib12)\), while another line of analysis studies which properties of a latent space make it easier for diffusion models to learn\([Skorokhodov et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib20);[Zhong et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib19);[Liu et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib22)\)\. These efforts primarily approach latent design through the lens of*diffusability*, i\.e\., how easily a diffusion model learns the latent distribution\. Under CFG, however, the guided score for a conditioncctakes the form∇zt​log​pt​\(zt\)\+ω​∇zt​log​pt​\(c∣zt\)\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)\+\\omega\\,\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(c\\mid z\_\{t\}\), where the second term introduces a condition\-dependent correction into the sampling dynamics\. Hence, diffusability alone does not fully capture how a latent space interacts with guidance\.

Intuitively, the correction term∇zt​log​pt​\(c∣zt\)\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(c\\mid z\_\{t\}\)pushesztz\_\{t\}toward regions whereccis more probable, and this push is shaped by how latents of different conditions are arranged relative to one another\. More precisely, sincept​\(c∣zt\)p\_\{t\}\(c\\mid z\_\{t\}\)is evaluated on noisy latents, what matters is how distinguishable this arrangement remains under noise corruption\. To characterize this arrangement more explicitly, we ground our analysis in discrete class conditions, where it reduces to the layout of classes, termed*class geometry*\. We thus ask:*what class geometry allows a latent space to work well with CFG?*

We address this question by analyzing the CFG correction term through the lens of class geometry \(Sec\.[2](https://arxiv.org/html/2610.22885#S2)\)\. Specifically, under a Gaussian class model, we derive the oracle CFG correction in closed form: it contrasts the target class mean against competing class means, weighted by their noisy posteriors\. We then show that, at fixed total variance, reallocating variance from intra\-class to inter\-class makes classes more distinguishable under noise, so the target class posterior concentrates earlier in denoising\. Finally, we express this geometric preference as a cross\-entropy objective on the noisy class posterior, whose per\-sample descent direction is proportional to the oracle correction\.

Guided by our analysis, we propose Prototype Regularization \(PR\), a tractable surrogate for the posterior objective on clean latents \(Sec\.[3](https://arxiv.org/html/2610.22885#S3)\)\. In practice, PR applies a cosine prototype classifier to spatially pooled latents, and we show that its gradient forms a target\-versus\-competitor contrast analogous to that in the oracle CFG correction\. On ImageNet class\-conditional generation, adding PR to an LDM baseline reduces guided gFID from 3\.31 to 2\.28, outperforming DINOv3\-aligned\([Siméoni et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib32)\)VA\-VAE \(2\.44\)\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6)\)with negligible overhead \(Sec\.[4\.2](https://arxiv.org/html/2610.22885#S4.SS2)\)\.

To isolate the source of this gain, we transplant PR class centers into LDM latents and recover most of PR’s guided improvement, indicating that the gain stems largely from class geometry \(Sec\.[4\.3](https://arxiv.org/html/2610.22885#S4.SS3)\)\. We further demonstrate the versatility of PR through extensions to pseudo\-labels, video latents, and continuous text conditions \(Sec\.[4\.4](https://arxiv.org/html/2610.22885#S4.SS4),[4\.5](https://arxiv.org/html/2610.22885#S4.SS5),[4\.6](https://arxiv.org/html/2610.22885#S4.SS6)\)\. Since PR relies only on discrete partitions to promote a cluster geometry, these results suggest that it applies beyond ground\-truth labels whenever this geometry stays informative under noise\. Together, our work highlights class geometry as a critical factor in guided generation and takes a first step toward CFG\-compatible representation learning\.

Our contributions are threefold:

- •We analyze CFG through the lens of class geometry\. Under a Gaussian class model, we derive the closed\-form oracle correction, show that shifting variance from intra\- to inter\-class at fixed total variance improves class distinguishability under noise, and formulate a noisy posterior objective whose per\-sample descent direction is proportional to this correction\.
- •We propose Prototype Regularization \(PR\), a lightweight surrogate for the noisy posterior objective that shapes class geometry\. By classifying clean latents against prototypes, PR yields gradients analogous to the target\-versus\-competitor contrast in the oracle correction\.
- •We show that PR generalizes across label sources, modalities, and condition types, suggesting its effectiveness as a CFG\-compatible geometry regularizer over discrete partitions\.

## 2From CFG Geometry to Representation Learning

### 2\.1Preliminaries

Latent Diffusion Models\.A latent diffusion model \(LDM\)\([Rombach et al\., 2022](https://arxiv.org/html/2610.22885#bib.bib1)\)operates in a compressed latent space learned by an autoencoder\. An encoderℰθ\\mathcal\{E\}\_\{\\theta\}maps the raw datax∈ℝDx\\in\\mathbb\{R\}^\{D\}to a compact latent codez=ℰθ​\(x\)∈ℝdz=\\mathcal\{E\}\_\{\\theta\}\(x\)\\in\\mathbb\{R\}^\{d\}withd≪Dd\\ll D, and a decoder𝒟ϕ\\mathcal\{D\}\_\{\\phi\}reconstructsx^=𝒟ϕ​\(z\)\\hat\{x\}=\\mathcal\{D\}\_\{\\phi\}\(z\)\. For images, the autoencoder is trained withℒLDM=ℒRecon\+λKL​ℒKL\+λLPIPS​ℒLPIPS\+λGAN​ℒGAN\\mathcal\{L\}\_\{\\mathrm\{LDM\}\}=\\mathcal\{L\}\_\{\\mathrm\{Recon\}\}\+\\lambda\_\{\\mathrm\{KL\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{KL\}\}\+\\lambda\_\{\\mathrm\{LPIPS\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{LPIPS\}\}\+\\lambda\_\{\\mathrm\{GAN\}\}\\,\\mathcal\{L\}\_\{\\mathrm\{GAN\}\}\. The diffusion model is then trained in the resulting latent space, where the latents are typically standardized via scalar factors or per\-channel statistics\.

Rectified Flow\.Rectified flow\([Liu et al\., 2023](https://arxiv.org/html/2610.22885#bib.bib29);[Lipman et al\., 2023](https://arxiv.org/html/2610.22885#bib.bib28)\)is a widely adopted training objective for diffusion models\. For a clean latentz∈ℝdz\\in\\mathbb\{R\}^\{d\}, we construct the linear interpolantzt=t​z\+\(1−t\)​ϵz\_\{t\}=t\\,z\+\(1\{\-\}t\)\\,\\epsilon, with independent noiseϵ∼𝒩⁡\(0,I\),t∈\(0,1\)\\epsilon\\sim\\mathcal\{N\}\(0,I\),\\,t\\in\(0,1\), and train a neural networkvψ​\(zt,t\)v\_\{\\psi\}\(z\_\{t\},t\)to predict the velocity targetv=z−ϵv=z\-\\epsilonby minimizingℒRF=𝔼z,ϵ,t​\[∥vψ​\(zt,t\)−\(z−ϵ\)∥2\]\\mathcal\{L\}\_\{\\mathrm\{RF\}\}=\\mathbb\{E\}\_\{z,\\,\\epsilon,\\,t\}\\bigl\[\\lVert v\_\{\\psi\}\(z\_\{t\},t\)\-\(z\-\\epsilon\)\\rVert^\{2\}\\bigr\]\.

Classifier\-Free Guidance\.Classifier\-free guidance \(CFG\)\([Ho and Salimans, 2021](https://arxiv.org/html/2610.22885#bib.bib23)\)amplifies the effect of conditioning during sampling to obtain higher\-fidelity generations\. A single network is trained with condition dropout to produce both a conditional predictionv^c\\widehat\{v\}\_\{c\}and an unconditional predictionv^u\\widehat\{v\}\_\{u\}\. The guided velocity isv^cfg=v^u\+ω⁡\(v^c−v^u\)\\widehat\{v\}\_\{\\mathrm\{cfg\}\}=\\widehat\{v\}\_\{u\}\+\\omega\\,\(\\widehat\{v\}\_\{c\}\-\\widehat\{v\}\_\{u\}\), whereω\>1\\omega\>1is the guidance scale\.

Figure 1:Simulations under the Gaussian class model withK=1000K=1000classes and latent dimensionsd=8192d=8192\(Eq\.[4](https://arxiv.org/html/2610.22885#S2.E4)\)\. \(a\) Oracle CFG correction strength along timett\(t=1t=1is clean\), for varying inter\-class variance fractionα\\alpha\. The correction vanishes once the target\-class posterior concentrates\. \(b\) Monte Carlo average of the analytic target\-class posterior under forward corruption\. Largerα\\alphayields earlier posterior concentration\. \(c\) Posterior losses \(normalized bylog⁡K\\log K\) versusα\\alpha, on clean\-only and uniform\-time corrupted latents\. Both posterior losses are non\-increasing inα\\alpha\.
### 2\.2A Geometric View of Classifier\-Free Guidance

How Class Geometry Enters CFG\.Under rectified flow\([Liu et al\., 2023](https://arxiv.org/html/2610.22885#bib.bib29)\), letvc⋆=𝔼\[v∣zt,t,C=c\]v\_\{c\}^\{\\star\}=\\mathbb\{E\}\[v\\mid z\_\{t\},t,C=c\]andvu⋆=𝔼\[v∣zt,t\]v\_\{u\}^\{\\star\}=\\mathbb\{E\}\[v\\mid z\_\{t\},t\]be the Bayes\-optimal conditional and unconditional predictors\. Substitutingv=\(z−zt\)/\(1−t\)v=\(z\-z\_\{t\}\)/\(1\-t\), the oracle CFG correctiongc⋆​\(zt,t\)g\_\{c\}^\{\\star\}\(z\_\{t\},t\)reduces to:

gc⋆​\(zt,t\):=vc⋆−vu⋆=𝔼\[z∣zt,t,C=c\]−𝔼\[z∣zt,t\]1−t\.g\_\{c\}^\{\\star\}\(z\_\{t\},t\):=v\_\{c\}^\{\\star\}\-v\_\{u\}^\{\\star\}=\\frac\{\\mathbb\{E\}\[z\\mid z\_\{t\},t,C=c\]\-\\mathbb\{E\}\[z\\mid z\_\{t\},t\]\}\{1\-t\}\.\(1\)Overall, the correction captures how conditioning on the target class changes the posterior estimate of the clean latent\. To gain concrete intuition on how class geometry shapesgc⋆g\_\{c\}^\{\\star\}, we derive a closed form of Eq\.[1](https://arxiv.org/html/2610.22885#S2.E1)under a simplified setting\. Suppose the conditionc∈\{1,…,K\}c\\in\\\{1,\\dots,K\\\}indexesKKdiscrete classes, each with a Gaussian class\-conditional priorp⁡\(z∣c\)=𝒩⁡\(z,μc,σ2​I\)p\(z\\mid c\)=\\mathcal\{N\}\(z;\\,\\mu\_\{c\},\\,\\sigma^\{2\}I\)with shared within\-class varianceσ2\\sigma^\{2\}, which we refer to as the*Gaussian class model*\. Letqj=Pr⁡\(C=j∣zt,t\)q\_\{j\}=\\Pr\(C=j\\mid z\_\{t\},t\)denote the posterior class probability\. We obtain:

gc⋆​\(zt,t\)=1−t\(1−t\)2\+t2​σ2​∑j≠cqj​\(μc−μj\)\.g\_\{c\}^\{\\star\}\(z\_\{t\},t\)=\\frac\{1\-t\}\{\(1\{\-\}t\)^\{2\}\+t^\{2\}\\sigma^\{2\}\}\\sum\_\{j\\neq c\}q\_\{j\}\\,\(\\mu\_\{c\}\-\\mu\_\{j\}\)\.\(2\)Therefore, the oracle CFG correction can be expressed as a posterior\-weighted sum of displacements from the competing class means toward the target mean, withσ2\\sigma^\{2\}shaping both the prefactor and the posterior weights\. The center displacementsμc−μj\\mu\_\{c\}\-\\mu\_\{j\}define the class contrasts, while the posterior weightsqjq\_\{j\}determine which contrasts are emphasized at each noisy stateztz\_\{t\}\. When the target class dominates the posterior \(i\.e\.,qc→1q\_\{c\}\\to 1\), the weights of all competing classes vanish, eliminating the target\-versus\-competitor contrast\. This behavior is illustrated in Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(a\) and Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(b\)\.

Class Distinguishability under Noise\.To understand how class geometry governs this posterior concentration, we examine class distinguishability under noise corruption\. Under the same Gaussian model, the class\-conditional distribution ofztz\_\{t\}isp⁡\(zt∣c\)=𝒩⁡\(zt,t​μc,\(t2​σ2\+\(1−t\)2\)​I\)p\(z\_\{t\}\\mid c\)=\\mathcal\{N\}\\bigl\(z\_\{t\};t\\mu\_\{c\},\(t^\{2\}\\sigma^\{2\}\+\(1\-t\)^\{2\}\)I\\bigr\), so the squared standardized distance between classesccandjjis given by:

δc​j2​\(t\)=t2​‖μc−μj‖2t2​σ2\+\(1−t\)2\.\\delta\_\{cj\}^\{2\}\(t\)=\\frac\{t^\{2\}\\\|\\mu\_\{c\}\-\\mu\_\{j\}\\\|^\{2\}\}\{t^\{2\}\\sigma^\{2\}\+\(1\-t\)^\{2\}\}\.\(3\)To isolate the effect of class geometry from changes in latent scale, we consider a controlled family of latent distributions\. With class priors held fixed, we choose the reference class means satisfying𝔼C​\[μCref\]=0\\mathbb\{E\}\_\{C\}\[\\mu\_\{C\}^\{\\mathrm\{ref\}\}\]=0and𝔼C​‖μCref‖2=d\\mathbb\{E\}\_\{C\}\\\|\\mu\_\{C\}^\{\\mathrm\{ref\}\}\\\|^\{2\}=d, whereddis the latent dimension, and set:

μc​\(α\)=α​μcref,σ2​\(α\)=1−α,α∈\(0,1\)\.\\mu\_\{c\}\(\\alpha\)=\\sqrt\{\\alpha\}\\,\\mu\_\{c\}^\{\\mathrm\{ref\}\},\\qquad\\sigma^\{2\}\(\\alpha\)=1\-\\alpha,\\qquad\\alpha\\in\(0,1\)\.\(4\)Under this controlled family, the total variance remainsdd, with inter\-class and intra\-class contributions ofα​d\\alpha dand\(1−α\)​d\(1\-\\alpha\)d, respectively\. Increasingα\\alphathus reallocates variance from intra\-class to inter\-class, scaling allδc​j2​\(t\)\\delta\_\{cj\}^\{2\}\(t\)by a common factor at everyt∈\(0,1\)t\\in\(0,1\), so the noisy classification problem only becomes easier\. Classes therefore remain distinguishable under stronger corruption, and the target\-class posterior concentrates earlier along the denoising process, as illustrated in Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(b\)\.

