Variational Inference

Also known as VI

Latest papers 140

14 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Apr 28, 2026

Variational Neural Belief Parameterizations for Robust Dexterous Grasping under Multimodal Uncertainty

Contact variability, sensing uncertainty, and external disturbances make grasp execution stochastic. Expected-quality objectives ignore tail outcomes and often select grasps that fail under adverse contact realizations. Risk-sensitive POMDPs address this failure mode, but many use particle-filter beliefs that scale poorly, obstruct gradient-based optimization, and estimate Conditional Value-at-Risk (CVaR) with high-variance approximations. We instead formulate grasp acquisition as variational inference over latent contact parameters and object pose, representing the belief with a differentiable Gaussian mixture. We use Gumbel-Softmax component selection and location-scale reparameterization to express samples as smooth functions of the belief parameters, enabling pathwise gradients through a differentiable CVaR surrogate for direct optimization of tail robustness. In simulation, our variational neural belief improves robust grasp success under contact-parameter uncertainty and exogenous force perturbations while reducing planning time by roughly an order of magnitude relative to particle-filter model-predictive control. On a serial-chain robot arm with a multifingered hand, we validate grasp-and-lift success under object-pose uncertainty against a Gaussian baseline. Both methods succeed on the tested perturbations, but our controller terminates in fewer steps and less wall-clock time while achieving a higher tactile grasp-quality proxy. Our learned belief also calibrates risk more accurately, keeping mean absolute calibration error below 0.14 across tested simulation regimes, compared with 0.58 for a Cross-Entropy Method planner.
Apr 28, 2026

Occam's Razor is Only as Sharp as Your ELBO

The marginal likelihood, also known as the evidence, is regarded as a mathematical embodiment of Occam's razor, enabling hyperparameter selection that avoids overfitting. The evidence lower bound (ELBO) objective from variational inference has also been used for similar purposes. In this paper, we define a signal-to-noise ratio (SNR) for Bayesian linear regression in terms of key hyperparameters controlling signal and noise variance. We then study how ELBO-based hyperparameter learning can under or overestimate SNR depending on the assumed covariance matrix structure in a Gaussian approximate posterior. We prove that assuming a diagonal or low-rank plus diagonal covariance matrix underestimates the SNR, while a low-rank covariance matrix overestimates the SNR. Underestimating the SNR leads to higher than optimal mean squared error on the training set (underfitting), while overestimating SNR leads to lower error (overfitting). Bayesian practitioners hoping to scale to large models should be cautious about how covariance assumptions needed for tractability may impact model selection.
Apr 27, 2026

Heterogeneous Variational Inference for Markov Degradation Hazard Models: Discretized Mixture with Interpretable Clusters

Bayesian finite mixture models can identify discrete risk clusters (low-risk vs. high-risk equipment), but face three critical bottlenecks: (1) insufficient degradation signals from coarse state discretization, (2) unstable cluster identification when data inherently supports fewer clusters than explored, and (3) computational infeasibility of Markov Chain Monte Carlo (MCMC) methods for production deployment (7+ hours per model). We propose a practical framework combining (1) 8-state global percentile discretization that amplifies degradation events, (2) 30-dimensional feature engineering integrating statistical trends (22 features), continuous health indicators, and text embeddings (PCA-compressed to 3 dimensions), (3) interpretable model selection rules enforcing minimum cluster share and separation alongside WAIC, and (4) Automatic Differentiation Variational Inference (ADVI) with full-rank covariance for stable, fast estimation. Applied to 280 industrial pump equipment with 104,703 inspection records, we demonstrate: (1) Random effect models (baseline) show ADVI and NUTS produce nearly identical estimates with 15×\times speedup, validating ADVI accuracy. (2) Finite mixture models identify optimal number of clusters with interpretability constraints. (3) NUTS exhibits severe convergence issues and label switching, while ADVI provides stable results in 84×\times less time. We contributed that (1) First demonstration that fine-grained state discretization (8-state) is essential for mixture model stability in survival analysis.(2) Comprehensive feature engineering strategy combining statistical, continuous, and semantic signals. (3) Practical interpretability rules preventing overfitting in automated model selection. (4) Empirical evidence that ADVI outperforms NUTS for finite mixture models in terms of convergence, stability, and computational efficiency.
Apr 26, 2026

