Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.
Despite their widespread use, large language models (LLMs) are known to hallucinate incorrect information and be poorly calibrated. This makes the uncertainty quantification of these models of critical importance, especially in high-stakes domains, such as autonomy and healthcare. Prior work has made Bayesian deep learning-based approaches to this problem more tractable by performing inference over the low-rank adaptation (LoRA) parameters of a fine-tuned model. While effective, these approaches struggle to scale to larger LLMs due to requiring further additional parameters compared to LoRA. In this work we present $\textbf{Scala}$ble $\textbf{B}$ayesian $\textbf{L}$ow-Rank Adaptation via Stochastic Variational Subspace Inference (ScalaBL). We perform Bayesian inference in an $r$-dimensional subspace, for LoRA rank $r$. By repurposing the LoRA parameters as projection matrices, we are able to map samples from this subspace into the full weight space of the LLM. This allows us to learn all the parameters of our approach using stochastic variational inference. Despite the low dimensionality of our subspace, we are able to achieve competitive performance with state-of-the-art approaches while only requiring ${\sim}1000$ additional parameters. Furthermore, it allows us to scale up to the largest Bayesian LLM to date, with four times as a many base parameters as prior work.
Contrastive learning has proven to be highly efficient and adaptable in shaping representation spaces across diverse modalities by pulling similar samples together and pushing dissimilar ones apart. However, two key limitations persist: (1) Without explicit regulation of the embedding distribution, semantically related instances can inadvertently be pushed apart unless complementary signals guide pair selection, and (2) excessive reliance on large in-batch negatives and tailored augmentations hinders generalization. To address these limitations, we propose Variational Supervised Contrastive Learning (VarCon), which reformulates supervised contrastive learning as variational inference over latent class variables and maximizes a posterior-weighted evidence lower bound (ELBO) that replaces exhaustive pair-wise comparisons for efficient class-aware matching and grants fine-grained control over intra-class dispersion in the embedding space. Trained exclusively on image data, our experiments on CIFAR-10, CIFAR-100, ImageNet-100, and ImageNet-1K show that VarCon (1) achieves state-of-the-art performance for contrastive learning frameworks, reaching 79.36% Top-1 accuracy on ImageNet-1K and 78.29% on CIFAR-100 with a ResNet-50 encoder while converging in just 200 epochs; (2) yields substantially clearer decision boundaries and semantic organization in the embedding space, as evidenced by KNN classification, hierarchical clustering results, and transfer-learning assessments; and (3) demonstrates superior performance in few-shot learning than supervised baseline and superior robustness across various augmentation strategies.
State estimation or filtering serves as a fundamental task to enable intelligent decision-making in applications such as autonomous vehicles, robotics, healthcare monitoring, smart grids, intelligent transportation, and predictive maintenance. Standard filtering assumes prior knowledge of noise statistics to extract latent system states from noisy sensor data. However, real-world scenarios involve abnormalities like outliers, biases, drifts, and missing observations with unknown or partially known statistics, limiting conventional approaches. This thesis presents novel robust nonlinear filtering methods to mitigate these challenges. Based on insights from our filtering proposals, we extend the formulations to offline estimation/learning setups and propose smoothing extensions. Our methods leverage Bayesian inference frameworks, employing both deterministic and stochastic approximation techniques including Variational Inference (VI) and Particle Filters/Sequential Monte Carlo (SMC). We also study theoretical estimation limits using Bayesian Cram\'er-Rao bounds (BCRBs) in the context of measurement abnormalities. To validate the performance gains of the proposed methods, we perform simulations and experiments in scenarios including target tracking, indoor localization, 3D point cloud registration, mesh registration, and pose graph optimization. The fundamental nature of the work makes it useful in diverse applications, with possible future extensions toward developing outlier-robust machine learning pipelines, learning system dynamics from anomalous data, and addressing challenges in generative AI where standard diffusion models struggle with outliers, imbalanced datasets, and mode collapse.