### 2\.3A CFG\-Motivated Representation Objective

Motivated by how CFG contrasts classes through the noisy posterior probabilities in Eq\.[2](https://arxiv.org/html/2610.22885#S2.E2), we seek representations that make the target class more distinguishable from its competitors under corruption\. We express this geometric preference through an ideal posterior cross\-entropy objective:

ℒpost​\(t\)=𝔼z,c,ϵ​\[−log⁡qc​\(t​z\+\(1−t\)​ϵ,t\)\]=H⁡\(C∣Zt\),\\mathcal\{L\}\_\{\\mathrm\{post\}\}\(t\)=\\mathbb\{E\}\_\{z,c,\\epsilon\}\\left\[\-\\log q\_\{c\}\\bigl\(tz\+\(1\-t\)\\epsilon,t\\bigr\)\\right\]=H\(C\\mid Z\_\{t\}\),\(5\)whereqcq\_\{c\}is the true noisy class posterior, andH⁡\(C∣Zt\)H\(C\\mid Z\_\{t\}\)denotes the conditional entropy of the class label given the corrupted latent\. This objective is directly related to the oracle correctiongc⋆g\_\{c\}^\{\\star\}via its input gradient\. With the posterior function held fixed, the per\-sample descent direction satisfies:

∇z​log​qc​\(t​z\+\(1−t\)​ϵ,t\)=t​∇zt​log​qc​\(zt,t\)=t21−t​gc⋆​\(zt,t\),t∈\(0,1\)\.\\nabla\_\{z\}\\log q\_\{c\}\\bigl\(tz\+\(1\-t\)\\epsilon,t\\bigr\)=t\\nabla\_\{z\_\{t\}\}\\log q\_\{c\}\(z\_\{t\},t\)=\\frac\{t^\{2\}\}\{1\-t\}\\,g\_\{c\}^\{\\star\}\(z\_\{t\},t\),\\quad t\\in\(0,1\)\.\(6\)Therefore, the ideal posterior objective provides a per\-sample latent descent direction proportional to the oracle CFG correction\. Note that Eq\.[5](https://arxiv.org/html/2610.22885#S2.E5)is defined at a single noise level, whereas diffusion training and sampling span all noise levels\. We therefore average it over a chosen time distributionρ\\rho, obtaining a family of posterior objectivesℒρ=𝔼t∼ρ​\[ℒpost​\(t\)\]\\mathcal\{L\}\_\{\\rho\}=\\mathbb\{E\}\_\{t\\sim\\rho\}\[\\mathcal\{L\}\_\{\\mathrm\{post\}\}\(t\)\]\. Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(c\) plots the losses for two choices ofρ\\rhoagainstα\\alpha: the posterior loss on clean latents \(t=1t=1\) and its average overt∼𝒰⁡\(0,1\)t\\sim\\mathcal\{U\}\(0,1\)\. Both losses are divided bylog⁡K\\log K\(i\.e\., the entropy of a uniform distribution over theKKclasses\), so a value of one corresponds to complete class uncertainty in this equal\-prior setting\.

Within the family of class geometries in Eq\.[4](https://arxiv.org/html/2610.22885#S2.E4), everyℒρ\\mathcal\{L\}\_\{\\rho\}is non\-increasing inα\\alpha\(Appendix[E\.8](https://arxiv.org/html/2610.22885#A5.SS8)\): lower loss corresponds to larger inter\-class separation and smaller intra\-class spread, which keep classes distinguishable under stronger noise\. Both losses in Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(c\) decrease accordingly, with the clean loss saturating at a smallerα\\alpha, while the noise\-averaged loss favors further separation more persistently\. In line with this, we empirically find that uniform\-time supervision can over\-regularize the representation \(e\.g\., over\-smoothing its spatial variation\), whereas clean\-only supervision is sufficient to yield notable gains in guided generation \(Appendix[D\.1](https://arxiv.org/html/2610.22885#A4.SS1)\)\.

In summary, we obtain a family of posterior objectives whose per\-sample descent directions follow the oracle CFG correction \(Eq\.[6](https://arxiv.org/html/2610.22885#S2.E6)\)\. Our fixed\-variance analysis further motivates minimizing these objectives to promote class separation and compactness\. To translate these objectives into practical regularizers, one needs to chooseρ\\rhoand a tractable approximation ofqcq\_\{c\}\.

## 3Prototype Regularization

Sec\.[2\.3](https://arxiv.org/html/2610.22885#S2.SS3)motivatesℒρ\\mathcal\{L\}\_\{\\rho\}as ideal objectives for CFG\-compatible representations, but they cannot be optimized directly: the true posteriorqcq\_\{c\}is defined by the very latent distribution being learned\. Since the diffusion model is trained in the latent space fixed by the autoencoder, we instead optimize a practical surrogate during autoencoder training, termed*Prototype Regularization*\(PR\)\. PR addresses both requirements identified above: it adopts clean\-only supervision \(t=1t=1\), and it approximatesqcq\_\{c\}with a lightweight prototype classifier that mirrors the posterior of a Gaussian class model\.

Prototype classifier on pooled latents\.Given a latentz∈ℝdc×h×wz\\in\\mathbb\{R\}^\{d\_\{c\}\\times h\\times w\}, we obtain a global featurez¯=AvgPool⁡\(z\)∈ℝdc\\bar\{z\}=\\mathrm\{AvgPool\}\(z\)\\in\\mathbb\{R\}^\{d\_\{c\}\}and associate each classjjwith a learnable prototypepj∈ℝdcp\_\{j\}\\in\\mathbb\{R\}^\{d\_\{c\}\}, a proxy for its class mean\. Withz~=z¯/‖z¯‖\\tilde\{z\}=\\bar\{z\}/\\\|\\bar\{z\}\\\|andp~j=pj/‖pj‖\\tilde\{p\}\_\{j\}=p\_\{j\}/\\\|p\_\{j\}\\\|, the prototype posterior and the PR loss are:

πj​\(z\)=exp⁡\(z~⊤​p~j/τ\)∑k=1Kexp⁡\(z~⊤​p~k/τ\),ℒPR=−log⁡πc​\(z\),\\pi\_\{j\}\(z\)=\\frac\{\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{j\}/\\tau\)\}\{\\sum\_\{k=1\}^\{K\}\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{k\}/\\tau\)\},\\qquad\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=\-\\log\\pi\_\{c\}\(z\),\(7\)whereccis the class label andτ\>0\\tau\>0is a fixed temperature\. Average pooling restricts PR to the spatial mean ofzz, leaving the residualz−z¯z\-\\bar\{z\}unconstrained; this avoids pulling the full latent toward a class template, which may constrain layout and reduce diversity\. Normalization makes the objective depend only on angular alignment, which excludes the trivial minimizer of inflating feature norms\.

Connection to CFG correction\.The posterior objective in Eq\.[5](https://arxiv.org/html/2610.22885#S2.E5)is well\-defined on clean latents \(t=1t=1\), where it equalsH⁡\(C∣Z\)H\(C\\mid Z\), whereas the gradient identity in Eq\.[6](https://arxiv.org/html/2610.22885#S2.E6)applies fort∈\(0,1\)t\\in\(0,1\)\. Under the Gaussian class model with equal priors and equal\-norm means, the clean posterior satisfiesqj∝exp⁡\(μj⊤​z/σ2\)q\_\{j\}\\propto\\exp\(\\mu\_\{j\}^\{\\top\}z/\\sigma^\{2\}\)\. Its gradient∇z​log​qc=σ−2​∑j≠cqj​\(μc−μj\)\\nabla\_\{z\}\\log q\_\{c\}=\\sigma^\{\-2\}\\sum\_\{j\\neq c\}q\_\{j\}\(\\mu\_\{c\}\-\\mu\_\{j\}\)is thet→1t\\to 1limit of the posterior gradient in Eq\.[6](https://arxiv.org/html/2610.22885#S2.E6), providing a clean\-latent counterpart to the CFG\-related descent direction\. PR adopts a similar softmax form on normalized features, yielding the negative gradient:

−∇z~ℒPR=1τ​\(p~c−∑j=1Kπj​p~j\)=1τ​∑j≠cπj​\(p~c−p~j\)\.\-\\nabla\_\{\\tilde\{z\}\}\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=\\frac\{1\}\{\\tau\}\\Bigl\(\\tilde\{p\}\_\{c\}\-\\sum\_\{j=1\}^\{K\}\\pi\_\{j\}\\tilde\{p\}\_\{j\}\\Bigr\)=\\frac\{1\}\{\\tau\}\\sum\_\{j\\neq c\}\\pi\_\{j\}\\,\(\\tilde\{p\}\_\{c\}\-\\tilde\{p\}\_\{j\}\)\.\(8\)This gradient shares the class\-contrast form of Eq\.[2](https://arxiv.org/html/2610.22885#S2.E2), with normalized prototypesp~j\\tilde\{p\}\_\{j\}replacing class meansμj\\mu\_\{j\}and classifier probabilitiesπj\\pi\_\{j\}replacing posterior probabilitiesqjq\_\{j\}, giving greater weight to more probable competitors\. This correspondence holds at the normalized featurez~\\tilde\{z\}; gradients to the original latent additionally pass through normalization and pooling\. Appendix[D](https://arxiv.org/html/2610.22885#A4)presents ablations on PR variants, such as the prototype shape, the classifier type, and supervision on noisy latents\.

Training objective\.We train the autoencoder with:

ℒ=ℒLDM\+λPR​ℒPR,\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{LDM\}\}\+\\lambda\_\{\\mathrm\{PR\}\}\\mathcal\{L\}\_\{\\mathrm\{PR\}\},\(9\)where the reconstructive terms inℒLDM\\mathcal\{L\}\_\{\\mathrm\{LDM\}\}retain instance\-specific information alongside the class structure from PR\. Following[Esser et al\. \(2021\)](https://arxiv.org/html/2610.22885#bib.bib31)and[Rombach et al\. \(2022\)](https://arxiv.org/html/2610.22885#bib.bib1),λPR\\lambda\_\{\\mathrm\{PR\}\}is set adaptively by balancing encoder last\-layer gradient norms against the reconstruction term\.

Beyond class labels\.PR only requires a partition of the data, so ground\-truth labels can be replaced by pseudo\-labels from clustering pretrained features, with one prototype per cluster\. Moreover, the regularization labels need not match the generation condition: once the autoencoder is trained with PR, the diffusion model can be trained on other downstream conditions, discrete or continuous\. We show that PR remains effective in these settings \(Sec\.[4\.4](https://arxiv.org/html/2610.22885#S4.SS4),[4\.6](https://arxiv.org/html/2610.22885#S4.SS6)\)\.

## 4Experiments

### 4\.1Experimental Setup

Our primary testbed is class\-conditional image generation on ImageNet\([Deng et al\., 2009](https://arxiv.org/html/2610.22885#bib.bib30)\)at256×256256\\times 256resolution\. We build on the VA\-VAE codebase\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6)\)and use a convolution\-based encoder/decoder with a latent shape of32×16×1632\\times 16\\times 16, paired with LightningDiT\-XL/1 as the stage\-2 backbone\. We use the same number of training epochs for all methods \(20 epochs for stage\-1 and 80 epochs for stage\-2\) for fair comparison\. All ImageNet experiments in the main text are trained on8×8\\timesNVIDIA H100 GPUs; a subset of appendix experiments uses 2 GPUs \(with 10 epochs for stage\-1 and 30 epochs for stage\-2\)\. Full hyper\-parameter details are provided in Appendix[C](https://arxiv.org/html/2610.22885#A3)\.

We use generation FID \(gFID\)\([Heusel et al\., 2017](https://arxiv.org/html/2610.22885#bib.bib37)\)as our primary metric, which measures how closely generated images match the training distribution\. For all results with classifier\-free guidance, we report gFID at the best guidance scaleω⋆\\omega^\{\\star\}, selected via a guidance scale sweep using gFID computed on 5K samples\. When comparing against baselines, we follow the standard practice and additionally report sFID, Inception Score \(IS\), Precision, and Recall using the ADM evaluation toolkit\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2610.22885#bib.bib35)\)\. All metrics are computed over 50K samples unless otherwise stated\. Setups for Sec\.[4\.5](https://arxiv.org/html/2610.22885#S4.SS5)and Sec\.[4\.6](https://arxiv.org/html/2610.22885#S4.SS6)are described in their respective subsections\.

### 4\.2Class\-Conditional Image Generation with Ground\-truth Labels

We compare PR against LDM\([Rombach et al\., 2022](https://arxiv.org/html/2610.22885#bib.bib1)\)and VA\-VAE\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6)\)\. We trained two VA\-VAE variants, aligned with DINOv3\-B and DINOv3\-L\([Siméoni et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib32)\), respectively\. Our method integrates PR into LDM training\. On reconstruction fidelity, we report PSNR, LPIPS, and rFID\. Overall, all methods perform comparably, with VA\-VAE holding a marginal edge\. PR therefore does not sacrifice reconstruction quality for class geometry\. Meanwhile, PR raises Purity@10 \(i\.e\., the fraction of each latent’s 10 nearest neighbors that share its class\) from0\.0200\.020\(LDM\) to0\.2790\.279, about4×4\\timesthat of the strongest VA\-VAE variant \(0\.0710\.071\), indicating substantially more class\-consistent local neighborhoods\. PR achieves this at negligible cost, whereas VA\-VAE slows stage\-1 training throughput by about25%25\\%due to the additional representation alignment\.

On the generation side, the gains of PR concentrate on guided settings\. Without CFG, PR improves over LDM \(13\.09→11\.5913\.09\\rightarrow 11\.59\) but trails both VA\-VAE variants\. With CFG, PR takes the lead, achieving the best gFID \(2\.282\.28\), IS \(269\.85269\.85\), and Recall \(0\.5850\.585\) among all methods\. Moreover, the gFID ranking with CFG is consistent with the Purity ranking, while the ranking without CFG differs\. This suggests that class geometry is particularly relevant to guided generation quality, which we investigate further in Section[4\.3](https://arxiv.org/html/2610.22885#S4.SS3)\. Both VA\-VAE and PR exhibit lower Precision and higher Recall than LDM under CFG, suggesting broader coverage of the data distribution at a cost in sample fidelity\.

Figure[2](https://arxiv.org/html/2610.22885#S4.F2)examines the latent spaces from three complementary angles: \(a\) We decompose the energy of per\-channel normalized latents into a global mean, inter\-class variation, and intra\-class variation\. In LDM, over90%90\\%of the energy belongs to the intra\-class portion\. VA\-VAE mildly increases inter\-class energy, but shifts a large portion to the global mean, a spatial template shared by all latents\. PR reallocates intra\-class energy to inter\-class variance, yielding the largest inter\-class share among all methods\. \(b\) We measure the average posterior𝔼​\[qc​\(zt\)\]\\mathbb\{E\}\[q\_\{c\}\(z\_\{t\}\)\]of the target class on both forward\-noised validation latents and intermediate states along CFG sampling trajectories, withqc​\(zt\)q\_\{c\}\(z\_\{t\}\)computed from the empirical training\-latent distribution under Gaussian forward corruption\. PR yields substantially higher target\-class posteriors across a broad range of noise levels, demonstrating improved class distinguishability under noise\. \(c\) Sweeping the guidance scale, PR reaches its optimum at a largerω⋆\\omega^\{\\star\}and degrades more slowly beyond it, resulting in a wider near\-optimal range\. This suggests that class geometry shaped via PR allows the model to benefit from stronger guidance\.