Reparameterization through Coverings and Topological Weight Priors

We generalise the reparameterization trick (RT) applied in variational autoencoders (VAEs) letting these have latent spaces of non-trivial topology - i.e. that of base manifolds covered with other ones, on which some technique for RT is available. That is possible since covering maps are measurable - moreover, this allows to establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, bounding it with KL-divergence between pullbacks on the cover, in some cases making the KL-term of VAE's ELBO analytically tractable, despite the topological non-triviality of the supporting latent manifold. Our development follows a route close but somewhat alternative to reparameterization on Lie groups, the latest proposal for which is to reparameterize PFs of normal densities from the Lie algebra - "through" the exponential map, seen by us as a particular case of what we propose to call reparameterization via covering (RVC). We demonstrate the working of our approach by constructing a VAE with the latent space of Klein bottle (not a Lie group) topology, which we call KleinVAE, successfully learning an appropriate artificial dataset. We discuss potential applicability of such topology-informed generative models as weight priors in Bayesian learning, particularly for convolutional vision models, where said manifold was peculiarly shown to have some relevance.
Apr 23, 2026

Even More Guarantees for Variational Inference in the Presence of Symmetries

When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous theoretical results on robust VI with location-scale families under target symmetries in two substantial ways: (1) We open them up to a wider range of divergences by providing sufficient conditions for exact recovery of the target mean and correlation matrix when using the forward Kullback-Leibler divergence and αα-divergences. (2) By doing so, we find that we can drop the restrictive assumption of a log-concave target made in previous work, allowing us to give guarantees for a wider range of targets, including multi-modal ones. In our experiments, we show how our guarantees can serve as guidelines for the choice of the variational family and αα-value and we illustrate on a diverse set of examples how and why optimization can fail in the absence of our sufficient conditions.
Apr 20, 2026

Symmetry Guarantees Statistic Recovery in Variational Inference

Variational inference (VI) is a central tool in modern machine learning, used to approximate an intractable target density by optimising over a tractable family of distributions. As the variational family cannot typically represent the target exactly, guarantees on the quality of the resulting approximation are crucial for understanding which of its properties VI can faithfully capture. Recent work has identified instances in which symmetries of the target and the variational family enable the recovery of certain statistics, even under model misspecification. However, these guarantees are inherently problem-specific and offer little insight into the fundamental mechanism by which symmetry forces statistic recovery. In this paper, we overcome this limitation by developing a general theory of symmetry-induced statistic recovery in variational inference. First, we characterise when variational minimisers inherit the symmetries of the target and establish conditions under which these pin down identifiable statistics. Second, we unify existing results by showing that previously known statistic recovery guarantees in location-scale families arise as special cases of our theory. Third, we apply our framework to distributions on the sphere to obtain novel guarantees for directional statistics in von Mises-Fisher families. Together, these results provide a modular blueprint for deriving new recovery guarantees for VI in a broad range of symmetry settings.
Apr 17, 2026

How to Approximate Inference with Subtractive Mixture Models

Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.
Apr 16, 2026

A Synonymous Variational Perspective on the Rate-Distortion-Perception Tradeoff

The fundamental limit of natural signal compression has traditionally been characterized by classical rate-distortion (RD) theory through the tradeoff between coding rate and reconstruction distortion, while the rate-distortion-perception (RDP) framework introduces a divergence-based measure of perceptual quality as a modeling principle, leaving its theoretical origin unclear. In this paper, motivated by a synonymity-based semantic information perspective, we reformulate perceptual reconstruction as recovering any admissible sample within an ideal synonymous set (synset) associated with the source, rather than the source sample itself, and establish a synonymous source coding architecture. On this basis, we develop a synonymous variational inference (SVI) analysis framework with a synonymous variational lower bound (SVLBO) for tractable analysis of synset-oriented compression. Within this framework, we establish a synonymity-perception consistency principle, showing that optimal identification of semantic information is theoretically consistent with perceptual optimization. Based on this result, we further derive a tight-bound synonymous source coding rate characterization and show that its Jensen-limit relaxation leads to a synonymous rate-distortion-perception form for practical optimization. These analytical results show that the distributional divergence term arises naturally from the synset-based reconstruction objective, clarify its compatibility with existing RDP formulations and classical RD theory, and suggest the potential advantages of synonymous source coding.
Apr 14, 2026