Bias field artifacts in magnetic resonance imaging (MRI) scans introduce spatially smooth intensity inhomogeneities that degrade image quality and hinder downstream analysis. To address this challenge, we propose a novel variational Hadamard U-Net (VHU-Net) for effective body MRI bias field correction. The encoder comprises multiple convolutional Hadamard transform blocks (ConvHTBlocks), each integrating convolutional layers with a Hadamard transform (HT) layer. Specifically, the HT layer performs channel-wise frequency decomposition to isolate low-frequency components, while a subsequent scaling layer and semi-soft thresholding mechanism suppress redundant high-frequency noise. To compensate for the HT layer's inability to model inter-channel dependencies, the decoder incorporates an inverse HT-reconstructed transformer block, enabling global, frequency-aware attention for the recovery of spatially consistent bias fields. The stacked decoder ConvHTBlocks further enhance the capacity to reconstruct the underlying ground-truth bias field. Building on the principles of variational inference, we formulate a new evidence lower bound (ELBO) as the training objective, promoting sparsity in the latent space while ensuring accurate bias field estimation. Comprehensive experiments on abdominal and prostate MRI datasets demonstrate the superiority of VHU-Net over existing state-of-the-art methods in terms of intensity uniformity, signal fidelity, and tissue contrast. Moreover, the corrected images yield substantial downstream improvements in segmentation accuracy. Our framework offers computational efficiency, interpretability, and robust performance across multi-center datasets, making it suitable for clinical deployment.
Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is typically the (reverse) Kullback-Leibler (KL) divergence. In this paper, we focus on the following parametric family: mixtures of isotropic Gaussians (i.e., with diagonal covariance matrices proportional to the identity) and uniform weights. We develop a variational framework and provide efficient algorithms suited for this family. In contrast with mixtures of Gaussian with generic covariance matrices, this choice presents a balance between accurate approximations of multimodal Bayesian posteriors, while being memory and computationally efficient. Our algorithms implement gradient descent on the location of the mixture components (the modes of the Gaussians), and either (an entropic) Mirror or Bures descent on their variance parameters. We illustrate the performance of our algorithms on numerical experiments.
Multiresolution image fusion is a key problem for real-time satellite imaging and plays a central role in detecting and monitoring natural phenomena such as floods. It aims to solve the trade-off between temporal and spatial resolution in remote sensing instruments. Although several algorithms have been proposed for this problem, the presence of outliers such as clouds downgrades their performance. Moreover, strategies that integrate robustness, recursive operation and learned models are missing. In this paper, a robust recursive image fusion framework leveraging location-aware neural networks (NN) to model the image dynamics is proposed. Outliers are modeled by representing the probability of contamination of a given pixel and band. A NN model trained on a small dataset provides accurate predictions of the stochastic image time evolution, which improves both the accuracy and robustness of the method. A recursive solution is proposed to estimate the high-resolution images using a Bayesian variational inference framework. Experiments fusing images from the Landsat 8 and MODIS instruments show that the proposed approach is significantly more robust against cloud cover, without losing performance when no clouds are present.
We propose a novel particle-based variational inference method designed to work with multimodal distributions. Our approach, referred to as Branched Stein Variational Gradient Descent (BSVGD), extends the classical Stein Variational Gradient Descent (SVGD) algorithm by incorporating a random branching mechanism that encourages the exploration of the state space. In this work, a theoretical guarantee for the convergence in distribution is presented, as well as numerical experiments to validate the suitability of our algorithm. Performance comparisons between the BSVGD and the SVGD are presented using the Wasserstein distance between samples and the corresponding computational times.
Variational inference (VI) is a popular method for approximating intractable posterior distributions in Bayesian inference and probabilistic machine learning. In this paper, we introduce a general framework for quantifying the statistical accuracy of mean-field variational inference (MFVI) for posterior approximation in Bayesian latent variable models with categorical local latent variables. Utilizing our general framework, we capture the exact asymptotic regime where MFVI `works' for the celebrated latent Dirichlet allocation (LDA) model. Focusing on the mixed membership stochastic blockmodel (MMSB), we show that the vanilla fully factorized MFVI, often used in the literature, is suboptimal. We propose a partially grouped VI algorithm for this model and show that it works, and derive its exact asymptotic performance. We further illustrate that our bounds are tight for both the above models.
Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target distribution varies within the class of strongly log-concave measures. We establish dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in the 2-Wasserstein distance, with Lipschitz constant inversely proportional to the log-concavity parameter. Under additional regularity conditions, we further show that the MFVI optimizer depends differentiably on the target potential and characterize the derivative by a partial differential equation. Methodologically, we follow a novel approach to MFVI via linearized optimal transport: the non-convex MFVI problem is lifted to a convex optimization over transport maps with a fixed base measure, enabling the use of calculus of variations and functional analysis. We discuss several applications of our results to robust Bayesian inference and empirical Bayes, including a quantitative Bernstein--von Mises theorem for MFVI, as well as to distributed stochastic control.