Table 1:Results of class\-conditional generation on ImageNet\. All quality metrics are evaluated with 50K samples\. Purity measures the fraction of each latent’sk=10k=10nearest neighbors that share its class, averaged over 50K normalized validation latents\. Best results are highlighted inbold\.Figure 2:\(a\) Energy decomposition of per\-channel normalized latents into global mean‖μ‖2\\\|\\mu\\\|^\{2\}, inter\-class variance𝔼c​‖μc−μ‖2\\mathbb\{E\}\_\{c\}\\\|\\mu\_\{c\}\-\\mu\\\|^\{2\}, and intra\-class variance𝔼​‖z−μc‖2\\mathbb\{E\}\\\|z\-\\mu\_\{c\}\\\|^\{2\}\. \(b\) Average target\-class posterior𝔼​\[qc​\(zt\)\]\\mathbb\{E\}\[q\_\{c\}\(z\_\{t\}\)\]on noised validation latents \(solid\) and CFG sampling trajectories \(dotted\), whereqc​\(zt\)q\_\{c\}\(z\_\{t\}\)is computed analytically from the empirical distribution of ImageNet training latents under Gaussian forward corruption\. \(c\) gFID \(10K samples\) versus guidance scaleω\\omega; diamonds mark the minima of the displayed curves \(ω⋆\\omega^\{\\star\}\)\. PR allocates more energy to inter\-class variance, makes the target class identifiable earlier and more confidently, and reaches a lower best gFID at a largerω⋆\\omega^\{\\star\}\.Table 2:Center transplantation on LDM latents \(z′=z−μcLDM\+μcRefz^\{\\prime\}=z\-\\mu\_\{c\}^\{\\mathrm\{LDM\}\}\+\\mu\_\{c\}^\{\\mathrm\{Ref\}\}\)\. Rescaled LDM centers match the inter\-class energy of full PR centers\. Random centers match the pooled PR energy\. Spatially constant centers are broadcast over the spatial grid\.ω⋆\\omega^\{\\star\}denotes the best CFG scale\.Base LatentCenter ConstructionCenter ShapegFID@50K↓\{\\downarrow\}ω⋆\\omega^\{\\star\}w/o CFGw/ CFGLDM \+ PRNative–11\.592\.282\.5LDMNative–13\.093\.312\.0LDMLDM \(Rescaled\)32×16×1632\\times 16\\times 1612\.582\.892\.0LDMPR32×16×1632\\times 16\\times 1612\.792\.633\.5LDMPR32×1×132\\times 1\\times 112\.652\.524\.0LDMRandom Directions \+ PR Norms32×1×132\\times 1\\times 113\.743\.164\.5
### 4\.3Understanding the Gains through Center Transplantation

Figure[2](https://arxiv.org/html/2610.22885#S4.F2)\(a\) shows that PR primarily reallocates latent energy to inter\-class variance \(i\.e\., more separated class centers\)\. This raises a natural question:*can an unregularized latent space benefit from the class geometry of PR in a plug\-and\-play manner?*To answer this, we transplant reference class centers into the LDM latent space\. For a latentzzof classcc, we replace its native center with a reference center,z′=z−μcLDM\+μcRefz^\{\\prime\}=z\-\\mu\_\{c\}^\{\\mathrm\{LDM\}\}\+\\mu\_\{c\}^\{\\mathrm\{Ref\}\}, which preserves intra\-class deviations, and train a stage\-2 model onz′z^\{\\prime\}\. To ensure the decoder receives latents in its native space, we apply the inverse mapping before decoding\. We use two PR references: the full center and its spatially pooled version\. As controls, we use LDM’s own centers rescaled to match the inter\-class energy of the full PR centers, and random directions with PR’s per\-class norms\. As presented in Tab\.[2](https://arxiv.org/html/2610.22885#S4.T2), transplanting PR centers improves guided generation \(gFID3\.31→2\.523\.31\\rightarrow 2\.52with pooled centers\), while the unguided setting changes only marginally \(13\.09→12\.6513\.09\\rightarrow 12\.65\)\. Increasing the inter\-class energy of LDM alone yields a partial gain \(2\.892\.89\), and PR centers improve further beyond this energy\-matched control\. In contrast, random directions provide little benefit with CFG and even hurt unguided generation\. These results indicate that PR centers have a better arrangement, and the gain does not stem from increased inter\-class energy alone\. Besides, pooled centers outperform spatially resolved ones \(2\.522\.52vs\.2\.632\.63\), so the useful class signal is mainly a global channel\-wise offset\.

Interestingly, transplanting PR or random centers shiftsω⋆\\omega^\{\\star\}upward \(3\.5 to 4\.5\), whereas rescaling LDM’s own centers does not \(2\.0\)\. However, a largerω⋆\\omega^\{\\star\}does not by itself imply better geometry: random centers admit the strongest guidance yet bring little gain in guided quality, while rescaled LDM centers improve quality without changingω⋆\\omega^\{\\star\}\. Hence, the optimal guidance scale and the quality attained at it should be considered separately\. Finally, center transplantation recovers most of PR’s guided gain but only a small portion of its unguided gain, and thus does not fully reproduce the benefits of PR\. This is likely because transplantation only alters inter\-class separation and leaves intra\-class spread untouched, whereas PR reshapes both during stage\-1 training\.

### 4\.4Generalization under Pseudo\-Labels and Non\-identical Label Sets

Building on PR’s performance gains on ground\-truth class labels, we further investigate whether PR remains effective on other types of labels, and whether the two stages must share the same label set\. To this end, we construct unsupervised labels by clustering the\[CLS\]tokens of DINOv3\-L\([Siméoni et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib32)\)with sphericalKK\-means, settingK=1000K\{=\}1000to match the number of ImageNet classes\. We then evaluate all combinations of stage\-1 PR labels and stage\-2 conditioning labels \(Tab\.[3](https://arxiv.org/html/2610.22885#S4.T3)\)\. When conditioning on DINO clusters, we sample cluster labels in proportion to cluster sizes during evaluation\.

Table 3:PR with different label set pairings for stage\-1 regularization and stage\-2 conditioning\. We report guided gFID//unguided gFID below\. The best result per\-column is inbold\.
Using ground\-truth labels in both stages yields the best guided generation \(2\.282\.28\)\. Replacing the stage\-1 labels with DINO clusters recovers most of this gain \(2\.342\.34, about94%94\\%of the improvement over the no\-PR baseline\) and even improves unguided generation \(11\.4411\.44vs\.11\.5911\.59\)\. When stage\-2 conditions on DINO clusters, PR also substantially improves over the no\-PR baseline \(3\.58→2\.423\.58\\rightarrow 2\.42and2\.552\.55with CFG\)\. Notably, under DINO conditioning, PR regularized with ground\-truth labels performs best with CFG \(2\.422\.42\), even though its stage\-1 labels differ from the stage\-2 condition\. These results show that PR neither relies on ground\-truth labels nor requires the two stages to share the same label set, making it applicable to settings without human annotations\.

### 4\.5Generalization to Video Latents

We further examine whether PR generalizes to video latents\. We conduct experiments on UCF\-101\([Soomro et al\., 2012](https://arxiv.org/html/2610.22885#bib.bib39)\), a human action video dataset spanning 101 action classes, and adopt LeanVAE\-16\([Cheng and Yuan, 2025](https://arxiv.org/html/2610.22885#bib.bib5)\)as the base autoencoder\. This VAE is trained with reconstruction objectives and compresses each 17\-frame256×256256\\times 256clip into a latent of shape5×16×32×325\\times 16\\times 32\\times 32\(T×C×H×WT\\times C\\times H\\times W\)\. Starting from the official VAE checkpoint, we fine\-tune for 3 epochs using either reconstruction alone or reconstruction with PR; for video latents, PR’s average pooling spans both the spatial and temporal dimensions\. We then fine\-tune a Latte\-XL/2 model\([Ma et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib38)\)on each of three latent spaces: the official VAE, the VAE fine\-tuned with reconstruction alone, and the VAE fine\-tuned with PR\. All three Stage 2 runs are initialized from the EMA weights of the same Latte\-XL/2 checkpoint provided by LeanVAE and fine\-tuned for 50K steps with a global batch size of 10\. We also evaluate the VAE and Latte\-XL/2 checkpoints provided by LeanVAE without further training as a reference\. We report FVD \(over 2048 clips\), which measures the distance between the generated and real video distributions, and action classification accuracy from a pre\-trained UCF\-101 classifier, which measures the class consistency of conditional generation\. As shown in Tab\.[4](https://arxiv.org/html/2610.22885#S4.T4), PR improves FVD in both settings with on\-par accuracy, reducing FVD from152\.7152\.7to140\.1140\.1with CFG and from287\.3287\.3to275\.7275\.7without CFG\. These results suggest that the benefit of PR is not specific to images and extends to latent spaces of other modalities\.

Table 4:Generalization to video latents on UCF\-101\([Soomro et al\., 2012](https://arxiv.org/html/2610.22885#bib.bib39)\)using LeanVAE\-16\([Cheng and Yuan, 2025](https://arxiv.org/html/2610.22885#bib.bib5)\)and Latte\-XL/2\([Ma et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib38)\)\.*Official*denotes the VAE and Latte\-XL/2 checkpoints provided by LeanVAE, used without further training\. We report FVD and action classification accuracy with and without CFG\. The best results are inbold\.Table 5:ImageNet text\-to\-image generation using different Stage\-1 latent representations\. All metrics use 50K images generated from held\-out validation captions\. All alignment to CLIP\-L adopts its text embeddings\. CLIP denotes mean cosine similarity of CLIP embeddings calculated from text captions and generated images\. All cluster labels useK=1000K=1000\. The best results are inbold\.
### 4\.6Generalization to Continuous Conditioning

Finally, we examine whether PR benefits generation under continuous conditioning signals\. We consider text\-to\-image generation on ImageNet: we caption each image with Florence\-2\-large\([Xiao et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib40)\)and encode the caption into a CLIP ViT\-L/14 text embedding\([Radford et al\., 2021](https://arxiv.org/html/2610.22885#bib.bib33)\), which replaces the class embedding in LightningDiT\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6)\)through a small MLP adapter\. Besides LDM and VA\-VAE \(DINOv3\-L\), we compare against two continuous alignment baselines, which align latents via cosine similarity to pre\-trained features projected to the channel dimension by PCA\. We apply 1000\-class PR with ground\-truth classes, clusters of CLIP\-L text embeddings, and DINOv3\-L\[CLS\]tokens\. All metrics are computed on 50K images generated from held\-out validation captions\. As shown in Tab\.[5](https://arxiv.org/html/2610.22885#S4.T5), PR variants lead on all three metrics under guidance, with cluster labels consistently outperforming cosine alignment to the same representation\. Meanwhile, clusters of CLIP\-L text or DINOv3\-L image embeddings yield performance comparable to ground\-truth classes\. Without CFG, VA\-VAE performs best, consistent with its leading gFID in Tab\.[1](https://arxiv.org/html/2610.22885#S4.T1)\. Since captions typically group around shared objects or scenes, the latent geometry induced by PR may help organize representations along semantic groupings reflected in the captions, making it informative for text\-conditioned generation\. Together, these results demonstrate that PR improves guided text\-to\-image generation without requiring ground\-truth class labels\.

## 5Related Work

Latent Representation for Diffusion Models\.Latent diffusion models \(LDMs\) adopt a two\-stage paradigm\([Rombach et al\., 2022](https://arxiv.org/html/2610.22885#bib.bib1)\): an autoencoder maps high\-dimensional data to a compact latent space, and a diffusion model learns to generate within it\. Subsequent work expands the design space by widening the latent channels\([Esser et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib2);[Labs et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib3)\)and raising the compression ratio\([Chen et al\., 2025e](https://arxiv.org/html/2610.22885#bib.bib4)\)\. Recent work focuses on improving latent representations\. One line of work uses pretrained representations as supervision: VA\-VAE\([Yao et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib6)\)aligns latents with pretrained features during autoencoder training; REPA\-E\([Leng et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib8)\)enables end\-to\-end tuning through REPA alignment\([Yu et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib15)\); MAETok\([Chen et al\., 2025b](https://arxiv.org/html/2610.22885#bib.bib9)\)uses masked modeling with auxiliary decoders\. Another line builds on pretrained encoders: AlignTok\([Chen et al\., 2025a](https://arxiv.org/html/2610.22885#bib.bib10)\)adapts a pretrained encoder through multi\-stage training; RAE\([Zheng et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib11);[Singh et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib13)\)uses frozen pretrained representations as diffusion latents; and SVG\([Shi et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib12)\)augments frozen DINO features with learned residual features\. VTP\([Yao et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib7)\)further explores joint training objectives, combining image\-text contrastive, self\-supervised, and reconstruction losses\.

Diffusability of Latent Representations\.Recent work studies latent*diffusability*\(i\.e\., how easily diffusion models learn a latent distribution\)\. Some approaches focus on spectral structure and spatial equivariance:[Skorokhodov et al\. \(2025\)](https://arxiv.org/html/2610.22885#bib.bib20)use scale\-equivariance regularization to suppress excessive high\-frequency content; EQ\-VAE\([Kouzelis et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib21)\)promotes equivariance by matching reconstructions of transformed latents to transformed images; SSVAE\([Liu et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib22)\)studies latent spectral properties in video\. Others regularize the latent distribution directly: UL\([Heek et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib16)\)uses a learned diffusion prior; DMVAE\([Ye et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib17)\)aligns the aggregate latent distribution with a reference via score\-based distribution matching\. Complementary studies examine multiple properties jointly: PAE\([Yue et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib18)\)highlights spatial structure, local manifold continuity, and global semantics;[Zhong et al\. \(2026\)](https://arxiv.org/html/2610.22885#bib.bib19)identify velocity irreducible variance, semantic separability, and spatial structure as consistent predictors of generation quality in a large\-scale study\.

Classifier\-Free Guidance\.Classifier\-free guidance \(CFG\)\([Ho and Salimans, 2021](https://arxiv.org/html/2610.22885#bib.bib23)\)strengthens conditioning at inference at the cost of reduced diversity, with subsequent works refining this trade\-off in sampling\([Karras et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib14);[Kynkäänniemi et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib24)\)\. Analyses show that CFG does not sample from the intended tilted distribution\([Bradley and Nakkiran, 2024](https://arxiv.org/html/2610.22885#bib.bib43);[Chidambaram et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib42)\)and raises classification confidence while reducing diversity\([Wu et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib41)\)\. It can be decomposed into a class\-mean shift and contrastive components in linear models\([Li et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib25)\), or viewed as pushing trajectories away from decision boundaries\([Zhao and Schwing, 2026](https://arxiv.org/html/2610.22885#bib.bib26)\)\.[Pavasovic et al\. \(2025\)](https://arxiv.org/html/2610.22885#bib.bib44)show that in high dimensions CFG acts mainly during class selection, consistent with the vanishing correction in Fig\.[1](https://arxiv.org/html/2610.22885#S2.F1)\(a\)\. These works primarily analyze CFG under a given data distribution; we instead treat the latent space as a design variable and study which class geometry facilitates CFG\.

