Variational Inference

Electronic health records (EHR) pose large-scale multi-disease modeling problems in which many outcomes are rare and strongly influenced by shared risk factors. While modern approaches achieve strong predictive performance, they often treat diseases independently or rely on black-box architectures, offering limited insight into how risk factors organize disease risk and little principled uncertainty quantification. We introduce a Bayesian hypergraph inference framework that reframes multi-disease modeling around **latent, risk-factor-modulated disease pathways**. Risk factors act on hyperedges, latent disease subsets with shared risk patterns, allowing diseases to participate in multiple distinct pathways and enabling interpretable, higher-order structure beyond pairwise associations. A repulsion prior encourages parsimonious and identifiable structure, while posterior inference provides calibrated uncertainty over both disease groupings and risk-factor influence. To enable scalable inference on large EHR datasets, we develop a structured variational inference algorithm that preserves logical dependencies among hyperedge existence, disease membership, and pathway-level effects. Experiments on simulated data and UK Biobank demonstrate stable and interpretable disease pathway structure, well-calibrated uncertainty, improved estimation for rare diseases, and competitive predictive performance.

Dynamic 3D content representation is crucial for generating moving 3D objects and scenes. Existing 4D variational autoencoders (VAEs) are mainly based on projected 2D pointmaps, which are only incomplete and view-dependent observations that do not model the native 4D positional relations between points. This often leads to projection-induced distortions and irreversible token dislocation. In this paper, we introduce a novel 4D VAE that operates directly in native 4D space, that is dynamic colored voxel space, without 2D projection. This preserves explicit spatio-temporal coordinates throughout the learned encoder and decoder, enabling both partial and complete 4D content encoding. To support a flexible temporal compression ratio, we also design a novel spatio-temporal window attention module that performs attention within local 4D windows. Additionally, we propose a differentiable voxel rendering loss based on sparse voxel rasterization to improve the geometry and color reconstruction quality. On 4D reconstruction tasks, our approach improves reconstruction fidelity over pointmap VAEs and flow-based VAEs while learning a more structurally consistent latent space. We further demonstrate the generative potential of our method by training a video-conditioned 4D diffusion model.

The choice of training objective is central to diffusion-based generative modeling in terms of both sample quality and distribution coverage. While standard maximum likelihood training provides a principled objective with strong theoretical grounding, empirical studies indicate that previous training objectives in diffusion models often face an inverse correlation between likelihood optimization and perceptual evaluations. We propose the Rényi diffusion model, a unified generative framework that formulates training objectives using Rényi divergence. This yields a generalized score matching objective providing explicit control over the trade-off between sample quality and distribution coverage. Experiments demonstrate improved balance between density estimation and sample generation performances across multiple datasets without modifying model architectures or sampling procedures.

Many modern generative models can be viewed as minimizing divergences between probability distributions, yet they rely on different algorithmic and geometric principles. Wasserstein gradient flows provide a continuous-time formulation for optimizing over distributions, and can be approximated through their implicit discretization via the Jordan–Kinderlehrer–Otto (JKO) scheme. In this work, we present a unified theoretical framework for generative modeling based on Wasserstein gradient flows, which we refer to as Generative Wasserstein Flows (GWF). We show that a broad class of existing methods can be derived as instances of parametric JKO schemes for $f$-divergence objectives, and we establish equivalences between several recently proposed algorithms. We extend this framework beyond $f$-divergences to Integral Probability Metrics and squared Maximum Mean Discrepancy, deriving new JKO-based generative algorithms, and clarifying their connections with GANs. We study empirically the impact of the JKO regularization for a wide set of objectives. Finally, we analyze parametric Wasserstein flows, where the dynamics are restricted to distributions induced by parametrized maps.

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 a gFID of 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.

Cross-domain recommendation (CDR) serves as a pivotal solution to data sparsity and cold-start problems by transferring knowledge across distinct domains. However, existing approaches predominately rely on Euclidean embedding spaces, which suffer from a fundamental geometry-distribution mismatch: real-world user-item interactions typically exhibit power-law distributions and latent hierarchical structures that flat Euclidean spaces cannot accurately represent without significant distortion. This geometric limitation not only compromises representation quality but, more critically, hinders the effective disentanglement of domain-invariant user preferences from domain-specific interests, limiting transferability in low-overlap scenarios. To bridge this gap, we introduce the Mixed-Curvature Hyperbolic Variational Auto-Encoder (HVAE), a principled framework that unifies knowledge extraction and transfer within a hyperbolic manifold. By leveraging the exponential expansion capacity of hyperbolic geometry, HVAE naturally accommodates hierarchical data structures, enabling precise disentanglement of user intents without the need for strict domain overlap constraints. Furthermore, we propose a rigorous hyperbolic Wasserstein barycenter mechanism to align invariant distributions across heterogeneous domains. Extensive experiments on large-scale industrial and public datasets demonstrate that HVAE achieves superior performance, particularly in challenging scenarios with long-tail distributions and minimal domain overlap.

