Pro-t-VAE: Closed-Form Student's t Mixtures for Robust Prototype Discovery

Zhikang Xu; Jiarui Xing; Jian Wang

cs.LGOct 10, 2026

Pro-t-VAE: Closed-Form Student's t Mixtures for Robust Prototype Discovery

Authors: Zhikang Xu, Jiarui Xing, Jian Wang

Organizations: Institute of Intelligent Information Processing, Shanxi University · School of Medicine, Yale University · Boston Children’s Hospital, Harvard Medical School

Abstract

We present a robust deep generative framework for unsupervised image prototype discovery, built on a heavy-tailed Student's t mixture prior, that bridges statistical robustness and mathematical tractability. Existing generative approaches typically rely on noise-prone Gaussian priors or computationally intensive diffusion models; neither simultaneously provides explicit latent structure nor exact analytic optimization. While diffusion models can yield strong priors, iterative sampling and intractable likelihoods limit scalability and interpretability for prototype discovery. We address these issues by constructing a Student's t latent space via a Bayesian scale-mixture formulation. This yields a fully closed-form augmented ELBO for unsupervised Student's t mixture variational autoencoders, which, to our knowledge, is the first such objective applied to robust image prototype discovery, avoiding biased variational approximations and high computational costs in prior heavy-tailed or diffusion-based frameworks. Treating precision as a latent variable automatically modulates each data point's influence, downweighting outliers to learn a data-dependent measure of trust. The result is improved robustness, accuracy, and efficiency, with well-separated latent representations and more stable manifold learning than state-of-the-art methods. We validate our model on standard vision benchmarks and challenging neuroimaging tasks. In brain MRI-driven neurodegenerative disease analysis, our model recovers clean, anatomically coherent prototypes where clinical noise often collapses rare subtypes. These results highlight the ability of our framework to preserve anatomy-associated patterns for precise discovery in clinical settings. Our code is available at: https://github.com/xuzhikangnba/torch-tvae.

Figures & tables

Figure 1: Prototype generation and representation separability versus training cost. Left: Noise-corrupted inputs, diffusion-based prototype estimates, and our method on MNIST, Fashion-MNIST, and brain MRI. Right: Mean class separability vs. per-update training time on brain scans; shading shows population variance.

Figure 2: Probabilistic graphical model of ( 5 ) as a scale mixture of Gaussians . Shaded nodes are observed, white nodes latent, red squares global parameters, and plate

N

denotes per-observation sampling. Given cluster

c

, an auxiliary precision

u\sim\mathrm{Gamma}(\nu_{c}/2,\nu_{c}/2)

scales the conditional covariance

(u\bm{\Lambda}_{c})^{-1}

for

\bm{z}

: small

u

captures tail observations, while large

u

concentrates around the component center. Marginalizing

u

recovers the Student’s

t

prior

\bm{z}\mid c\sim\mathcal{T}_{\nu_{c}}(\bm{\mu}_{c},\bm{\Lambda}_{c}^{-1})

. The encoder maps

\bm{x}

q_{\phi}(\bm{z}\mid\bm{x})

, the mixture yields soft prototype assignments, and the shared decoder maps

\bm{z}

to reconstructions and

\bm{\mu}_{c}

to image prototypes.

Figure 3: Left: Mean of the learned per-cluster degrees of freedom

\nu_{c}

across training epochs on MNIST and Fashion-MNIST under four Cauchy-noise levels

\gamma\in\{0.00,0.04,0.08,0.14\}

; all runs initialize

\nu_{c}=5

. Right: Prototype assignment accuracy (ACC, ARI, NMI) averaged over noise levels

\gamma\in\{0.02,\ldots,0.14\}

on MNIST and Fashion-MNIST. Best in bold , second underlined .

Figure 4: Top: quantitative evaluation of all models under increasing noise corruption of images. Bottom: prototype visualization across all models when image is under severe corruption $\gamma=0.12$ . Our approach preserves crisp, distinguishable prototypes without duplication.

Figure 5: Top: $t$ -SNE of latent representations from VaDE, LDDPM-VaDE, t-VaDE-log, and ours on MNIST and Fashion-MNIST with clean ( $\gamma=0.0$ ) and noisy inputs ( $\gamma=0.12$ MNIST, $0.14$ Fashion-MNIST). Bottom-left: Silhouette on OASIS Brain MRI under nine realistic corruptions (Gaussian, Rician, bias-field; mild/medium/severe). Bottom-right: per-voxel uncertainty maps from our model across four groups of anatomy-associated patterns.

Figure 6: Visual comparison of learned atlases on OASIS dataset under Cauchy noise ( $\gamma=0.12$ ). Col. 1 shows corrupted inputs; cols. 2–8 show cluster-mean atlases reconstructed by different methods. Rows 1–2 are sagittal atlases for two clusters; rows 3–4 are coronal atlases. White numbers (bottom right) report per-cluster PSNR (dB) vs. each method’s clean atlas after one-to-one cluster matching (higher is better); best per row is bold. Red arrows mark baseline artifacts. The blue box shows an LDDPM-VaDE failure: the two coronal cluster atlases are nearly identical, indicating prototype collapse. Aggregate PSNR and Silhouette results are reported in Tab. 1 (a).