Class Separation for Guided Generation\.Recent work links CFG to class separation:[Ventura et al\. \(2026\)](https://arxiv.org/html/2610.22885#bib.bib45)show that CFG distorts the conditional target when classes are poorly separated\. MCLR\([Li et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib46)\)attributes the reliance on CFG to weak separation between learned conditionals and enlarges it by maximizing inter\-class likelihood ratios when training the generator; more broadly, guidance\-free training internalizes the effect of CFG into the generator\([Chen et al\., 2025c](https://arxiv.org/html/2610.22885#bib.bib47);[Tang et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib48);[Chen et al\., 2025d](https://arxiv.org/html/2610.22885#bib.bib49)\)\. ProtoDiffusion\([Baykal et al\., 2024](https://arxiv.org/html/2610.22885#bib.bib27)\)instead modifies the conditioning, replacing learned class embeddings with pretrained class prototypes\. These methods change how the generator is trained or conditioned while keeping the data representation fixed, whereas PR reshapes the latent space itself to promote class separation in the diffusion target\.

## 6Discussion

Unguided performance\.PR primarily targets guided generation, where its gains concentrate; without guidance, it still improves over LDM but trails VA\-VAE \(Tab\.[1](https://arxiv.org/html/2610.22885#S4.T1)\)\. The remaining gap may stem from average pooling, which constrains only the channel dimension and leaves spatial structure unregularized, while suppressing high\-frequency spatial components has been shown to be helpful for diffusability\([Skorokhodov et al\., 2025](https://arxiv.org/html/2610.22885#bib.bib20);[Liu et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib22)\)\. Since PR leaves the spatial residual unconstrained, it can be combined with regularizers that shape other aspects of the latent space: applying PR on top of VA\-VAE improves both guided and unguided generation \(Appendix[D\.5](https://arxiv.org/html/2610.22885#A4.SS5)\)\.

Scope of analysis\.Under Gaussian forward corruption, Tweedie’s formula links the posterior gradient to the Bayes\-optimal CFG correction \(Eq\.[6](https://arxiv.org/html/2610.22885#S2.E6)\) without assuming Gaussian latents \(Appendix[E\.7](https://arxiv.org/html/2610.22885#A5.SS7)\)\. The Gaussian class model additionally provides a closed\-form correction and predicts how variance reallocation affects noisy class separation and posterior uncertainty\. The empirical posterior trend is consistent with these predictions \(Fig\.[2](https://arxiv.org/html/2610.22885#S4.F2)\(b\)\)\. Appendix[F](https://arxiv.org/html/2610.22885#A6)further discusses how class geometry relates to other CFG quantities, complementing the posterior\-based analysis\.

Design space of PR\.PR instantiates the posterior objective through two design choices, the time distributionρ\\rhoand the approximation ofqcq\_\{c\}, both of which leave room for exploration \(Appendix[D\.1](https://arxiv.org/html/2610.22885#A4.SS1)\)\. Forρ\\rho, clean\-latent supervision avoids the over\-regularization of noisy supervision, though intermediate distributions may offer better trade\-offs\. Forqcq\_\{c\}, the pooled prototype classifier is one of many ways to realize the target\-versus\-competitor contrast\.

## 7Conclusion

In this work, we study CFG\-compatible representation learning through the lens of class geometry\. Our analysis links the oracle CFG correction to how distinguishable classes remain under noise, which motivates Prototype Regularization \(PR\), a lightweight regularizer that shapes class geometry and improves guided generation\. PR generalizes across label choices, data modalities, and condition signals, suggesting that aligning latent geometry through categorical priors is a simple yet effective way to design latent spaces for guided generation\. Given that CFG is used almost universally at sampling time, we hope this work encourages treating CFG compatibility as an explicit and principled axis of latent design, alongside reconstruction and generation quality\.

## Acknowledgments

This work is supported by the NYU Shanghai Center for Data Science\. This work is also supported in part through the NYU IT High Performance Computing resources, services, and staff expertise\.

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## Appendix

The appendix is organized as follows:

- •Appendix[A](https://arxiv.org/html/2610.22885#A1)presents UMAP visualizations of the learned VAE latents\.
- •Appendix[B](https://arxiv.org/html/2610.22885#A2)provides additional visualizations of noisy posterior and oracle velocity field\.
- •Appendix[C](https://arxiv.org/html/2610.22885#A3)provides full details of the experimental configurations\.
- •Appendix[D](https://arxiv.org/html/2610.22885#A4)reports additional experimental results\.
- •Appendix[E](https://arxiv.org/html/2610.22885#A5)derives the equations and claims stated in the main text\.
- •Appendix[F](https://arxiv.org/html/2610.22885#A6)derives complementary perspectives on classifier\-free guidance \(CFG\)\.

## Appendix ALatent Visualizations

![Refer to caption](https://arxiv.org/html/2610.22885v1/vis_latent.png)Figure 3:UMAP visualization of full and spatially pooled latents from LDM, VA\-VAE, and PR \(ours\)\. We encode 50K validation images and plot latents from 50 selected classes, colored by class label\. Latents are normalized per dimension before UMAP\. PR produces noticeably tighter and better separated class clusters\. Zoom in for the best view\. Better view with colors\.
## Appendix BClassifier\-Free Guidance Visualizations

![Refer to caption](https://arxiv.org/html/2610.22885v1/vis_posterior.png)Figure 4:Noisy marginals ofztz\_\{t\}at severalt∈\(0,1\)t\\in\(0,1\)for toy Gaussian mixtures with fixed total variance but different allocations between inter\-class and intra\-class variance\. Background colors mark the Maximum A Posteriori \(MAP\) decision regionsarg⁡maxc​qc​\(zt\)\\arg\\max\_\{c\}q\_\{c\}\(z\_\{t\}\), with opacity proportional to the confidencemaxc⁡qc​\(zt\)\\max\_\{c\}q\_\{c\}\(z\_\{t\}\)\. With larger inter\-class variance, classes become separable at smallertt, i\.e\., at noisier levels, and the confident MAP regions align more closely with the true class supports\.![Refer to caption](https://arxiv.org/html/2610.22885v1/vis_velocity.png)Figure 5:Oracle velocity fields on toy Gaussian mixtures with fixed total variance\. From top to bottom, the fraction of variance allocated to class means increases\. For each space, we show the conditional velocityvc⋆v\_\{c\}^\{\\star\}\(conditioned on the red class\), the unconditional velocityvu⋆v\_\{u\}^\{\\star\}, and the correctiongc⋆=vc⋆−vu⋆g\_\{c\}^\{\\star\}=v\_\{c\}^\{\\star\}\-v\_\{u\}^\{\\star\}att∈\{0\.2,0\.5,0\.8\}t\\in\\\{0\.2,0\.5,0\.8\\\}\. Gray contours mark the 95% density region ofztz\_\{t\}, and dots mark the class means ofztz\_\{t\}\. With clearer class geometry, the conditional and correction velocities point more consistently toward the target class mean, and the correction vanishes where the posterior concentrates on the target class\. Zoom in for the best view\. Better view with colors\.
## Appendix CExperiment Configurations

Table 6:Training and evaluation configurations for ImageNet experiments\. Full\-budget experiments use 8 GPUs; reduced\-budget ablations use 2 GPUs\. Both budgets use the same architectures and evaluation protocol, with experiment\-specific CFG search ranges listed below\.ConfigurationFull BudgetReduced BudgetData and ArchitectureDatasetImageNet,256×256256\\times 256GPUs per Run82Stage 1 ArchitectureKL\-VAE, downsampling factor 16VAE Base Channels / Multipliers128 /\(1,1,2,2,4\)\(1,1,2,2,4\)VAE Residual Blocks / Attention Resolution2 per level /16×1616\\times 16Latent Shape \(C×H×WC\\times H\\times W\)32×16×1632\\times 16\\times 16Stage 2 ArchitectureLightningDiT\-XL/1DiT Depth / Hidden Dimension / Heads28 / 1152 / 16DiT Patch Size1×11\\times 1DiT ComponentsQK\-Norm, SwiGLU, RoPE, RMSNormStage 1 TrainingImage PreprocessingShortest\-side resize to 256, random crop, no flipBase ObjectiveL1 \+ LPIPS \+ KL \+ adaptive adversarial lossKL Weight / Adversarial Weight Factor10−610^\{\-6\}/0\.50\.5Discriminator Start2,500 stepsOptimizer \(Generator and Discriminator\)Adam,\(β1,β2\)=\(0\.5,0\.9\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.5,0\.9\)Learning Rate / Schedule10−410^\{\-4\}/ constant, no warmupBatch Size per GPU3232Global Batch Size25664Training Epochs2010Recorded Training Steps100,080200,180Stage 2 TrainingLatent ExtractionFrozen EMA encoder, posterior mode, center cropLatent NormalizationPer\-channel mean and standard deviationObjective / PredictionLinear\-path flow matching / velocityTraining Time SamplerLogit\-normalOptimizerAdamWOptimizer\(β1,β2\)\(\\beta\_\{1\},\\beta\_\{2\}\)\(0\.9,0\.95\)\(0\.9,0\.95\)\(0\.9,0\.995\)\(0\.9,0\.995\)Learning Rate / Schedule10−410^\{\-4\}/ constant, no warmupWeight Decay / Max Gradient Norm0 / 1\.0Batch Size per GPU128128Global Batch Size1,024256Training Duration80 epochs∼\\sim30 epochsRecorded Training Steps100,080150,000Condition Dropout0\.1Shared Training SettingsPrecisionBF16 mixed precisionEMA Decay0\.9999Training Seed42Sampling and EvaluationEvaluation WeightsEMAStage 1 Evaluation50K validation images; rFID, PSNR, LPIPS, L1ODE Solver / StepsEuler / 50Sampling Time GridUniform, no timestep shiftCFG IntervalEnabled fort≥0\.11t\\geq 0\.11\(t=0t=0denotes noise\)CFG SelectionMinimum torch\-FID over 5K generated samplesDefault CFG Grid\{1\.25,1\.50,…,3\.50\}\\\{1\.25,1\.50,\\ldots,3\.50\\\}\(extended by step of 0\.5 if still improving\)Final Generation Evaluation50K samples with selected CFG; 50K without CFGGeneration Metrics / EvaluatorFID, sFID, IS, Precision, Recall / OpenAI ADMDefault FID ReferenceImageNet training reference batchAdditional T2I EvaluationValidation\-reference FID and image–prompt CLIP cosine
## Appendix DAdditional Experiment Results

### D\.1Ablation on Classifier Type in PR

Table 7:Ablation on the classifier type in PR \(reduced compute budget\)\.We vary the classifier head \(cosine vs\. linear\) and the latent on which PR is applied \(clean latents only vs\. noisy latents at uniformly sampled timesteps\)\. The cosine head on clean\-only latents achieves the best rFID and gFID, both with and without CFG\. Applying PR on uniform\-time noisy latents over\-regularizes the latent space: the top PCA component degenerates into a near\-constant channel that fails to capture spatial variations \(i\.e\., over\-smoothed latents\)\. The linear head, optimized with additional weight decay, collapses to a trivial solution that arbitrarily inflates the latent norm to enlarge the logits\.
### D\.2Ablation on Prototype Resolution

Table 8:Ablation on prototype shape \(reduced compute budget\)\.All variants use the default PR setting \(cosine prototype classifier on clean\-only latents\) and differ only in the output shape of spatial average pooling applied before the classifier, where16×1616\{\\times\}16corresponds to no pooling\. Overall, prototypes with lower resolutions perform best:1×11\{\\times\}1prototypes \(i\.e\., global average pooling\) achieve the best rFID and gFID without CFG, and2×22\{\\times\}2prototypes achieve the best gFID with CFG, while further increasing prototype resolution consistently degrades generation quality\. This suggests that reducing the spatial resolution of prototypes benefits generation, likely because it avoids imposing class\-level layout constraints on the latent space\.
### D\.3Ablation on Prototype Temperature

Table 9:Ablation on the temperature of PR \(reduced compute budget\)\.τ\\tauscales the cosine\-similarity logits of the prototype classifier, where a smallerτ\\tauyields a sharper softmax\. The defaultτ=0\.1\\tau=0\.1was fixed prior to this ablation and is used throughout the main text\. All PR variants improve over LDM both with and without CFG, and the differences amongτ\\tauvalues are small compared with the gain from adding PR, indicating thatτ=0\.1\\tau=0\.1is a reasonable default\. Adjustingτ\\tau\(e\.g\.,τ=0\.3\\tau=0\.3here\) may bring further improvements, though the best value is likely setting\-dependent\.
### D\.4Ablation on Pseudo\-label Granularity

Table 10:Ablation on DINO pseudo\-label granularity \(reduced compute budget\)\.This group of experiments uses a separate hyper\-parameter setup, so the results are comparable within this table but not with other tables\. Reconstruction is largely insensitive to the choice of targets\. Unguided gFID degrades asKKincreases, while guided gFID remains comparable acrossKKexcept for overly fine clusters\. The nested variant \(i\.e\., partitioning each of the 64 clusters into 16 sub\-clusters\) achieves the best generation among unsupervised targets \(best unguided gFID and guided gFID close to GT labels\), showing the feasibility of hierarchical PR\.
### D\.5Composability of PR and VA\-VAE

Table 11:Composability of PR and VA\-VAE\.This group of experiments uses the same separate hyper\-parameter setup as Tab\.[10](https://arxiv.org/html/2610.22885#A4.T10), so the results are comparable with that table but not with the others\. We apply PR on top of VA\-VAE, which aligns to DINOv3\-L features\. Combining the two yields better generation both with and without CFG, at the cost of mildly degraded reconstruction, indicating that PR is complementary to external feature alignment for generation\.