This study advances the Variational Autoencoder (VAE) framework by addressing challenges in Independent Component Analysis (ICA) under both determined and underdetermined conditions, focusing on enhancing the independence and interpretability of latent variables. Traditional VAEs map observed data to latent variables and back via an encoder-decoder architecture, but struggle with underdetermined ICA where the number of latent variables exceeds observed signals. The proposed Half Adversarial VAE (Half-AVAE) builds on the encoder-free Half-VAE framework, eliminating explicit inverse mapping to tackle underdetermined scenarios. By integrating adversarial networks and External Enhancement (EE) terms, Half-AVAE promotes mutual independence among latent dimensions, achieving factorized and interpretable representations. Experiments with synthetic signals demonstrate that Half-AVAE outperforms baseline models, including GP-AVAE and Half-VAE, in recovering independent components under underdetermined conditions, as evidenced by lower root mean square errors. The study highlights the flexibility of VAEs in variational inference, showing that encoder omission, combined with adversarial training and structured priors, enables effective solutions for complex ICA tasks, advancing applications in disentanglement, causal inference, and generative modeling.
In the fifth-generation communication system (5G), multipath-assisted positioning (MAP) has emerged as a promising approach. With the enhancement of signal resolution, multipath component (MPC) are no longer regarded as noise but rather as valuable information that can contribute to positioning. However, existing research often treats reflective surfaces as ideal reflectors, while being powerless in the face of indistinguishable multipath caused by diffuse reflectors. This study approaches diffuse reflectors from the perspective of uncertainty, investigating the statistical distribution characteristics of indoor diffuse and specular reflectors. Based on these insights, a task-guided disentangled representation learning method leveraging multi-time channel impulse response (CIR) observations is designed to directly map CIRs to positions, while mitigating the adverse effects of components that contribute minimally to localization accuracy (e.g., diffuse multipath).In this semi-supervised learning framework, a global feature extraction architecture based on self-attention is proposed to capture location-independent wireless environmental information, while an MLP is employed to extract the time-varying features related to user equipment (UE) positions. Variational inference based on a latent variable model (LVM) is applied to separate independent features within the CIR, with position labels guiding the LVM to express components more beneficial for localization. Additionally, we provide a feasibility proof for the separability of diffuse and specular environmental features in CIRs. Simulation results demonstrate that the proposed method achieves higher localization accuracy compared to conventional search-based localization methods, with enhanced robustness against indistinguishable multipath from diffuse reflectors.
Binary Neural Networks (BiNNs), which employ single-bit precision weights, have emerged as a promising solution to reduce memory usage and power consumption while maintaining competitive performance in large-scale systems. However, training BiNNs remains a significant challenge due to the limitations of conventional training algorithms. Quantum HyperNetworks offer a novel paradigm for enhancing the optimization of BiNN by leveraging quantum computing. Specifically, a Variational Quantum Algorithm is employed to generate binary weights through quantum circuit measurements, while key quantum phenomena such as superposition and entanglement facilitate the exploration of a broader solution space. In this work, we establish a connection between this approach and Bayesian inference by deriving the Evidence Lower Bound (ELBO), when direct access to the output distribution is available (i.e., in simulations), and introducing a surrogate ELBO based on the Maximum Mean Discrepancy (MMD) metric for scenarios involving implicit distributions, as commonly encountered in practice. Our experimental results demonstrate that the proposed methods outperform standard Maximum Likelihood Estimation (MLE), improving trainability and generalization.
We introduce an optimization framework for variational inference based on the coupled free energy, extending variational inference techniques to account for the curved geometry of the coupled exponential family. This family includes important heavy-tailed distributions such as the generalized Pareto and the Student's t. By leveraging the coupled free energy, which is equal to the coupled evidence lower bound (ELBO) of the inverted probabilities, we improve the accuracy and robustness of the learned model. The coupled generalization of Fisher Information metric and the affine connection. The method is applied to the design of a coupled variational autoencoder (CVAE). By using the coupling for both the distributions and cost functions, the reconstruction metric is derived to still be the mean-square average loss with modified constants. The novelty comes from sampling the heavy-tailed latent distribution with its associated coupled probability, which has faster decaying tails. The result is the ability to train a model robust against severe outliers, while assuring that the training process is stable. The Wasserstein-2 or Fr\'echet Inception Distance of the reconstructed CelebA images shows the CVAE has a 3\% improvement over the VAE after 5 epochs of training.