Monte Carlo Stochastic Depth for Uncertainty Estimation in Deep Learning

The deployment of deep neural networks in safety-critical systems necessitates reliable and efficient uncertainty quantification (UQ). A practical and widespread strategy for UQ is repurposing stochastic regularizers as scalable approximate Bayesian inference methods, such as Monte Carlo Dropout (MCD) and MC-DropBlock (MCDB). However, this paradigm remains under-explored for Stochastic Depth (SD), a regularizer integral to the residual-based backbones of most modern architectures. While prior work demonstrated its empirical promise for segmentation, a formal theoretical connection to Bayesian variational inference and a benchmark on complex, multi-task problems like object detection are missing. In this paper, we first provide theoretical insights connecting Monte Carlo Stochastic Depth (MCSD) to principled approximate variational inference. We then present the first comprehensive empirical benchmark of MCSD against MCD and MCDB on state-of-the-art detectors (YOLO, RT-DETR) using the COCO and COCO-O datasets. Our results position MCSD as a robust and computationally efficient method that achieves highly competitive predictive accuracy (mAP), notably yielding slight improvements in calibration (ECE) and uncertainty ranking (AUARC) compared to MCD. We thus establish MCSD as a theoretically-grounded and empirically-validated tool for efficient Bayesian approximation in modern deep learning.
Apr 10, 2026

Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models

Rare events are central to the evolution of complex many-body systems, characterized as key transitional configurations on the free energy surface (FES). Conventional methods require adequate sampling of rare event transitions to obtain the FES, which is computationally very demanding. Here we introduce the variational free energy surface (VaFES), a dataset-free framework that directly models FESs using tractable-density generative models. Rare events can then be immediately identified from the FES with their configurations generated directly via one-shot sampling of generative models. By extending a coarse-grained collective variable (CV) into its reversible equivalent, VaFES constructs a latent space of intermediate representation in which the CVs explicitly occupy a subset of dimensions. This latent-space construction preserves the physical interpretability and transparent controllability of the CVs by design, while accommodating arbitrary CV formulations. The reversibility makes the system energy exactly accessible, enabling variational optimization of the FES without pre-generated simulation data. A single optimization yields a continuous, differentiable FES together with one-shot generation of rare-event configurations. Our method can reproduce the exact analytical solution for the bistable dimer potential and identify a chignolin native folded state in close alignment with the experimental NMR structure. Our approach thus establishes a scalable, systematic framework for advancing the study of complex statistical systems.
Apr 9, 2026

Practical Bayesian Inference for Speech SNNs: Uncertainty and Loss-Landscape Smoothing

Spiking Neural Networks (SNNs) are naturally suited for speech processing tasks due to their specific dynamics, which allows them to handle temporal data. However, the threshold-based generation of spikes in SNNs intuitively causes an angular or irregular predictive landscape. We explore the effect of using the Bayesian learning approach for the weights on the irregular predictive landscape. For the surrogate-gradient SNNs, we also explore the application of the Improved Variational Online Newton (IVON) approach, which is an efficient variational approach. The performance of the proposed approach is evaluated on the Heidelberg Digits and Speech Commands datasets. The hypothesis is that the Bayesian approach will result in a smoother and more regular predictive landscape, given the angular nature of the deterministic predictive landscape. The experimental evaluation of the proposed approach shows improved performance on the negative log-likelihood and Brier score. Furthermore, the proposed approach has resulted in a smoother and more regular predictive landscape compared to the deterministic approach, based on the one-dimensional slices of the weight space
Apr 8, 2026

Variational Approximated Restricted Maximum Likelihood Estimation for Spatial Data

This research considers a scalable inference for spatial data modeled through Gaussian intrinsic conditional autoregressive (ICAR) structures. The classical estimation method, restricted maximum likelihood (REML), requires repeated inversion and factorization of large, sparse precision matrices, which makes this computation costly. To sort this problem out, we propose a variational restricted maximum likelihood (VREML) framework that approximates the intractable marginal likelihood using a Gaussian variational distribution. By constructing an evidence lower bound (ELBO) on the restricted likelihood, we derive a computationally efficient coordinate-ascent algorithm for jointly estimating the spatial random effects and variance components. In this article, we theoretically establish the monotone convergence of ELBO and mathematically exhibit that the variational family is exact under Gaussian ICAR settings, which is an indication of nullifying approximation error at the posterior level. We empirically establish the supremacy of our VREML over MLE and INLA.
Apr 8, 2026