## Appendix EDerivation of Equations and Claims in Main Text

### E\.1Notation and setup

#### Setup\.

LetC∈\{1,…,K\}C\\in\\\{1,\\ldots,K\\\}denote the class label andZ∈ℝdZ\\in\\mathbb\{R\}^\{d\}the clean latent, with𝔼​‖Z‖2<∞\\mathbb\{E\}\\\|Z\\\|^\{2\}<\\infty\. Letϵ∼𝒩⁡\(0,I\)\\epsilon\\sim\\mathcal\{N\}\(0,I\)be independent of\(Z,C\)\(Z,C\)\. For a fixedt∈\(0,1\)t\\in\(0,1\), defineZt=t​Z\+\(1−t\)​ϵZ\_\{t\}=tZ\+\(1\-t\)\\epsilonandv=Z−ϵv=Z\-\\epsilon\. We usezz,cc, andztz\_\{t\}for realizations, writeqj​\(zt,t\)=Pr⁡\(C=j∣Zt=zt,t\)q\_\{j\}\(z\_\{t\},t\)=\\Pr\(C=j\\mid Z\_\{t\}=z\_\{t\},t\), and abbreviate it asqjq\_\{j\}when the arguments are clear\. The Bayes\-optimal conditional and unconditional velocity predictors are denoted byvc⋆v\_\{c\}^\{\\star\}andvu⋆v\_\{u\}^\{\\star\}, respectively, andgc⋆=vc⋆−vu⋆g\_\{c\}^\{\\star\}=v\_\{c\}^\{\\star\}\-v\_\{u\}^\{\\star\}\. All expectations and entropies involvingZtZ\_\{t\}are evaluated at the stated fixed time unless time is explicitly averaged over\. Under the Gaussian class model,Z\|C=j∼𝒩⁡\(μj,σ2​I\)Z\\mid C=j\\sim\\mathcal\{N\}\(\\mu\_\{j\},\\sigma^\{2\}I\)withσ2\>0\\sigma^\{2\}\>0, and we writest2=t2​σ2\+\(1−t\)2s\_\{t\}^\{2\}=t^\{2\}\\sigma^\{2\}\+\(1\-t\)^\{2\}\. Class priors are fixed and positive, but need not be equal unless stated otherwise\. For PR, the latent is viewed as a tensorz∈ℝdc×h×wz\\in\\mathbb\{R\}^\{d\_\{c\}\\times h\\times w\}, withd=dc​h​wd=d\_\{c\}hw, pooled featurez¯=AvgPool⁡\(z\)\\bar\{z\}=\\mathrm\{AvgPool\}\(z\), and normalized featuresz~=z¯/‖z¯‖\\tilde\{z\}=\\bar\{z\}/\\\|\\bar\{z\}\\\|andp~j=pj/‖pj‖\\tilde\{p\}\_\{j\}=p\_\{j\}/\\\|p\_\{j\}\\\|\. These normalized vectors are assumed to have nonzero denominators\. We reserveqjq\_\{j\}for the true posterior andπj\\pi\_\{j\}for the prototype classifier\. All logarithms are natural\.

### E\.2Posterior\-mean expression for CFG \(Eq\.[1](https://arxiv.org/html/2610.22885#S2.E1)\)

#### Derivation\.

The interpolant givesϵ=\(zt−t​z\)/\(1−t\)\\epsilon=\(z\_\{t\}\-tz\)/\(1\-t\), so the velocity target can be written as

v=z−ϵ=z−zt1−t\.v=z\-\\epsilon=\\frac\{z\-z\_\{t\}\}\{1\-t\}\.Under squared\-error prediction, the Bayes\-optimal predictors are conditional expectations\. Sinceztz\_\{t\}andttare fixed within these expectations,

vc⋆=𝔼\[Z∣Zt=zt,t,C=c\]−zt1−t,vu⋆=𝔼\[Z∣Zt=zt,t\]−zt1−t\.v\_\{c\}^\{\\star\}=\\frac\{\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t,C=c\]\-z\_\{t\}\}\{1\-t\},\\qquad v\_\{u\}^\{\\star\}=\\frac\{\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t\]\-z\_\{t\}\}\{1\-t\}\.Subtracting cancels the common observed latent and yields

gc⋆​\(zt,t\)=𝔼\[Z∣Zt=zt,t,C=c\]−𝔼\[Z∣Zt=zt,t\]1−t\.g\_\{c\}^\{\\star\}\(z\_\{t\},t\)=\\frac\{\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t,C=c\]\-\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t\]\}\{1\-t\}\.\(10\)This identity does not require the Gaussian class model\.

### E\.3Gaussian class geometry of CFG \(Eq\.[2](https://arxiv.org/html/2610.22885#S2.E2)\)

#### Derivation\.

Under the Gaussian class model, the independent within\-class variation and corruption noise give

Zt∣C=j∼𝒩\(tμj,st2I\),st2=t2σ2\+\(1−t\)2\.Z\_\{t\}\\mid C=j\\sim\\mathcal\{N\}\(t\\mu\_\{j\},s\_\{t\}^\{2\}I\),\\qquad s\_\{t\}^\{2\}=t^\{2\}\\sigma^\{2\}\+\(1\-t\)^\{2\}\.\(11\)Bayes’ rule therefore gives

qj​\(zt,t\)=Pr⁡\(C=j\)​exp⁡\(−‖zt−t​μj‖22​st2\)∑k=1KPr⁡\(C=k\)​exp⁡\(−‖zt−t​μk‖22​st2\)\.q\_\{j\}\(z\_\{t\},t\)=\\frac\{\\Pr\(C=j\)\\exp\\\!\\left\(\-\\frac\{\\\|z\_\{t\}\-t\\mu\_\{j\}\\\|^\{2\}\}\{2s\_\{t\}^\{2\}\}\\right\)\}\{\\sum\_\{k=1\}^\{K\}\\Pr\(C=k\)\\exp\\\!\\left\(\-\\frac\{\\\|z\_\{t\}\-t\\mu\_\{k\}\\\|^\{2\}\}\{2s\_\{t\}^\{2\}\}\\right\)\}\.Within classjj, the clean posterior mean combines the Gaussian prior and corruption likelihood:

p⁡\(z∣zt,t,C=j\)∝exp⁡\(−‖z−μj‖22​σ2−‖zt−t​z‖22​\(1−t\)2\)\.p\(z\\mid z\_\{t\},t,C=j\)\\propto\\exp\\\!\\left\(\-\\frac\{\\\|z\-\\mu\_\{j\}\\\|^\{2\}\}\{2\\sigma^\{2\}\}\-\\frac\{\\\|z\_\{t\}\-tz\\\|^\{2\}\}\{2\(1\-t\)^\{2\}\}\\right\)\.Completing the square gives

𝔼\[Z∣Zt=zt,t,C=j\]\\displaystyle\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t,C=j\]=\(1σ2\+t2\(1−t\)2\)−1​\(μjσ2\+t​zt\(1−t\)2\)\\displaystyle=\\left\(\\frac\{1\}\{\\sigma^\{2\}\}\+\\frac\{t^\{2\}\}\{\(1\-t\)^\{2\}\}\\right\)^\{\-1\}\\left\(\\frac\{\\mu\_\{j\}\}\{\\sigma^\{2\}\}\+\\frac\{tz\_\{t\}\}\{\(1\-t\)^\{2\}\}\\right\)=t​σ2st2​zt\+\(1−t\)2st2​μj\.\\displaystyle=\\frac\{t\\sigma^\{2\}\}\{s\_\{t\}^\{2\}\}z\_\{t\}\+\\frac\{\(1\-t\)^\{2\}\}\{s\_\{t\}^\{2\}\}\\mu\_\{j\}\.Averaging over the posterior class probabilities gives

𝔼\[Z∣Zt=zt,t\]=t​σ2st2zt\+\(1−t\)2st2∑j=1Kqjμj\.\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t\]=\\frac\{t\\sigma^\{2\}\}\{s\_\{t\}^\{2\}\}z\_\{t\}\+\\frac\{\(1\-t\)^\{2\}\}\{s\_\{t\}^\{2\}\}\\sum\_\{j=1\}^\{K\}q\_\{j\}\\mu\_\{j\}\.Substituting both posterior means into Eq\.[10](https://arxiv.org/html/2610.22885#A5.E10)yields

gc⋆​\(zt,t\)\\displaystyle g\_\{c\}^\{\\star\}\(z\_\{t\},t\)=1−tst2​\(μc−∑j=1Kqj​μj\)\\displaystyle=\\frac\{1\-t\}\{s\_\{t\}^\{2\}\}\\left\(\\mu\_\{c\}\-\\sum\_\{j=1\}^\{K\}q\_\{j\}\\mu\_\{j\}\\right\)=1−t\(1−t\)2\+t2​σ2​∑j≠cqj​\(μc−μj\),\\displaystyle=\\frac\{1\-t\}\{\(1\-t\)^\{2\}\+t^\{2\}\\sigma^\{2\}\}\\sum\_\{j\\neq c\}q\_\{j\}\(\\mu\_\{c\}\-\\mu\_\{j\}\),where the last equality uses∑jqj=1\\sum\_\{j\}q\_\{j\}=1\.

### E\.4Class separation under corruption \(Eq\.[3](https://arxiv.org/html/2610.22885#S2.E3)\)

#### Derivation\.

By Eq\.[11](https://arxiv.org/html/2610.22885#A5.E11), the corrupted class means aret​μct\\mu\_\{c\}andt​μjt\\mu\_\{j\}, with shared covariancest2​Is\_\{t\}^\{2\}I\. Their squared standardized distance is therefore

δc​j2​\(t\)\\displaystyle\\delta\_\{cj\}^\{2\}\(t\)=\(t​μc−t​μj\)⊤​\(st2​I\)−1​\(t​μc−t​μj\)\\displaystyle=\(t\\mu\_\{c\}\-t\\mu\_\{j\}\)^\{\\top\}\(s\_\{t\}^\{2\}I\)^\{\-1\}\(t\\mu\_\{c\}\-t\\mu\_\{j\}\)=t2​‖μc−μj‖2t2​σ2\+\(1−t\)2\.\\displaystyle=\\frac\{t^\{2\}\\\|\\mu\_\{c\}\-\\mu\_\{j\}\\\|^\{2\}\}\{t^\{2\}\\sigma^\{2\}\+\(1\-t\)^\{2\}\}\.

### E\.5Fixed\-variance construction \(Eq\.[4](https://arxiv.org/html/2610.22885#S2.E4)\)

#### Verification\.

Letμc=α​μcref\\mu\_\{c\}=\\sqrt\{\\alpha\}\\,\\mu\_\{c\}^\{\\mathrm\{ref\}\}andσ2=1−α\\sigma^\{2\}=1\-\\alpha, where𝔼C​\[μCref\]=0\\mathbb\{E\}\_\{C\}\[\\mu\_\{C\}^\{\\mathrm\{ref\}\}\]=0and𝔼C​‖μCref‖2=d\\mathbb\{E\}\_\{C\}\\\|\\mu\_\{C\}^\{\\mathrm\{ref\}\}\\\|^\{2\}=d\. Then𝔼⁡\[Z\]=0\\mathbb\{E\}\[Z\]=0, and the inter\-class and intra\-class variance contributions are, respectively,

𝔼C​‖𝔼⁡\[Z∣C\]−𝔼⁡\[Z\]‖2=α​𝔼C​‖μCref‖2=α​d,\\mathbb\{E\}\_\{C\}\\\|\\mathbb\{E\}\[Z\\mid C\]\-\\mathbb\{E\}\[Z\]\\\|^\{2\}=\\alpha\\,\\mathbb\{E\}\_\{C\}\\\|\\mu\_\{C\}^\{\\mathrm\{ref\}\}\\\|^\{2\}=\\alpha d,𝔼C​tr⁡Cov⁡\(Z∣C\)=d​σ2=\(1−α\)​d\.\\mathbb\{E\}\_\{C\}\\operatorname\{tr\}\\operatorname\{Cov\}\(Z\\mid C\)=d\\sigma^\{2\}=\(1\-\\alpha\)d\.The law of total variance therefore gives

tr⁡Cov⁡\(Z\)=α​d\+\(1−α\)​d=d\.\\operatorname\{tr\}\\operatorname\{Cov\}\(Z\)=\\alpha d\+\(1\-\\alpha\)d=d\.This reallocates a fixed total variance between class means and within\-class variation\.

### E\.6Posterior cross\-entropy and conditional entropy \(Eq\.[5](https://arxiv.org/html/2610.22885#S2.E5)\)

#### Derivation\.

At fixedtt, sampling\(Z,C,ϵ\)\(Z,C,\\epsilon\)induces the joint distribution of\(Zt,C\)\(Z\_\{t\},C\)\. Conditioning first onZtZ\_\{t\}and using the true class posterior gives

ℒpost​\(t\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{post\}\}\(t\)=𝔼Z,C,ϵ​\[−log⁡qC​\(t​Z\+\(1−t\)​ϵ,t\)\]\\displaystyle=\\mathbb\{E\}\_\{Z,C,\\epsilon\}\\\!\\left\[\-\\log q\_\{C\}\(tZ\+\(1\-t\)\\epsilon,t\)\\right\]=𝔼Zt​\[𝔼C\|Zt,t​\[−log⁡qC​\(Zt,t\)\]\]\\displaystyle=\\mathbb\{E\}\_\{Z\_\{t\}\}\\\!\\left\[\\mathbb\{E\}\_\{C\\mid Z\_\{t\},t\}\\\!\\left\[\-\\log q\_\{C\}\(Z\_\{t\},t\)\\right\]\\right\]=𝔼Zt\[−∑j=1Kqj\(Zt,t\)logqj\(Zt,t\)\]\\displaystyle=\\mathbb\{E\}\_\{Z\_\{t\}\}\\\!\\left\[\-\\sum\_\{j=1\}^\{K\}q\_\{j\}\(Z\_\{t\},t\)\\log q\_\{j\}\(Z\_\{t\},t\)\\right\]=H⁡\(C∣Zt\)\.\\displaystyle=H\(C\\mid Z\_\{t\}\)\.The equality relies on using the true posterior, rather than an arbitrary classifier\.

### E\.7Posterior gradient and CFG \(Eq\.[6](https://arxiv.org/html/2610.22885#S2.E6)\)

#### Derivation\.

We apply Tweedie’s formula under Gaussian forward corruption without assuming a Gaussian class model for the clean latent distribution\.