Matrix factorization is a widely used technique in recommendation systems. Probabilistic Matrix Factorization (PMF) [1] extends traditional matrix factorization by incorporating probability distributions over latent factors, allowing for uncertainty quantification. However, computing the posterior distribution is intractable due to the high-dimensional integral. To address this, we employ two Bayesian inference methods: Markov Chain Monte Carlo (MCMC) [2] and Variational Inference (VI) [3] to approximate the posterior. We evaluate their performance on MovieLens dataset and compare their convergence speed, predictive accuracy, and computational efficiency. Experimental results demonstrate that VI offers faster convergence, while MCMC provides more accurate posterior estimates.
Latent Gaussian variables have been popularised in probabilistic machine learning. In turn, gradient estimators are the machinery that facilitates gradient-based optimisation for models with latent Gaussian variables. The reparameterisation trick is often used as the default estimator as it is simple to implement and yields low-variance gradients for variational inference. In this work, we propose the R2-G2 estimator as the Rao-Blackwellisation of the reparameterisation gradient estimator. Interestingly, we show that the local reparameterisation gradient estimator for Bayesian MLPs is an instance of the R2-G2 estimator and Rao-Blackwellisation. This lets us extend benefits of Rao-Blackwellised gradients to a suite of probabilistic models. We show that initial training with R2-G2 consistently yields better performance in models with multiple applications of the reparameterisation trick.
The performance of automatic speaker verification (ASV) and anti-spoofing drops seriously under real-world domain mismatch conditions. The relaxed instance frequency-wise normalization (RFN), which normalizes the frequency components based on the feature statistics along the time and channel axes, is a promising approach to reducing the domain dependence in the feature maps of a speaker embedding network. We advocate that the different frequencies should receive different weights and that the weights' uncertainty due to domain shift should be accounted for. To these ends, we propose leveraging variational inference to model the posterior distribution of the weights, which results in Bayesian weighted RFN (BWRFN). This approach overcomes the limitations of fixed-weight RFN, making it more effective under domain mismatch conditions. Extensive experiments on cross-dataset ASV, cross-TTS anti-spoofing, and spoofing-robust ASV show that BWRFN is significantly better than WRFN and RFN.
Predicting Click-Through Rates is a crucial function within recommendation and advertising platforms, as the output of CTR prediction determines the order of items shown to users. The Embedding \& MLP paradigm has become a standard approach for industrial recommendation systems and has been widely deployed. However, this paradigm suffers from cold-start problems, where there is either no or only limited user action data available, leading to poorly learned ID embeddings. The cold-start problem hampers the performance of new items. To address this problem, we designed a novel diffusion model to generate a warmed-up embedding for new items. Specifically, we define a novel diffusion process between the ID embedding space and the side information space. In addition, we can derive a sub-sequence from the diffusion steps to expedite training, given that our diffusion model is non-Markovian. Our diffusion model is supervised by both the variational inference and binary cross-entropy objectives, enabling it to generate warmed-up embeddings for items in both the cold-start and warm-up phases. Additionally, we have conducted extensive experiments on three recommendation datasets. The results confirmed the effectiveness of our approach.
Semi-implicit variational inference (SIVI) is a powerful framework for approximating complex posterior distributions, but training with the Kullback-Leibler (KL) divergence can be challenging due to high variance and bias in high-dimensional settings. While current state-of-the-art semi-implicit variational inference methods, particularly Kernel Semi-Implicit Variational Inference (KSIVI), have been shown to work in high dimensions, training remains moderately expensive. In this work, we propose a kernelized KL divergence estimator that stabilizes training through nonparametric smoothing. To further reduce the bias, we introduce an importance sampling correction. We provide a theoretical connection to the amortized version of the Stein variational gradient descent, which estimates the score gradient via Stein's identity, showing that both methods minimize the same objective, but our semi-implicit approach achieves lower gradient variance. In addition, our method's bias in function space is benign, leading to more stable and efficient optimization. Empirical results demonstrate that our method outperforms or matches state-of-the-art SIVI methods in both performance and training efficiency.
The expressiveness of flow-based models combined with stochastic variational inference (SVI) has, in recent years, expanded the application of optimization-based Bayesian inference to include problems with complex data relationships. However, until now, SVI using flow-based models has been limited to problems of fixed dimension. We introduce CoSMIC, normalizing flows (COntextually-Specified Masking for Identity-mapped Components), an extension to neural autoregressive conditional normalizing flow architectures that enables using a single amortized variational density for inference over a transdimensional target distribution. We propose a combined stochastic variational transdimensional inference (VTI) approach to training CoSMIC flows using techniques from Bayesian optimization and Monte Carlo gradient estimation. Numerical experiments demonstrate the performance of VTI on challenging problems that scale to high-cardinality model spaces.