Instance-Adaptive Parametrization for Amortized Variational Inference

Variational autoencoders (VAEs) rely on amortized variational inference to enable efficient posterior approximation, but this efficiency comes at the cost of a shared parametrization, giving rise to the amortization gap. We propose the instance-adaptive variational autoencoder (IA-VAE), an amortized inference framework in which a hypernetwork generates input-dependent modulations of a shared encoder. This enables input-specific adaptation of the inference model while preserving the efficiency of a single forward pass. From a theoretical perspective, we show that the variational family induced by IA-VAE contains that of standard amortized inference, implying that IA-VAE cannot yield a worse optimal ELBO. By leveraging instance-specific parameter modulations, the proposed approach can achieve performance comparable to standard encoders with substantially fewer parameters, indicating a more efficient use of model capacity. Experiments on synthetic data, where the true posterior is known, show that IA-VAE yields more accurate posterior approximations and reduces the amortization gap. Similarly, on standard image benchmarks, IA-VAE consistently improves held-out ELBO over baseline VAEs, with statistically significant gains across multiple runs. These results suggest that increasing the flexibility of the inference parametrization through instance-adaptive modulation is an effective strategy for mitigating amortization-induced suboptimality in deep generative models.
Apr 5, 2026

Avoiding Non-Integrable Beliefs in Expectation Propagation

Expectation Propagation (EP) is a widely used iterative message-passing algorithm that decomposes a global inference problem into multiple local ones. It approximates marginal distributions as beliefs'' using intermediate functions called messages''. It has been shown that the stationary points of EP are the same as corresponding constrained Bethe Free Energy (BFE) optimization problem. Therefore, EP is an iterative method of optimizing the constrained BFE. However, the iterative method may fall out of the feasible set of the BFE optimization problem, i.e., the beliefs are not integrable. In most literature, the authors use various methods to keep all the messages integrable. In most Bayesian estimation problems, limiting the messages to be integrable shrinks the actual feasible set. Furthermore, in extreme cases where the factors are not integrable, making the message itself integrable is not enough to have integrable beliefs. In this paper, two EP frameworks are proposed to ensure that EP has integrable beliefs. Both of the methods allows non-integrable messages. We then investigate the signal recovery problem in Generalized Linear Model (GLM) using our proposed methods.
Apr 3, 2026

State estimations and noise identifications with intermittent corrupted observations via Bayesian variational inference

This paper focuses on the state estimation problem in distributed sensor networks, where intermittent packet dropouts, corrupted observations, and unknown noise covariances coexist. To tackle this challenge, we formulate the joint estimation of system states, noise parameters, and network reliability as a Bayesian variational inference problem, and propose a novel variational Bayesian adaptive Kalman filter (VB-AKF) to approximate the joint posterior probability densities of the latent parameters. Unlike existing AKF that separately handle missing data and measurement outliers, the proposed VB-AKF adopts a dual-mask generative model with two independent Bernoulli random variables, explicitly characterizing both observable communication losses and latent data authenticity. Additionally, the VB-AKF integrates multiple concurrent multiple observations into the adaptive filtering framework, which significantly enhances statistical identifiability. Comprehensive numerical experiments verify the effectiveness and asymptotic optimality of the proposed method, showing that both parameter identification and state estimation asymptotically converge to the theoretical optimal lower bound with the increase in the number of sensors.
Apr 1, 2026

Inverse-Free Sparse Variational Gaussian Processes

Gaussian processes (GPs) offer appealing properties but are costly to train at scale. Sparse variational GP (SVGP) approximations reduce cost yet still rely on Cholesky decompositions of kernel matrices, ill-suited to low-precision, massively parallel hardware. While one can construct valid variational bounds that rely only on matrix multiplications (matmuls) via an auxiliary matrix parameter, optimising them with off-the-shelf first-order methods is challenging. We make the inverse-free approach practical by proposing a better-conditioned bound and deriving a matmul-only natural-gradient update for the auxiliary parameter, markedly improving stability and convergence. We further provide simple heuristics, such as step-size schedules and stopping criteria, that make the overall optimisation routine fit seamlessly into existing workflows. Across regression and classification benchmarks, we demonstrate that our method 1) serves as a drop-in replacement in SVGP-based models (e.g., deep GPs), 2) recovers similar performance to traditional methods, and 3) can be faster than baselines when well tuned.
Mar 31, 2026

Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems

The increasingly wide use of deep machine learning techniques in computational mechanics has significantly accelerated simulations of problems that were considered unapproachable just a few years ago. However, in critical applications such as Digital Twins for engineering or medicine, fast responses are not enough; reliable results must also be provided. In certain cases, traditional deterministic methods may not be optimal as they do not provide a measure of confidence in their predictions or results, especially in inverse problems where the solution may not be unique or the initial data may not be entirely reliable due to the presence of noise, for instance. Classic deep neural networks also lack a clear measure to quantify the uncertainty of their predictions. In this work, we present a variational graph neural network (VGNN) architecture that integrates variational layers into its architecture to model the probability distribution of weights. Unlike computationally expensive full Bayesian networks, our approach strategically introduces variational layers exclusively in the decoder, allowing us to estimate cognitive uncertainty and statistical uncertainty at a relatively lower cost. In this work, we validate the proposed methodology in two cases of solid mechanics: the identification of the value of the elastic modulus with nonlinear distribution in a 2D elastic problem and the location and quantification of the loads applied to a 3D hyperelastic beam, in both cases using only the displacement field of each test as input data. The results show that the model not only recovers the physical parameters with high precision, but also provides confidence intervals consistent with the physics of the problem, as well as being able to locate the position of the applied load and estimate its value, giving a confidence interval for that experiment.
Mar 29, 2026

Variational Learning of Fractional Posteriors

We introduce a novel one-parameter variational objective that lower bounds the data evidence and enables the estimation of approximate fractional posteriors. We extend this framework to hierarchical construction and Bayes posteriors, offering a versatile tool for probabilistic modelling. We demonstrate two cases where gradients can be obtained analytically and a simulation study on mixture models showing that our fractional posteriors can be used to achieve better calibration compared to posteriors from the conventional variational bound. When applied to variational autoencoders (VAEs), our approach attains higher evidence bounds and enables learning of high-performing approximate Bayes posteriors jointly with fractional posteriors. We show that VAEs trained with fractional posteriors produce decoders that are better aligned for generation from the prior.
Mar 24, 2026

Stepwise Variational Inference with Vine Copulas

We propose stepwise variational inference (VI) with vine copulas: a universal VI procedure that combines vine copulas with a novel stepwise estimation procedure of the variational parameters. Vine copulas consist of a nested sequence of trees built from copulas, where more complex latent dependence can be modeled with increasing number of trees. We propose to estimate the vine copula approximate posterior in a stepwise fashion, tree by tree along the vine structure. Further, we show that the usual backward Kullback-Leibler divergence cannot recover the correct parameters in the vine copula model, thus the evidence lower bound is defined based on the Rényi divergence. Finally, an intuitive stopping criterion for adding further trees to the vine eliminates the need to pre-define a complexity parameter of the variational distribution, as required for most other approaches. Thus, our method interpolates between mean-field VI (MFVI) and full latent dependence. In many applications, in particular sparse Gaussian processes, our method is parsimonious with parameters, while outperforming MFVI.
Mar 22, 2026

Amortized Variational Inference for Logistic Regression with Missing Covariates

Missing covariate data pose a significant challenge to statistical inference and machine learning, particularly for classification tasks like logistic regression. Classical iterative approaches (EM, multiple imputation) are often computationally intensive, sensitive to high missingness rates, and limited in uncertainty propagation. Recent deep generative models based on VAEs show promise but rely on complex latent representations. We propose Amortized Variational Inference for Logistic Regression (AV-LR), a unified end-to-end framework for binary logistic regression with missing covariates. AV-LR integrates a probabilistic generative model with a simple amortized inference network, trained jointly by maximizing the evidence lower bound. Unlike competing methods, AV-LR performs inference directly in the space of missing data without additional latent variables, using a single inference network and a linear layer that jointly estimate regression parameters and the missingness mechanism. AV-LR achieves estimation accuracy comparable to or better than state-of-the-art EM-like algorithms, with significantly lower computational cost. It naturally extends to missing-not-at-random settings by explicitly modeling the missingness mechanism. Empirical results on synthetic and real-world datasets confirm its effectiveness and efficiency across various missing-data scenarios.
Mar 21, 2026