The corruption likelihood isp⁡\(zt∣z,t\)=𝒩⁡\(zt,t​z,\(1−t\)2​I\)p\(z\_\{t\}\\mid z,t\)=\\mathcal\{N\}\(z\_\{t\};tz,\(1\-t\)^\{2\}I\)\. Differentiating its marginal with respect toztz\_\{t\}gives

∇zt​log​p​\(zt∣C=c,t\)\\displaystyle\\nabla\_\{z\_\{t\}\}\\log p\(z\_\{t\}\\mid C=c,t\)=𝔼Z\|C=c​\[∇ztp​\(zt∣Z,t\)\]p⁡\(zt∣C=c,t\)\\displaystyle=\\frac\{\\mathbb\{E\}\_\{Z\\mid C=c\}\[\\nabla\_\{z\_\{t\}\}p\(z\_\{t\}\\mid Z,t\)\]\}\{p\(z\_\{t\}\\mid C=c,t\)\}=𝔼\[t​Z−zt\(1−t\)2\|Zt=zt,t,C=c\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\.\\frac\{tZ\-z\_\{t\}\}\{\(1\-t\)^\{2\}\}\\right\|Z\_\{t\}=z\_\{t\},t,C=c\\right\]=t𝔼\[Z∣Zt=zt,t,C=c\]−zt\(1−t\)2\.\\displaystyle=\\frac\{t\\,\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t,C=c\]\-z\_\{t\}\}\{\(1\-t\)^\{2\}\}\.The same calculation without conditioning on the class yields

∇zt​log​p​\(zt∣t\)=t𝔼\[Z∣Zt=zt,t\]−zt\(1−t\)2\.\\nabla\_\{z\_\{t\}\}\\log p\(z\_\{t\}\\mid t\)=\\frac\{t\\,\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t\]\-z\_\{t\}\}\{\(1\-t\)^\{2\}\}\.By Bayes’ rule, the class prior contributes no derivative with respect toztz\_\{t\}\. Subtracting the two scores and using Eq\.[10](https://arxiv.org/html/2610.22885#A5.E10), we obtain

∇zt​log​qc​\(zt,t\)\\displaystyle\\nabla\_\{z\_\{t\}\}\\log q\_\{c\}\(z\_\{t\},t\)=∇zt​log​p​\(zt∣C=c,t\)−∇zt​log​p​\(zt∣t\)\\displaystyle=\\nabla\_\{z\_\{t\}\}\\log p\(z\_\{t\}\\mid C=c,t\)\-\\nabla\_\{z\_\{t\}\}\\log p\(z\_\{t\}\\mid t\)=t\(1−t\)2\(𝔼\[Z∣Zt=zt,t,C=c\]−𝔼\[Z∣Zt=zt,t\]\)\\displaystyle=\\frac\{t\}\{\(1\-t\)^\{2\}\}\\left\(\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t,C=c\]\-\\mathbb\{E\}\[Z\\mid Z\_\{t\}=z\_\{t\},t\]\\right\)=t1−t​gc⋆​\(zt,t\)\.\\displaystyle=\\frac\{t\}\{1\-t\}g\_\{c\}^\{\\star\}\(z\_\{t\},t\)\.Finally, holdqcq\_\{c\},tt, andϵ\\epsilonfixed and differentiate throughzt=t​z\+\(1−t\)​ϵz\_\{t\}=tz\+\(1\-t\)\\epsilon:

∇z​log​qc​\(t​z\+\(1−t\)​ϵ,t\)=t​∇zt​log​qc​\(zt,t\)=t21−t​gc⋆​\(zt,t\)\.\\nabla\_\{z\}\\log q\_\{c\}\(tz\+\(1\-t\)\\epsilon,t\)=t\\nabla\_\{z\_\{t\}\}\\log q\_\{c\}\(z\_\{t\},t\)=\\frac\{t^\{2\}\}\{1\-t\}g\_\{c\}^\{\\star\}\(z\_\{t\},t\)\.This is the per\-sample negative cross\-entropy gradient with a fixed posterior function; it does not include changes in that function induced by changing the underlying latent distribution\. The identity does not require the Gaussian class model\.

### E\.8Monotonic posterior loss under variance allocation \(Eqs\.[4](https://arxiv.org/html/2610.22885#S2.E4)and[5](https://arxiv.org/html/2610.22885#S2.E5)\)

#### Derivation\.

Under the Gaussian family in Appendix[E\.5](https://arxiv.org/html/2610.22885#A5.SS5), keep the class priors and reference means\{μcref\}\\\{\\mu\_\{c\}^\{\\mathrm\{ref\}\}\\\}fixed, and write

Z=α​μCref\+1−α​η,α∈\(0,1\),Z=\\sqrt\{\\alpha\}\\,\\mu\_\{C\}^\{\\mathrm\{ref\}\}\+\\sqrt\{1\-\\alpha\}\\,\\eta,\\qquad\\alpha\\in\(0,1\),whereη\\etaand the corruption noiseϵ\\epsilonare independent standard Gaussian vectors, both independent ofCC\. For any fixedt∈\(0,1\]t\\in\(0,1\], the corrupted latent is

Zt=t​α​μCref\+t​1−α​η\+\(1−t\)​ϵ\.Z\_\{t\}=t\\sqrt\{\\alpha\}\\,\\mu\_\{C\}^\{\\mathrm\{ref\}\}\+t\\sqrt\{1\-\\alpha\}\\,\\eta\+\(1\-t\)\\epsilon\.Sincet2​\(1−α\)\+\(1−t\)2\>0t^\{2\}\(1\-\\alpha\)\+\(1\-t\)^\{2\}\>0throughout this domain, combining the two Gaussian noise terms and rescaling the observation gives the equivalent channel

Yα:=Ztt2​\(1−α\)\+\(1−t\)2=γt​\(α\)​μCref\+ξ,γt​\(α\):=t2​αt2​\(1−α\)\+\(1−t\)2,Y\_\{\\alpha\}:=\\frac\{Z\_\{t\}\}\{\\sqrt\{t^\{2\}\(1\-\\alpha\)\+\(1\-t\)^\{2\}\}\}=\\sqrt\{\\gamma\_\{t\}\(\\alpha\)\}\\,\\mu\_\{C\}^\{\\mathrm\{ref\}\}\+\\xi,\\qquad\\gamma\_\{t\}\(\\alpha\):=\\frac\{t^\{2\}\\alpha\}\{t^\{2\}\(1\-\\alpha\)\+\(1\-t\)^\{2\}\},whereξ∼𝒩⁡\(0,I\)\\xi\\sim\\mathcal\{N\}\(0,I\)is independent ofCC\. For eachα\\alpha, the rescaling is invertible, soH⁡\(C∣Zt\)=H⁡\(C∣Yα\)H\(C\\mid Z\_\{t\}\)=H\(C\\mid Y\_\{\\alpha\}\)\. Moreover,

d​γt​\(α\)d​α=t2​\(t2\+\(1−t\)2\)\(t2​\(1−α\)\+\(1−t\)2\)2\>0\.\\frac\{d\\gamma\_\{t\}\(\\alpha\)\}\{d\\alpha\}=\\frac\{t^\{2\}\\bigl\(t^\{2\}\+\(1\-t\)^\{2\}\\bigr\)\}\{\\bigl\(t^\{2\}\(1\-\\alpha\)\+\(1\-t\)^\{2\}\\bigr\)^\{2\}\}\>0\.
To compare0<α1<α2<10<\\alpha\_\{1\}<\\alpha\_\{2\}<1, writeγi=γt​\(αi\)\\gamma\_\{i\}=\\gamma\_\{t\}\(\\alpha\_\{i\}\), so that0<γ1<γ20<\\gamma\_\{1\}<\\gamma\_\{2\}, and construct

Y2=γ2​μCref\+ξ2,Y^1=γ1γ2​Y2\+1−γ1γ2​ξ′,Y\_\{2\}=\\sqrt\{\\gamma\_\{2\}\}\\,\\mu\_\{C\}^\{\\mathrm\{ref\}\}\+\\xi\_\{2\},\\qquad\\widehat\{Y\}\_\{1\}=\\sqrt\{\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{2\}\}\}\\,Y\_\{2\}\+\\sqrt\{1\-\\frac\{\\gamma\_\{1\}\}\{\\gamma\_\{2\}\}\}\\,\\xi^\{\\prime\},whereξ2\\xi\_\{2\}andξ′\\xi^\{\\prime\}are independent standard Gaussian vectors, both independent ofCC\. Conditional onCC,Y^1\\widehat\{Y\}\_\{1\}is Gaussian with meanγ1​μCref\\sqrt\{\\gamma\_\{1\}\}\\,\\mu\_\{C\}^\{\\mathrm\{ref\}\}and covarianceII\. Thus\(C,Y^1\)\(C,\\widehat\{Y\}\_\{1\}\)has the same distribution as\(C,Yα1\)\(C,Y\_\{\\alpha\_\{1\}\}\)\. Moreover,C→Y2→Y^1C\\to Y\_\{2\}\\to\\widehat\{Y\}\_\{1\}is a Markov chain\. The data\-processing inequality thus gives

I⁡\(C,Yα1\)=I⁡\(C,Y^1\)≤I⁡\(C,Y2\)=I⁡\(C,Yα2\)\.I\(C;Y\_\{\\alpha\_\{1\}\}\)=I\(C;\\widehat\{Y\}\_\{1\}\)\\leq I\(C;Y\_\{2\}\)=I\(C;Y\_\{\\alpha\_\{2\}\}\)\.Since the class priors and henceH⁡\(C\)H\(C\)are fixed,

H⁡\(C∣Yα2\)=H⁡\(C\)−I⁡\(C,Yα2\)≤H⁡\(C\)−I⁡\(C,Yα1\)=H⁡\(C∣Yα1\)\.H\(C\\mid Y\_\{\\alpha\_\{2\}\}\)=H\(C\)\-I\(C;Y\_\{\\alpha\_\{2\}\}\)\\leq H\(C\)\-I\(C;Y\_\{\\alpha\_\{1\}\}\)=H\(C\\mid Y\_\{\\alpha\_\{1\}\}\)\.Using the entropy identity in Appendix[E\.6](https://arxiv.org/html/2610.22885#A5.SS6), we conclude thatℒpost​\(t\)\\mathcal\{L\}\_\{\\mathrm\{post\}\}\(t\)is non\-increasing inα\\alphaat every fixedt∈\(0,1\]t\\in\(0,1\]\.

In particular, att=1t=1we haveZt=ZZ\_\{t\}=Zand

Yα=Z1−α,γ1​\(α\)=α1−α\.Y\_\{\\alpha\}=\\frac\{Z\}\{\\sqrt\{1\-\\alpha\}\},\\qquad\\gamma\_\{1\}\(\\alpha\)=\\frac\{\\alpha\}\{1\-\\alpha\}\.The rescaling remains invertible forα∈\(0,1\)\\alpha\\in\(0,1\), so the same argument directly establishes that the clean posterior lossH⁡\(C∣Z\)H\(C\\mid Z\)is non\-increasing inα\\alpha\.

### E\.9Prototype posterior and class\-contrast gradient \(Eqs\.[7](https://arxiv.org/html/2610.22885#S3.E7)and[8](https://arxiv.org/html/2610.22885#S3.E8)\)

#### Gaussian posterior form\.

For a clean latent under the Gaussian class model, Bayes’ rule and expansion of the squared distance give

qj​\(z,1\)=exp⁡\(μj⊤​zσ2−‖μj‖22​σ2\+log⁡Pr⁡\(C=j\)\)∑k=1Kexp⁡\(μk⊤​zσ2−‖μk‖22​σ2\+log⁡Pr⁡\(C=k\)\)\.q\_\{j\}\(z,1\)=\\frac\{\\exp\\\!\\left\(\\frac\{\\mu\_\{j\}^\{\\top\}z\}\{\\sigma^\{2\}\}\-\\frac\{\\\|\\mu\_\{j\}\\\|^\{2\}\}\{2\\sigma^\{2\}\}\+\\log\\Pr\(C=j\)\\right\)\}\{\\sum\_\{k=1\}^\{K\}\\exp\\\!\\left\(\\frac\{\\mu\_\{k\}^\{\\top\}z\}\{\\sigma^\{2\}\}\-\\frac\{\\\|\\mu\_\{k\}\\\|^\{2\}\}\{2\\sigma^\{2\}\}\+\\log\\Pr\(C=k\)\\right\)\}\.With equal priors and equal\-norm class means, the bias terms are common across classes and cancel:

qj​\(z,1\)=exp⁡\(μj⊤​z/σ2\)∑k=1Kexp⁡\(μk⊤​z/σ2\)\.q\_\{j\}\(z,1\)=\\frac\{\\exp\(\\mu\_\{j\}^\{\\top\}z/\\sigma^\{2\}\)\}\{\\sum\_\{k=1\}^\{K\}\\exp\(\\mu\_\{k\}^\{\\top\}z/\\sigma^\{2\}\)\}\.PR adopts this softmax inner\-product form on normalized pooled features, with learnable normalized prototypes and temperatureτ\>0\\tau\>0\. Its posterior and cross\-entropy are defined as

πj​\(z\)=exp⁡\(z~⊤​p~j/τ\)∑k=1Kexp⁡\(z~⊤​p~k/τ\),ℒPR=−log⁡πc​\(z\)\.\\pi\_\{j\}\(z\)=\\frac\{\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{j\}/\\tau\)\}\{\\sum\_\{k=1\}^\{K\}\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{k\}/\\tau\)\},\\qquad\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=\-\\log\\pi\_\{c\}\(z\)\.This surrogate matches the Gaussian discriminant form without assuming that normalized pooled features themselves follow the original Gaussian class model\.

#### Gradient derivation\.

Expand the cross\-entropy as

ℒPR=−z~⊤​p~cτ\+log∑k=1Kexp\(z~⊤p~k/τ\)\.\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=\-\\frac\{\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{c\}\}\{\\tau\}\+\\log\\sum\_\{k=1\}^\{K\}\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{k\}/\\tau\)\.Holding the prototypes fixed and differentiating with respect to the normalized feature gives

∇z~ℒPR\\displaystyle\\nabla\_\{\\tilde\{z\}\}\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=−p~cτ\+∑j=1Kexp⁡\(z~⊤​p~j/τ\)​p~j/τ∑k=1Kexp⁡\(z~⊤​p~k/τ\)\\displaystyle=\-\\frac\{\\tilde\{p\}\_\{c\}\}\{\\tau\}\+\\frac\{\\sum\_\{j=1\}^\{K\}\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{j\}/\\tau\)\\,\\tilde\{p\}\_\{j\}/\\tau\}\{\\sum\_\{k=1\}^\{K\}\\exp\(\\tilde\{z\}^\{\\top\}\\tilde\{p\}\_\{k\}/\\tau\)\}=1τ​\(∑j=1Kπj​p~j−p~c\)\.\\displaystyle=\\frac\{1\}\{\\tau\}\\left\(\\sum\_\{j=1\}^\{K\}\\pi\_\{j\}\\tilde\{p\}\_\{j\}\-\\tilde\{p\}\_\{c\}\\right\)\.Using∑jπj=1\\sum\_\{j\}\\pi\_\{j\}=1, we obtain

−∇z~ℒPR\\displaystyle\-\\nabla\_\{\\tilde\{z\}\}\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=1τ​\(p~c−∑j=1Kπj​p~j\)\\displaystyle=\\frac\{1\}\{\\tau\}\\left\(\\tilde\{p\}\_\{c\}\-\\sum\_\{j=1\}^\{K\}\\pi\_\{j\}\\tilde\{p\}\_\{j\}\\right\)=1τ​∑j≠cπj​\(p~c−p~j\)\.\\displaystyle=\\frac\{1\}\{\\tau\}\\sum\_\{j\\neq c\}\\pi\_\{j\}\(\\tilde\{p\}\_\{c\}\-\\tilde\{p\}\_\{j\}\)\.This has the same posterior\-weighted class\-contrast structure as the expression in Appendix[E\.3](https://arxiv.org/html/2610.22885#A5.SS3), with normalized prototypes and classifier probabilities replacing class means and true posterior probabilities\. For completeness, we derive the gradient with respect to the clean latent\. Withz¯=N−1​∑s=1Nzs\\bar\{z\}=N^\{\-1\}\\sum\_\{s=1\}^\{N\}z\_\{s\}andN=h​wN=hw, backpropagation through normalization and pooling gives

−∇zsℒPR=I−z~​z~⊤N​τ​‖z¯‖∑j≠cπj\(p~c−p~j\),s=1,…,N\.\-\\nabla\_\{z\_\{s\}\}\\mathcal\{L\}\_\{\\mathrm\{PR\}\}=\\frac\{I\-\\tilde\{z\}\\tilde\{z\}^\{\\top\}\}\{N\\tau\\\|\\bar\{z\}\\\|\}\\sum\_\{j\\neq c\}\\pi\_\{j\}\(\\tilde\{p\}\_\{c\}\-\\tilde\{p\}\_\{j\}\),\\qquad s=1,\\ldots,N\.Thus, the class contrast is projected onto the tangent space atz~\\tilde\{z\}, rescaled, and distributed equally across spatial locations\. The resulting supervision promotes class discrimination in the pooled representation while leaving spatial residualszs−z¯z\_\{s\}\-\\bar\{z\}unconstrained\.