The Joint-Embedding Predictive Architecture (JEPA) is often seen as a non-generative alternative to likelihood-based self-supervised learning, emphasizing prediction in representation space rather than reconstruction in observation space. We argue that the resulting separation from probabilistic generative modeling is largely rhetorical rather than structural: the canonical JEPA design–coupled encoders with a context-to-target predictor–mirrors the variational posteriors and learned conditional priors obtained when variational inference is applied to a particular class of coupled latent-variable models, and standard JEPA can be viewed as a deterministic specialization in which regularization is imposed via architectural and training heuristics rather than an explicit likelihood. Building on this view, we derive the *Variational JEPA* (Var-JEPA), which makes the latent generative structure explicit by optimizing a single Evidence Lower Bound (ELBO). This yields meaningful representations without ad-hoc anti-collapse regularizers and allows principled uncertainty quantification in the latent space. We instantiate the framework for tabular data (Var-T-JEPA) and achieve strong representation learning and downstream performance, improving over T-JEPA across real-world tabular benchmarks while remaining competitive with strong raw-feature baselines.

Joint Embedding Predictive Architectures (JEPAs) avoid pixel reconstruction by predicting latent representations, but standard formulations remain deterministic and provide limited uncertainty estimates for planning and control. We introduce \emph{Variational JEPA (VJEPA)}, a probabilistic extension that learns predictive distributions over future latent states using a latent-space variational objective, without autoregressive observation likelihoods. We show that VJEPA links JEPA-style self-supervised learning to predictive state representations and Bayesian filtering, and that its latent variables can serve as sufficient information states for control when they preserve task-relevant predictive information. We also propose \emph{Bayesian JEPA (BJEPA)}, which combines a learned dynamics expert with modular prior experts through a Product of Experts, enabling constraint-aware prediction and zero-shot prior swapping. Experiments on Noisy-TV systems, nonlinear and image-based benchmarks, STL-10 with a ViT encoder, and DMC Cheetah-run show that predictive JEPA-family objectives are more robust to high-variance nuisance distractors than reconstruction-based world-model baselines. These results position probabilistic latent prediction as a principled framework for robust, uncertainty-aware, reconstruction-free world models.

Multi-view clustering aims to obtain a consensus clustering by integrating complementary and consistent information from multiple views. However, two critical challenges still exist in variational methods: (1) view heterogeneity and noise often make fusion unreliable; (2) ambiguous posteriors and misassigned boundary samples impact the clustering performance. To address these issues, we propose Dual-stage Contrastive Learning-enhanced Multi-view Variational Clustering (DCL-MVC), which integrates contrastive learning into both the fusion and representation stages. Firstly, at the fusion stage, we introduce a fusion-then-attention mechanism to capture cross-view interactions and learn view-level attention weights for building a unified and reliable fused representation, and further introduce instance-level contrastive learning to enforce cross-view alignment at the instance level. Secondly, we focus on boundary samples with uncertain posteriors and refine their cluster assignments by using cluster-center contrastive loss to enlarge inter-cluster margins, while leveraging prototypical contrastive learning with a confidence-aware curriculum to promote intra-cluster compactness at the representation stage. Extensive experiments on six real-world datasets demonstrate consistent improvements over strong baselines and validate the contribution of each component.

Computing the differential entropy of distributions known only up to a normalization constant is a fundamental challenge with broad theoretical and practical significance. While variational inference is highly scalable for density approximation from samples, its application to unnormalized densities remains under-explored due to the difficulty of constructing variational distributions that exploit the structure of the unnormalized density and are simultaneously expressive, tractable, efficiently samplable. Recently, Messaoud et al. (ICLR 24) introduced _P-SVGD_, a Stein variational inference method for this setting. We show, however, that \textit{P-SVGD} fails to scale to high dimensions due to incorrect invertibility assumptions, omission of a critical trace-of-Hessian term, and unstable divergence-control heuristics. We propose _MET-SVGD_, a principled extension of _P-SVGD_ that provides a general framework for stable _SVGD_ hyperparameter selection with global invertibility and convergence guarantees. Empirically, _MET-SVGD_ achieves up to $12\times$ and $16\times$ lower entropy estimation error than _P-SVGD_ and existing _SVGD_ baselines, respectively. On CIFAR-10 energy-based image generation, it improves FID by $80.4\%$ and yields $64\times$ higher training stability. In maximum-entropy reinforcement learning, it achieves up to $16\%$ higher returns than _P-SVGD_. Code is available at https://shorturl.at/fTG0G.