Stability of Sequential and Parallel Coordinate Ascent Variational Inference

We highlight a striking difference in behavior between two widely used variants of coordinate ascent variational inference: the sequential and parallel algorithms. While such differences were known in the numerical analysis literature in simpler settings, they remain largely unexplored in the optimization-focused literature on variational inference in more complex models. Focusing on the moderately high-dimensional linear regression problem, we show that the sequential algorithm, although typically slower, enjoys convergence guarantees under more relaxed conditions than the parallel variant, which is often employed to facilitate block-wise updates and improve computational efficiency.
Mar 20, 2026

Var-JEPA: A Variational Formulation of the Joint-Embedding Predictive Architecture - Bridging Predictive and Generative Self-Supervised Learning

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.
Mar 18, 2026

Variational Rectification Inference for Learning with Noisy Labels

Label noise has been broadly observed in real-world datasets. To mitigate the negative impact of overfitting to label noise for deep models, effective strategies (e.g., re-weighting, or loss rectification) have been broadly applied in prevailing approaches, which have been generally learned under the meta-learning scenario. Despite the robustness of noise achieved by the probabilistic meta-learning models, they usually suffer from model collapse that degenerates generalization performance. In this paper, we propose variational rectification inference (VRI) to formulate the adaptive rectification for loss functions as an amortized variational inference problem and derive the evidence lower bound under the meta-learning framework. Specifically, VRI is constructed as a hierarchical Bayes by treating the rectifying vector as a latent variable, which can rectify the loss of the noisy sample with the extra randomness regularization and is, therefore, more robust to label noise. To achieve the inference of the rectifying vector, we approximate its conditional posterior with an amortization meta-network. By introducing the variational term in VRI, the conditional posterior is estimated accurately and avoids collapsing to a Dirac delta function, which can significantly improve the generalization performance. The elaborated meta-network and prior network adhere to the smoothness assumption, enabling the generation of reliable rectification vectors. Given a set of clean meta-data, VRI can be efficiently meta-learned within the bi-level optimization programming. Besides, theoretical analysis guarantees that the meta-network can be efficiently learned with our algorithm. Comprehensive comparison experiments and analyses validate its effectiveness for robust learning with noisy labels, particularly in the presence of open-set noise.
Mar 17, 2026

Adaptive regularization parameter selection for high-dimensional inverse problems: A Bayesian approach with Tucker low-rank constraints

This paper introduces a novel variational Bayesian method that integrates Tucker decomposition for efficient high-dimensional inverse problem solving. The method reduces computational complexity by transforming variational inference from a high-dimensional space to a lower-dimensional core tensor space via Tucker decomposition. A key innovation is the introduction of per-mode precision parameters, enabling adaptive regularization for anisotropic structures. For instance, in directional image deblurring, learned parameters align with physical anisotropy, applying stronger regularization to critical directions (e.g., row vs. column axes). The method further estimates noise levels from data, eliminating reliance on prior knowledge of noise parameters (unlike conventional benchmarks such as the discrepancy principle (DP)). Experimental evaluations across 2D deblurring, 3D heat conduction, and Fredholm integral equations demonstrate consistent improvements in quantitative metrics (PSNR, SSIM) and qualitative visualizations (error maps, precision parameter trends) compared to L-curve criterion, generalized cross-validation (GCV), unbiased predictive risk estimator (UPRE), and DP. The approach scales to problems with 110,000 variables and outperforms existing methods by 0.73-2.09 dB in deblurring tasks and 6.75 dB in 3D heat conduction. Limitations include sensitivity to rank selection in Tucker decomposition and the need for theoretical analysis. Future work will explore automated rank selection and theoretical guarantees. This method bridges Bayesian theory and scalable computation, offering practical solutions for large-scale inverse problems in imaging, remote sensing, and scientific computing.
Mar 14, 2026