## Appendix FComplementary Perspectives on Classifier\-Free Guidance

We examine CFG from three complementary perspectives: the irreducible uncertainty of velocity prediction, the error in a learned guidance correction, and the estimation accuracy attainable under a controlled model\. The first isolates the role of latent geometry, the second separates conditional error contrasts from branch consistency, and the third connects these quantities through an explicit estimator\. Throughout,Zt=t​Z\+\(1−t\)​ϵZ\_\{t\}=tZ\+\(1\-t\)\\epsilon,v=Z−ϵv=Z\-\\epsilon, andt∈\(0,1\)t\\in\(0,1\)is fixed, withϵ∼𝒩⁡\(0,I\)\\epsilon\\sim\\mathcal\{N\}\(0,I\)independent of\(Z,C\)\(Z,C\)\. Expectations are taken over the forward process generated by\(Z,C,ϵ\)\(Z,C,\\epsilon\)\. We useqj=Pr⁡\(C=j∣Zt,t\)q\_\{j\}=\\Pr\(C=j\\mid Z\_\{t\},t\),vj⋆=𝔼\[v∣Zt,t,C=j\]v\_\{j\}^\{\\star\}=\\mathbb\{E\}\[v\\mid Z\_\{t\},t,C=j\],vu⋆=𝔼\[v∣Zt,t\]v\_\{u\}^\{\\star\}=\\mathbb\{E\}\[v\\mid Z\_\{t\},t\], andgc⋆=vc⋆−vu⋆g\_\{c\}^\{\\star\}=v\_\{c\}^\{\\star\}\-v\_\{u\}^\{\\star\}\.

### F\.1Class Geometry and Irreducible Velocity Uncertainty

#### Takeaway and setup\.

Under the Gaussian class model, reallocating a fixed total variance toward class means strictly reduces the conditional velocity Bayes risk and does not increase the unconditional risk\. AssumeZ\|C=c∼𝒩⁡\(μc,σ2​I\)Z\\mid C=c\\sim\\mathcal\{N\}\(\\mu\_\{c\},\\sigma^\{2\}I\)and writest2=\(1−t\)2\+t2​σ2s\_\{t\}^\{2\}=\(1\-t\)^\{2\}\+t^\{2\}\\sigma^\{2\}\. For the geometric comparison, fix the class priors and reference means satisfying𝔼C​\[μCref\]=0\\mathbb\{E\}\_\{C\}\[\\mu\_\{C\}^\{\\mathrm\{ref\}\}\]=0and𝔼C​‖μCref‖2=d\\mathbb\{E\}\_\{C\}\\\|\\mu\_\{C\}^\{\\mathrm\{ref\}\}\\\|^\{2\}=d, and setμc=α​μcref\\mu\_\{c\}=\\sqrt\{\\alpha\}\\,\\mu\_\{c\}^\{\\mathrm\{ref\}\}andσ2=1−α\\sigma^\{2\}=1\-\\alpha, withα∈\(0,1\)\\alpha\\in\(0,1\)\.

#### Velocity Bayes risks\.

The conditional and unconditional irreducible errors satisfy

ℛcond​\(t\)\\displaystyle\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\):=𝔼​‖v−vC⋆‖2=d​σ2st2,\\displaystyle:=\\mathbb\{E\}\\\|v\-v\_\{C\}^\{\\star\}\\\|^\{2\}=\\frac\{d\\sigma^\{2\}\}\{s\_\{t\}^\{2\}\},\(12\)ℛuncond​\(t\)\\displaystyle\\mathcal\{R\}\_\{\\mathrm\{uncond\}\}\(t\):=𝔼​‖v−vu⋆‖2=ℛcond​\(t\)\+𝔼​‖gC⋆‖2\.\\displaystyle:=\\mathbb\{E\}\\\|v\-v\_\{u\}^\{\\star\}\\\|^\{2\}=\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)\+\\mathbb\{E\}\\\|g\_\{C\}^\{\\star\}\\\|^\{2\}\.To obtain the first identity, Gaussian conditioning gives

Cov⁡\(Z∣Zt,t,C\)=\(1σ2\+t2\(1−t\)2\)−1​I=σ2​\(1−t\)2st2​I\.\\operatorname\{Cov\}\(Z\\mid Z\_\{t\},t,C\)=\\left\(\\frac\{1\}\{\\sigma^\{2\}\}\+\\frac\{t^\{2\}\}\{\(1\-t\)^\{2\}\}\\right\)^\{\-1\}I=\\frac\{\\sigma^\{2\}\(1\-t\)^\{2\}\}\{s\_\{t\}^\{2\}\}I\.Sincev=\(Z−Zt\)/\(1−t\)v=\(Z\-Z\_\{t\}\)/\(1\-t\), its conditional covariance isσ2​I/st2\\sigma^\{2\}I/s\_\{t\}^\{2\}, whose trace givesℛcond\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\. For the second identity, writev−vu⋆=\(v−vC⋆\)\+gC⋆v\-v\_\{u\}^\{\\star\}=\(v\-v\_\{C\}^\{\\star\}\)\+g\_\{C\}^\{\\star\}\. The cross term vanishes because𝔼\[v−vC⋆∣Zt,t,C\]=0\\mathbb\{E\}\[v\-v\_\{C\}^\{\\star\}\\mid Z\_\{t\},t,C\]=0\.

#### Effect of variance allocation\.

For any0<α1<α2<10<\\alpha\_\{1\}<\\alpha\_\{2\}<1, we have

ℛcond​\(t,α2\)\\displaystyle\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t;\\alpha\_\{2\}\)<ℛcond​\(t,α1\),\\displaystyle<\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t;\\alpha\_\{1\}\),\(13\)ℛuncond​\(t,α2\)\\displaystyle\\mathcal\{R\}\_\{\\mathrm\{uncond\}\}\(t;\\alpha\_\{2\}\)≤ℛuncond​\(t,α1\)\.\\displaystyle\\leq\\mathcal\{R\}\_\{\\mathrm\{uncond\}\}\(t;\\alpha\_\{1\}\)\.The conditional statement follows directly from

∂ℛcond​\(t,α\)∂α=−d​\(1−t\)2\(\(1−t\)2\+t2​\(1−α\)\)2<0\.\\frac\{\\partial\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t;\\alpha\)\}\{\\partial\\alpha\}=\-\\frac\{d\(1\-t\)^\{2\}\}\{\\bigl\(\(1\-t\)^\{2\}\+t^\{2\}\(1\-\\alpha\)\\bigr\)^\{2\}\}<0\.For the unconditional statement, couple the two models as follows\. LetZ2Z\_\{2\}follow the family atα2\\alpha\_\{2\}, letϵ2∼𝒩⁡\(0,I\)\\epsilon\_\{2\}\\sim\\mathcal\{N\}\(0,I\)be independent, and take independent standard Gaussiansη0,ϵ0\\eta\_\{0\},\\epsilon\_\{0\}independent of\(Z2,C,ϵ2\)\(Z\_\{2\},C,\\epsilon\_\{2\}\)\. Set

ρ=α1α2,Z1=ρ​Z2\+1−ρ2​η0,ϵ1=ρ​ϵ2\+1−ρ2​ϵ0\.\\rho=\\sqrt\{\\frac\{\\alpha\_\{1\}\}\{\\alpha\_\{2\}\}\},\\qquad Z\_\{1\}=\\rho Z\_\{2\}\+\\sqrt\{1\-\\rho^\{2\}\}\\,\\eta\_\{0\},\\qquad\\epsilon\_\{1\}=\\rho\\epsilon\_\{2\}\+\\sqrt\{1\-\\rho^\{2\}\}\\,\\epsilon\_\{0\}\.ThenZ1\|C=c∼𝒩⁡\(α1​μcref,\(1−α1\)​I\)Z\_\{1\}\\mid C=c\\sim\\mathcal\{N\}\(\\sqrt\{\\alpha\_\{1\}\}\\mu\_\{c\}^\{\\mathrm\{ref\}\},\(1\-\\alpha\_\{1\}\)I\), andϵ1\\epsilon\_\{1\}is independent of\(Z1,C\)\(Z\_\{1\},C\)\. WriteYi=t​Zi\+\(1−t\)​ϵiY\_\{i\}=tZ\_\{i\}\+\(1\-t\)\\epsilon\_\{i\}andVi=Zi−ϵiV\_\{i\}=Z\_\{i\}\-\\epsilon\_\{i\}fori=1,2i=1,2\. Remove a common linear predictor by defining

At=t2\+\(1−t\)2,κt=2​t−1At,ri=Vi−κt​Yi\.A\_\{t\}=t^\{2\}\+\(1\-t\)^\{2\},\\qquad\\kappa\_\{t\}=\\frac\{2t\-1\}\{A\_\{t\}\},\\qquad r\_\{i\}=V\_\{i\}\-\\kappa\_\{t\}Y\_\{i\}\.The fixed total variance implies

𝔼​‖Yi‖2=d​At,𝔼⁡\[Vi⊤​Yi\]=d⁡\(2​t−1\),𝔼​‖Vi‖2=2​d,\\mathbb\{E\}\\\|Y\_\{i\}\\\|^\{2\}=dA\_\{t\},\\qquad\\mathbb\{E\}\[V\_\{i\}^\{\\top\}Y\_\{i\}\]=d\(2t\-1\),\\qquad\\mathbb\{E\}\\\|V\_\{i\}\\\|^\{2\}=2d,and hence𝔼​‖ri‖2=d/At\\mathbb\{E\}\\\|r\_\{i\}\\\|^\{2\}=d/A\_\{t\}for both models\. Moreover,

Y1=ρ​Y2\+1−ρ2​Y0,r1=ρ​r2\+1−ρ2​r0,Y\_\{1\}=\\rho Y\_\{2\}\+\\sqrt\{1\-\\rho^\{2\}\}\\,Y\_\{0\},\\qquad r\_\{1\}=\\rho r\_\{2\}\+\\sqrt\{1\-\\rho^\{2\}\}\\,r\_\{0\},whereY0=t​η0\+\(1−t\)​ϵ0Y\_\{0\}=t\\eta\_\{0\}\+\(1\-t\)\\epsilon\_\{0\}andr0=η0−ϵ0−κt​Y0r\_\{0\}=\\eta\_\{0\}\-\\epsilon\_\{0\}\-\\kappa\_\{t\}Y\_\{0\}\. These two auxiliary variables are jointly Gaussian with zero cross\-covariance, sor0r\_\{0\}is independent ofY0Y\_\{0\}, and both are independent of\(Y2,r2\)\(Y\_\{2\},r\_\{2\}\)\. Therefore,

𝔼⁡\[r1∣Y1\]=ρ​𝔼​\[𝔼⁡\[r2∣Y2\]∣Y1\],𝔼​‖𝔼⁡\[r1∣Y1\]‖2≤ρ2​𝔼​‖𝔼⁡\[r2∣Y2\]‖2\.\\mathbb\{E\}\[r\_\{1\}\\mid Y\_\{1\}\]=\\rho\\,\\mathbb\{E\}\\\!\\left\[\\mathbb\{E\}\[r\_\{2\}\\mid Y\_\{2\}\]\\mid Y\_\{1\}\\right\],\\qquad\\mathbb\{E\}\\\|\\mathbb\{E\}\[r\_\{1\}\\mid Y\_\{1\}\]\\\|^\{2\}\\leq\\rho^\{2\}\\mathbb\{E\}\\\|\\mathbb\{E\}\[r\_\{2\}\\mid Y\_\{2\}\]\\\|^\{2\}\.Finally, subtracting a known function of the observation does not change the Bayes risk\. Orthogonality of conditional expectation gives

ℛuncond​\(t,α1\)\\displaystyle\\mathcal\{R\}\_\{\\mathrm\{uncond\}\}\(t;\\alpha\_\{1\}\)=dAt−𝔼​‖𝔼⁡\[r1∣Y1\]‖2\\displaystyle=\\frac\{d\}\{A\_\{t\}\}\-\\mathbb\{E\}\\\|\\mathbb\{E\}\[r\_\{1\}\\mid Y\_\{1\}\]\\\|^\{2\}≥dAt−ρ2​𝔼​‖𝔼⁡\[r2∣Y2\]‖2\\displaystyle\\geq\\frac\{d\}\{A\_\{t\}\}\-\\rho^\{2\}\\mathbb\{E\}\\\|\\mathbb\{E\}\[r\_\{2\}\\mid Y\_\{2\}\]\\\|^\{2\}≥dAt−𝔼​‖𝔼⁡\[r2∣Y2\]‖2=ℛuncond​\(t,α2\)\.\\displaystyle\\geq\\frac\{d\}\{A\_\{t\}\}\-\\mathbb\{E\}\\\|\\mathbb\{E\}\[r\_\{2\}\\mid Y\_\{2\}\]\\\|^\{2\}=\\mathcal\{R\}\_\{\\mathrm\{uncond\}\}\(t;\\alpha\_\{2\}\)\.

#### Connection to diffusability\.

Velocity irreducible variance has been studied as a statistical indicator of latent diffusability\([Zhong et al\., 2026](https://arxiv.org/html/2610.22885#bib.bib19)\)\. The conditional risk in Eq\.[12](https://arxiv.org/html/2610.22885#A6.E12)is the isotropic Gaussian instance of that quantity\. Equation[13](https://arxiv.org/html/2610.22885#A6.E13)connects this perspective to class geometry: at a fixed latent scale, increasing the inter\-class variance fraction reduces velocity\-target uncertainty in the conditional branch and preserves or reduces it in the unconditional branch\.

### F\.2Conditional Error Contrasts and Branch Consistency

#### Takeaway and setup\.

The error in a learned CFG correction separates exactly into differences between conditional prediction errors and a consistency term relating the two branches\. This decomposition holds without Gaussian assumptions\. Letej=v^j−vj⋆e\_\{j\}=\\widehat\{v\}\_\{j\}\-v\_\{j\}^\{\\star\}andeu=v^u−vu⋆e\_\{u\}=\\widehat\{v\}\_\{u\}\-v\_\{u\}^\{\\star\}, and defineg^c=v^c−v^u\\widehat\{g\}\_\{c\}=\\widehat\{v\}\_\{c\}\-\\widehat\{v\}\_\{u\}\. Treat the learned predictors as fixed when averaging over the forward distribution\.