Incomplete multi-view clustering (IMVC) aims to uncover shared cluster structures from data with partially observed views. Although recent imputation-free methods based on variational inference demonstrate robustness to missing views, they commonly rely on a conditional independence assumption across views, which fails to capture the inherently structured and potentially correlated nature of multi-view data. In this paper, we propose a variational framework that explicitly goes beyond this assumption by introducing a learnable cross-view correlation structure. Specifically, we explicitly model and learn correlations between views by utilizing the covariance structure of posterior estimation errors. To facilitate robust and efficient learning, the correlation matrix is parameterized through a normalized Cholesky decomposition, ensuring positive definiteness and enabling the entire model to be trained jointly through a unified variational objective. Extensive experiments on multiple IMVC benchmarks demonstrate that our method consistently outperforms state-of-the-art approaches across a wide range of missing-view settings. These results highlight the effectiveness of adaptive correlation modeling in variational IMVC, demonstrating the need to go beyond the independence assumption in IMVC.

Disentangled representations separate factors that are shared across conditions from those that are condition-specific. Such separation is needed for generalization to new domains, treatments, patients, or species. A dominant line of work pursues this goal through variational formulations. While these approaches achieve partial disentanglement, they often exhibit three common limitations: they either do not remove all condition-specific information from the shared representation, allow the shared representation to become uninformative, or impose independence assumptions that do not reflect the underlying generative process. In this work, we introduce DisCoVR, a variational framework that addresses these limitations. Its objective is aligned with the probabilistic structure of the data-generating process, and includes an adversarial term that prevents condition-specific information from being encoded in the shared representation. DisCoVR reconstructs the data from both shared and condition-specific representations, ensuring that each remains informative, and uses a structured prior that further reinforces the informativeness of both representations. We show that across synthetic, image, and single-cell RNA-sequencing datasets, DisCoVR achieves stronger disentanglement compared to previous approaches.

We study last-iterate convergence for stochastic smooth and monotone variational inequalities (VIs), a framework that captures convex-concave saddle points and Nash equilibrium computation in monotone games with noisy payoff feedback. In contrast to the well-understood average-iterate guarantees, anytime last-iterate guarantees in stochastic settings remain limited, despite their relevance for uncoupled learning dynamics that output a single current strategy. We analyze two single-call regularized methods, the \emph{regularized gradient (RG)} and the \emph{regularized optimistic gradient (ROG)} methods, and establish anytime last-iterate convergence rates in terms of the squared gap function. For monotone VIs, RG attains $O(t^{-2/5})$ while ROG achieves the variance-adaptive rate $O(\sigma^{4/5} t^{-2/5} + t^{-1})$, where $\sigma^2$ is the noise variance. For $\lambda$-strongly monotone VIs, ROG yields $O(\sigma^2 / (\lambda^2 t) + t^{-c})$ for any constant $c \ge 2$. These results give anytime last-iterate guarantees without knowing the horizon and show that optimism improves convergence in the low-noise regime.

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

We analyze algorithms for solving stochastic variational inequalities (VI) without the bounded variance or bounded domain assumptions, where our main focus is min-max optimization with possibly unbounded constraint sets. We focus on two classes of problems: monotone VIs; and structured nonmonotone VIs that admit a solution to the *weak Minty VI*. The latter assumption allows us to solve structured nonconvex-nonconcave min-max problems. For both classes of VIs, to make the expected residual norm less than $\varepsilon$, we show an oracle complexity of $\widetilde{O}(\varepsilon^{-4})$, which is the best-known for constrained VIs. In our setting, this complexity had been obtained with the bounded variance assumption in the literature, which is not even satisfied for bilinear min-max problems with an unbounded domain. We obtain this complexity for stochastic oracles whose variance can grow as fast as the squared norm of the optimization variable.