Exploring the Dimensions of a Variational Neuron

We introduce EVE (Elemental Variational Expanse), a variational distributional neuron formulated as a local probabilistic computational unit with an explicit prior, an amortized posterior, and unit-level variational regularization. In most modern architectures, uncertainty is modeled through global latent variables or parameter uncertainty, while the computational unit itself remains scalar. EVE instead relocates probabilistic structure to the neuron level, making it locally observable and controllable. In this paper, the term dimensions refers primarily to the neuron's internal latent dimensionality, denoted by k. We study how varying k, from the atomic case k = 1 to higher-dimensional latent spaces, changes the neuron's learned operating regime. We then examine how this main axis interacts with two additional structural properties: local capacity control and temporal persistence through a neuron-level autoregressive extension. To support this study, EVE is instrumented with internal diagnostics and constraints, including effective KL, a target band on mu^2, out-of-band fractions, and indicators of drift and collapse. Across selected forecasting and tabular settings, we show that latent dimensionality, control, and temporal extension shape the neuron's internal regime, and that some neuron-level variables are measurable, informative, and related to downstream behavior. Overall, the paper provides an experimentally grounded first map of the design space opened by a variational neuron.
Mar 13, 2026

Variational Garrote for Sparse Inverse Problems

Sparse regularization plays a central role in solving inverse problems arising from incomplete or corrupted measurements. Different regularizers correspond to different prior assumptions about the structure of the unknown signal, and reconstruction performance depends on how well these priors match the intrinsic sparsity of the data. This work investigates the effect of sparsity priors in inverse problems by comparing conventional L1 regularization with the Variational Garrote (VG), a probabilistic method that approximates L0 sparsity through variational binary gating variables. A unified experimental framework is constructed across multiple reconstruction tasks including signal resampling, signal denoising, and sparse-view computed tomography. To enable consistent comparison across models with different parameterizations, regularization strength is swept across wide ranges and reconstruction behavior is analyzed through train-generalization error curves. Experiments reveal characteristic bias-variance tradeoff patterns across tasks and demonstrate that VG frequently achieves lower minimum generalization error and improved stability in strongly underdetermined regimes where accurate support recovery is critical. These results suggest that sparsity priors closer to spike-and-slab structure can provide advantages when the underlying coefficient distribution is strongly sparse. The study highlights the importance of prior-data alignment in sparse inverse problems and provides empirical insights into the behavior of variational L0-type methods across different information bottlenecks.
Mar 12, 2026

Slack More, Predict Better: Proximal Relaxation for Probabilistic Latent Variable Model-based Soft Sensors

Nonlinear Probabilistic Latent Variable Models (NPLVMs) are a cornerstone of soft sensor modeling due to their capacity for uncertainty delineation. However, conventional NPLVMs are trained using amortized variational inference, where neural networks parameterize the variational posterior. While facilitating model implementation, this parameterization converts the distributional optimization problem within an infinite-dimensional function space to parameter optimization within a finite-dimensional parameter space, which introduces an approximation error gap, thereby degrading soft sensor modeling accuracy. To alleviate this issue, we introduce KProxNPLVM, a novel NPLVM that pivots to relaxing the objective itself and improving the NPLVM's performance. Specifically, we first prove the approximation error induced by the conventional approach. Based on this, we design the Wasserstein distance as the proximal operator to relax the learning objective, yielding a new variational inference strategy derived from solving this relaxed optimization problem. Based on this foundation, we provide a rigorous derivation of KProxNPLVM's optimization implementation, prove the convergence of our algorithm can finally sidestep the approximation error, and propose the KProxNPLVM by summarizing the abovementioned content. Finally, extensive experiments on synthetic and real-world industrial datasets are conducted to demonstrate the efficacy of the proposed KProxNPLVM.
Mar 10, 2026

Variational Routing: A Scalable Bayesian Framework for Calibrated Mixture-of-Experts Transformers

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 fine-tuning 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.
Mar 1, 2026

Active Flow Matching

Discrete diffusion and flow matching models capture complex, non-additive and non-autoregressive structure in high-dimensional objective landscapes through parallel, iterative refinement. However, their implicit generative nature precludes direct integration with principled variational frameworks for online black-box optimisation, such as variational search distributions (VSD) and conditioning by adaptive sampling (CbAS). We introduce Active Flow Matching (AFM), which reformulates variational objectives to operate on conditional endpoint distributions along the flow, enabling gradient-based steering of flow models toward high-fitness regions while preserving the rigour of VSD and CbAS. We derive forward and reverse Kullback-Leibler (KL) variants using self-normalised importance sampling. Across a suite of online protein and small molecule design tasks, forward-KL AFM consistently performs competitively compared to state-of-the-art baselines, demonstrating effective exploration-exploitation under tight experimental budgets.
Feb 27, 2026

Neural Diffusion Intensity Models for Point Process Data

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.