#### Exact correction\-error decomposition\.

The expected correction error is

ℰg​\(t\)\\displaystyle\\mathcal\{E\}\_\{g\}\(t\):=𝔼​‖g^C−gC⋆‖2=ℰg\(1\)​\(t\)\+ℰg\(2\)​\(t\),\\displaystyle:=\\mathbb\{E\}\\\|\\widehat\{g\}\_\{C\}\-g\_\{C\}^\{\\star\}\\\|^\{2\}=\\mathcal\{E\}\_\{g\}^\{\(1\)\}\(t\)\+\\mathcal\{E\}\_\{g\}^\{\(2\)\}\(t\),\(14\)ℰg\(1\)​\(t\)\\displaystyle\\mathcal\{E\}\_\{g\}^\{\(1\)\}\(t\):=12​𝔼​∑i,jqi​qj​‖ei−ej‖2,\\displaystyle:=\\frac\{1\}\{2\}\\mathbb\{E\}\\sum\_\{i,j\}q\_\{i\}q\_\{j\}\\\|e\_\{i\}\-e\_\{j\}\\\|^\{2\},ℰg\(2\)​\(t\)\\displaystyle\\mathcal\{E\}\_\{g\}^\{\(2\)\}\(t\):=𝔼​‖v^u−∑jqj​v^j‖2\.\\displaystyle:=\\mathbb\{E\}\\left\\\|\\widehat\{v\}\_\{u\}\-\\sum\_\{j\}q\_\{j\}\\widehat\{v\}\_\{j\}\\right\\\|^\{2\}\.To see this, the tower property givesvu⋆=∑jqj​vj⋆v\_\{u\}^\{\\star\}=\\sum\_\{j\}q\_\{j\}v\_\{j\}^\{\\star\}\. Definee¯=∑jqj​ej\\bar\{e\}=\\sum\_\{j\}q\_\{j\}e\_\{j\}andΔu=v^u−∑jqj​v^j=eu−e¯\\Delta\_\{u\}=\\widehat\{v\}\_\{u\}\-\\sum\_\{j\}q\_\{j\}\\widehat\{v\}\_\{j\}=e\_\{u\}\-\\bar\{e\}\. Then

g^C−gC⋆=eC−eu=\(eC−e¯\)−Δu\.\\widehat\{g\}\_\{C\}\-g\_\{C\}^\{\\star\}=e\_\{C\}\-e\_\{u\}=\(e\_\{C\}\-\\bar\{e\}\)\-\\Delta\_\{u\}\.Conditional onZtZ\_\{t\}, we have𝔼\[eC−e¯∣Zt,t\]=0\\mathbb\{E\}\[e\_\{C\}\-\\bar\{e\}\\mid Z\_\{t\},t\]=0, so the cross term vanishes\. The remaining conditional variance satisfies

∑jqj​‖ej−e¯‖2=∑jqj​‖ej‖2−‖e¯‖2=12​∑i,jqi​qj​‖ei−ej‖2,\\sum\_\{j\}q\_\{j\}\\\|e\_\{j\}\-\\bar\{e\}\\\|^\{2\}=\\sum\_\{j\}q\_\{j\}\\\|e\_\{j\}\\\|^\{2\}\-\\\|\\bar\{e\}\\\|^\{2\}=\\frac\{1\}\{2\}\\sum\_\{i,j\}q\_\{i\}q\_\{j\}\\\|e\_\{i\}\-e\_\{j\}\\\|^\{2\},which proves Eq\.[14](https://arxiv.org/html/2610.22885#A6.E14)\. The first term measures error differences across jointly plausible classes; any component shared by alleje\_\{j\}cancels\. The second measures agreement between the learned unconditional prediction and the posterior\-weighted conditional predictions\.

#### Posterior weighting and prediction uncertainty\.

Define the expected posterior ambiguity as

Ut:=𝔼⁡\[1−∑jqj2\]=𝔼​∑i≠jqi​qj\.U\_\{t\}:=\\mathbb\{E\}\\\!\\left\[1\-\\sum\_\{j\}q\_\{j\}^\{2\}\\right\]=\\mathbb\{E\}\\sum\_\{i\\neq j\}q\_\{i\}q\_\{j\}\.If‖ei−ej‖≤B\\\|e\_\{i\}\-e\_\{j\}\\\|\\leq Buniformly, thenℰg\(1\)​\(t\)≤B2​Ut/2\\mathcal\{E\}\_\{g\}^\{\(1\)\}\(t\)\\leq B^\{2\}U\_\{t\}/2\. Thus, for a common error\-difference bound, lower posterior ambiguity tightens the bound on the conditional contrast term\. These learned errors are distinct from the irreducible uncertainty in Appendix[F\.1](https://arxiv.org/html/2610.22885#A6.SS1); for example, conditional prediction obeys

𝔼​‖v−v^C‖2=ℛcond​\(t\)\+𝔼​‖eC‖2\.\\mathbb\{E\}\\\|v\-\\widehat\{v\}\_\{C\}\\\|^\{2\}=\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)\+\\mathbb\{E\}\\\|e\_\{C\}\\\|^\{2\}\.Here the first term is determined by the data distribution, while the second measures approximation of its oracle predictor\.

### F\.3Geometry\-Dependent Estimation Error under Consistent Branches

#### Takeaway and setup\.

Under a controlled Gaussian estimator, within\-class compactness and low posterior ambiguity jointly reduce correction and guided\-field estimation errors\. Use the same fixed\-variance family as in Appendix[F\.1](https://arxiv.org/html/2610.22885#A6.SS1)\. Assume that the common linear term of the conditional predictors is known, estimate each class offset fromnnindependent residual samples, and construct the unconditional estimate using the true posterior weights\. Estimation samples are independent across classes and independent of the test sample\. Expectations below also average over these estimation samples, withnnand the guidance scaleω≥1\\omega\\geq 1held fixed across the family\.

#### Estimating the class offsets\.

The Gaussian conditional predictor has the form

vj⋆​\(zt,t\)=at​zt\+hj,at=t​σ2−\(1−t\)st2,hj=1−tst2​μj\.v\_\{j\}^\{\\star\}\(z\_\{t\},t\)=a\_\{t\}z\_\{t\}\+h\_\{j\},\\qquad a\_\{t\}=\\frac\{t\\sigma^\{2\}\-\(1\-t\)\}\{s\_\{t\}^\{2\}\},\\qquad h\_\{j\}=\\frac\{1\-t\}\{s\_\{t\}^\{2\}\}\\mu\_\{j\}\.For classjj, a velocity residualv−at​Ztv\-a\_\{t\}Z\_\{t\}has meanhjh\_\{j\}and covarianceσ2​I/st2\\sigma^\{2\}I/s\_\{t\}^\{2\}: subtractinghjh\_\{j\}givesv−vj⋆v\-v\_\{j\}^\{\\star\}, whose conditional mean is zero and whose conditional covariance was computed in Appendix[F\.1](https://arxiv.org/html/2610.22885#A6.SS1)\. Estimate

h^j=1n​∑ℓ=1n\(vj​ℓ−at​Zt,j​ℓ\),v^j​\(zt,t\)=at​zt\+h^j,v^u=∑jqj​v^j\.\\widehat\{h\}\_\{j\}=\\frac\{1\}\{n\}\\sum\_\{\\ell=1\}^\{n\}\\bigl\(v\_\{j\\ell\}\-a\_\{t\}Z\_\{t,j\\ell\}\\bigr\),\\qquad\\widehat\{v\}\_\{j\}\(z\_\{t\},t\)=a\_\{t\}z\_\{t\}\+\\widehat\{h\}\_\{j\},\\qquad\\widehat\{v\}\_\{u\}=\\sum\_\{j\}q\_\{j\}\\widehat\{v\}\_\{j\}\.Consequently,ej=h^j−hje\_\{j\}=\\widehat\{h\}\_\{j\}\-h\_\{j\}is independent of the test sample and satisfies

𝔼\[ej\]=0,𝔼∥ej∥2=d​σ2n​st2=ℛcond​\(t\)n,𝔼\[ei⊤ej\]=0\(i≠j\)\.\\mathbb\{E\}\[e\_\{j\}\]=0,\\qquad\\mathbb\{E\}\\\|e\_\{j\}\\\|^\{2\}=\\frac\{d\\sigma^\{2\}\}\{ns\_\{t\}^\{2\}\}=\\frac\{\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)\}\{n\},\\qquad\\mathbb\{E\}\[e\_\{i\}^\{\\top\}e\_\{j\}\]=0\\quad\(i\\neq j\)\.

#### Correction and guided\-field errors\.

The construction makesℰg\(2\)​\(t\)=0\\mathcal\{E\}\_\{g\}^\{\(2\)\}\(t\)=0in Eq\.[14](https://arxiv.org/html/2610.22885#A6.E14)\. Since𝔼​‖ei−ej‖2=2​ℛcond​\(t\)/n\\mathbb\{E\}\\\|e\_\{i\}\-e\_\{j\}\\\|^\{2\}=2\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)/nfori≠ji\\neq j, its first term gives the correction error below\. For guidance, definevcfg⋆=vu⋆\+ω⁡\(vC⋆−vu⋆\)v\_\{\\mathrm\{cfg\}\}^\{\\star\}=v\_\{u\}^\{\\star\}\+\\omega\(v\_\{C\}^\{\\star\}\-v\_\{u\}^\{\\star\}\)andv^cfg=v^u\+ω⁡\(v^C−v^u\)\\widehat\{v\}\_\{\\mathrm\{cfg\}\}=\\widehat\{v\}\_\{u\}\+\\omega\(\\widehat\{v\}\_\{C\}\-\\widehat\{v\}\_\{u\}\)\. Then

v^cfg−vcfg⋆=ωeC\+\(1−ω\)e¯=∑j\(ω𝟏\{j=C\}\+\(1−ω\)qj\)ej\.\\widehat\{v\}\_\{\\mathrm\{cfg\}\}\-v\_\{\\mathrm\{cfg\}\}^\{\\star\}=\\omega e\_\{C\}\+\(1\-\\omega\)\\bar\{e\}=\\sum\_\{j\}\\bigl\(\\omega\\mathbf\{1\}\_\{\\\{j=C\\\}\}\+\(1\-\\omega\)q\_\{j\}\\bigr\)e\_\{j\}\.Independence and zero\-mean estimation errors remove cross terms\. Averaging the squared coefficients overC\|ZtC\\mid Z\_\{t\}gives

∑cqc∑j\(ω𝟏\{j=c\}\+\(1−ω\)qj\)2=ω2\+\(1−ω2\)∑jqj2\.\\sum\_\{c\}q\_\{c\}\\sum\_\{j\}\\bigl\(\\omega\\mathbf\{1\}\_\{\\\{j=c\\\}\}\+\(1\-\\omega\)q\_\{j\}\\bigr\)^\{2\}=\\omega^\{2\}\+\(1\-\\omega^\{2\}\)\\sum\_\{j\}q\_\{j\}^\{2\}\.Thus, the two estimation errors are

ℰg​\(t\)\\displaystyle\\mathcal\{E\}\_\{g\}\(t\)=ℛcond​\(t\)n​Ut,\\displaystyle=\\frac\{\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)\}\{n\}\\,U\_\{t\},\(15\)ℰω​\(t\)\\displaystyle\\mathcal\{E\}\_\{\\omega\}\(t\):=𝔼​‖v^cfg−vcfg⋆‖2=ℛcond​\(t\)n​\[1\+\(ω2−1\)​Ut\]\.\\displaystyle:=\\mathbb\{E\}\\\|\\widehat\{v\}\_\{\\mathrm\{cfg\}\}\-v\_\{\\mathrm\{cfg\}\}^\{\\star\}\\\|^\{2\}=\\frac\{\\mathcal\{R\}\_\{\\mathrm\{cond\}\}\(t\)\}\{n\}\\bigl\[1\+\(\\omega^\{2\}\-1\)U\_\{t\}\\bigr\]\.

#### Monotonicity under variance allocation\.

It remains to show thatUtU\_\{t\}is non\-increasing inα\\alpha\. Rescaling the noisy latent gives an equivalent observationYα=γt​\(α\)​μCref\+ξY\_\{\\alpha\}=\\sqrt\{\\gamma\_\{t\}\(\\alpha\)\}\\mu\_\{C\}^\{\\mathrm\{ref\}\}\+\\xi, whereξ∼𝒩⁡\(0,I\)\\xi\\sim\\mathcal\{N\}\(0,I\)is independent ofCCand

γt​\(α\)=t2​αt2​\(1−α\)\+\(1−t\)2\.\\gamma\_\{t\}\(\\alpha\)=\\frac\{t^\{2\}\\alpha\}\{t^\{2\}\(1\-\\alpha\)\+\(1\-t\)^\{2\}\}\.This signal\-to\-noise ratio increases withα\\alpha\. Forα1<α2\\alpha\_\{1\}<\\alpha\_\{2\}, the Gaussian channel coupling in Appendix[E\.8](https://arxiv.org/html/2610.22885#A5.SS8)constructs observations with the corresponding marginals andC→Yα2→Yα1C\\to Y\_\{\\alpha\_\{2\}\}\\to Y\_\{\\alpha\_\{1\}\}\. Writing𝐪i=Pr\(C=⋅∣Yαi\)\\mathbf\{q\}\_\{i\}=\\Pr\(C=\\cdot\\mid Y\_\{\\alpha\_\{i\}\}\)for their posterior probability vectors, the tower property and Jensen’s inequality give

𝐪1=𝔼⁡\[𝐪2∣Yα1\],𝔼​‖𝐪1‖2≤𝔼​‖𝐪2‖2,\\mathbf\{q\}\_\{1\}=\\mathbb\{E\}\[\\mathbf\{q\}\_\{2\}\\mid Y\_\{\\alpha\_\{1\}\}\],\\qquad\\mathbb\{E\}\\\|\\mathbf\{q\}\_\{1\}\\\|^\{2\}\\leq\\mathbb\{E\}\\\|\\mathbf\{q\}\_\{2\}\\\|^\{2\},and henceUt​\(α2\)≤Ut​\(α1\)U\_\{t\}\(\\alpha\_\{2\}\)\\leq U\_\{t\}\(\\alpha\_\{1\}\)\. Together with the decrease ofℛcond\\mathcal\{R\}\_\{\\mathrm\{cond\}\}in Eq\.[13](https://arxiv.org/html/2610.22885#A6.E13), Eq\.[15](https://arxiv.org/html/2610.22885#A6.E15)shows that bothℰg​\(t\)\\mathcal\{E\}\_\{g\}\(t\)andℰω​\(t\)\\mathcal\{E\}\_\{\\omega\}\(t\)are non\-increasing inα\\alpha\. Within this estimator, allocating more variance to class means lowers the cost of estimating each conditional predictor and reduces the posterior ambiguity that weights class contrasts\.