Foundation models are increasingly being deployed in contexts where understanding the uncertainty of their outputs is critical to ensuring responsible deployment. While Bayesian methods offer a principled approach to uncertainty quantification, their computational overhead renders their use impractical for training or inference at foundation model scale. State-of-the-art models achieve parameter counts in the trillions through carefully engineered sparsity including Mixture-of-Experts (MoE) layers. In this work, we demonstrate calibrated uncertainty at scale by introducing Variational Mixture-of-Experts Routing (VMoER), a structured Bayesian approach for modelling uncertainty in MoE layers. VMoER confines Bayesian inference to the expert-selection stage which is typically done by a deterministic routing network. We instantiate VMoER using two inference strategies: amortised variational inference over routing logits and inferring a temperature parameter for stochastic expert selection. Across tested foundation models, VMoER improves routing stability under noise by 38%, reduces calibration error by 94%, and increases out-of-distribution AUROC by 12%, while incurring less than 1% additional FLOPs. These results suggest VMoER offers a scalable path toward robust and uncertainty-aware foundation models.

Poisson-distributed latent variable models are widely used in computational neuroscience, but differentiating through discrete stochastic samples remains challenging. Two approaches address this: *Exponential Arrival Time* (EAT) simulation and *Gumbel-SoftMax* (GSM) relaxation. We provide the first systematic comparison of these methods, along with practical guidance for practitioners. Our main technical contribution is a modification to the EAT method that theoretically guarantees an unbiased first moment (exactly matching the firing rate), and reduces second-moment bias. We evaluate these methods on their distributional fidelity, gradient quality, and performance on two tasks: (1) variational autoencoders with Poisson latents, and (2) partially observable generalized linear models, where latent neural connectivity must be inferred from observed spike trains. Across all metrics, our modified EAT method exhibits better overall performance (often comparable to exact gradients), and substantially higher robustness to hyperparameter choices. These results extend to over-dispersed Negative Binomial latents, where modified EAT again performs best. However, only GSM generalizes to arbitrary non-Poisson distributions, including the under-dispersed regime. Together, our results clarify the trade-offs between these methods and offer concrete recommendations for practitioners working with Poisson latent variable models.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian linear regression, we show that this characterisation is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on i.i.d. data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

Multi-modal prompt learning is a parameter-efficient approach to adapting large vision-language models to downstream classification tasks. However, prompts can inadvertently evolve into a high-capacity pathway that encodes environment-dependent spurious correlations, which are predictive only in the source domain and thereby undermine transferability. To address this issue, this paper introduces **Do-Prompt**, a *compress-and-intervene* framework that brings together variational bottlenecks and causal interventions for robust prompt tuning. We model prompts as stochastic latent variables and impose a *variational prompt bottleneck* to explicitly regulate the information transmitted through prompts, effectively mitigating their tendency to memorize spurious nuisance cues. Building on this capacity constraint, we propose lightweight *prompt-level interventions* by perturbing the environment-related prompt components and enforcing prediction consistency under these *do*-style perturbations. This synergistic integration encourages reliance on task-stable, invariant semantics rather than spurious prompt content. Notably, Do-Prompt is plug-and-play compatible with existing multi-modal prompt tuning pipelines and introduces negligible computational overhead. Extensive experiments on base-to-novel generalization, cross-dataset transfer, and ImageNet distribution shifts demonstrate consistent performance gains, with particularly notable improvements on datasets exhibiting pronounced domain or texture biases.

Heavy-tailed distributions are ubiquitous in real-world data, where rare but extreme events dominate risk and variability. However, standard Variational Autoencoders (VAEs) employ simple decoder distributions (e.g., Gaussian) that fail to capture heavy-tailed behavior, while existing heavy-tail-aware extensions remain restricted to predefined parametric families whose tail behavior is fixed a priori. We propose the Phase-Type Variational Autoencoder (PH-VAE), whose decoder distribution is a latent-conditioned Phase-Type (PH) distribution—defined as the absorption time of a continuous-time Markov chain (CTMC). This formulation composes multiple exponential time scales, yielding a flexible, analytically tractable decoder that adapts its finite-range tail behavior directly from the observed data. Experiments on synthetic and real-world benchmarks demonstrate that PH-VAE accurately approximates diverse heavy-tailed distributions, significantly outperforming Gaussian, Student-t, and extreme-value-based VAE decoders in modeling observed tail behavior and extreme quantiles. In multivariate settings, PH-VAE captures realistic cross-dimensional tail dependence through its shared latent representation. To our knowledge, this is the first work to integrate Phase-Type distributions into deep generative modeling, bridging applied probability and representation learning.