Table 1: Prototype separability and reconstruction quality on OASIS at

K=2

, averaged over sagittal/axial/coronal views. (a) Sil, DB, and PSNR at four noise levels; PSNR is reported only for corrupted conditions relative to matched clean-run prototypes (clean shown as -- ). Collapsed runs (single cluster; Sil undefined) are excluded; cells with any excluded run are italic † . (b) Collapse count and rate per method over 118 runs. Best in bold .

Variational Autoencoders

Variational autoencoders are probabilistic encoder–decoder models for learning latent representations, generating data, and estimating uncertainty.

See all 209 papers

Jun 17, 2026

Deep Image Prototype Learning with Geometric Heat-Kernel Priors

Learning unsupervised representations of medical imaging cohorts can reveal anatomically meaningful prototypes without expert labels, which are often noisy and fail to capture true pathological heterogeneity. However, existing deep latent-variable models estimate Gaussian mixture priors via Euclidean averaging, producing prototypes that drift off the curved data manifold and degenerate as the number of sub-populations grows. We propose a manifold-anchored variational framework built on a geometry-aware Expectation-Maximization (EM) algorithm, whose M-step selects each sub-population prototype as the graph medoid with the highest diffusion centrality on a heat-kernel-weighted latent graph, ensuring that every prototype remains on-manifold. A Dirichlet energy regularizer enforces geometric smoothness of the latent space, and a per-sub-population uncertainty score enables label-free quality assessment. The manifold-anchored EM is a general-purpose geometric tool that extends standard EM and applies readily to other latent-variable models beyond this setting. On cardiac scar and brain MRI benchmarks, our framework attains the highest accuracy among all compared methods, produces the sharpest prototypes reported to date, and remains stable at large sub-population counts where all baselines degenerate. Code and implementation details are available at https://github.com/jr-xing/On-Manifold-Variational-Learning-with-Heat-Kernel-Priors.

Jiarui Xing, Tal Zeevi, Nian Wu +1

Yale School of Medicine, New Haven CT, 06510, USA · University of Virginia, Charlottesville, VA, 22903, USA · Harvard Medical School, Boston MA, 02115, USA

Medical Image Analysis Prototype Learning Unsupervised Representation Learning Geometric Representation Learning Medical Image Representation Learning

Ask PDF

Dec 8, 2025

Distribution Matching Variational AutoEncoder

Most visual generative models compress images into a latent space before applying diffusion or autoregressive modelling. Yet, existing approaches such as VAEs and foundation model aligned encoders implicitly constrain the latent space without explicitly shaping its distribution, making it unclear which types of distributions are optimal for modeling. We introduce Distribution-Matching VAE (DMVAE), which explicitly aligns the encoder's latent distribution with an arbitrary reference distribution via a distribution matching constraint. This generalizes beyond the Gaussian prior of conventional VAEs, enabling alignment with distributions derived from self-supervised features, diffusion noise, or other prior distributions. With DMVAE, we can systematically investigate which latent distributions are more conducive to modeling, and we find that SSL-derived distributions provide an excellent balance between reconstruction fidelity and modeling efficiency, reaching gFID equals 3.2 on ImageNet with only 64 training epochs. Our results suggest that choosing a suitable latent distribution structure (achieved via distribution-level alignment), rather than relying on fixed priors, is key to bridging the gap between easy-to-model latents and high-fidelity image synthesis. Code is avaliable at https://github.com/sen-ye/dmvae.

Sen Ye, Jianning Pei, Mengde Xu +4

State Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University · University of Chinese Academy of Sciences · Tencent +1

Variational Autoencoders Generative Modeling

Ask PDF

Feb 25, 2026

Function-Space Empirical Bayes Regularisation with Student's t Priors

Bayesian deep learning (BDL) has emerged as a principled approach to produce reliable uncertainty estimates by integrating deep neural networks with Bayesian inference, and the selection of informative prior distributions remains a significant challenge. Various function-space variational inference (FSVI) regularisation methods have been presented, assigning meaningful priors over model predictions. However, these methods typically rely on a Gaussian prior, which fails to capture the heavy-tailed statistical characteristics inherent in neural network outputs. By contrast, this work proposes a novel function-space empirical Bayes regularisation framework -- termed ST-FS-EB -- which employs heavy-tailed Student's

t

priors in both parameter and function spaces. Also, we approximate the posterior distribution through variational inference (VI), inducing an evidence lower bound (ELBO) objective based on Monte Carlo (MC) dropout. Furthermore, the proposed method is evaluated against various VI-based BDL baselines, and the results demonstrate its robust performance in in-distribution prediction, out-of-distribution (OOD) detection and handling distribution shifts.

Pengcheng Hao, Ercan Engin Kuruoglu

Institute of Data and Information, Tsinghua Shenzhen International Graduate School, Shenzhen, China

Bayesian Neural Networks Bayesian Inference Empirical Bayes Variational Inference

Ask PDF

Abstract

Variational Autoencoders

Explore similar work

Deep Image Prototype Learning with Geometric Heat-Kernel Priors

Distribution Matching Variational AutoEncoder

Function-Space Empirical Bayes Regularisation with Student's t Priors