Bayesian Computation in Deep Learning

This review paper is intended for the 2nd edition of the Handbook of Markov chain Monte Carlo. We provide an introduction to approximate inference techniques as Bayesian computation methods applied to deep learning models. We organize the chapter by presenting popular computational methods for Bayesian neural networks and deep generative models, explaining their unique challenges in posterior inference as well as the solutions.

Unsupervised Accelerated MRI Reconstruction via Ground-Truth-Free Flow Matching

Accelerated magnetic resonance imaging involves reconstructing fully sampled images from undersampled k-space measurements. Current state-of-the-art approaches have mainly focused on either end-to-end supervised training inspired by compressed sensing formulations, or posterior sampling methods built on modern generative models. However, their efficacy heavily relies on large datasets of fully sampled images, which may not always be available in practice. To address this issue, we propose an unsupervised MRI reconstruction method based on ground-truth-free flow matching (GTF$^2$M). Particularly, the GTF$^2$M learns a prior denoising process of fully sampled ground-truth images using only undersampled data. Based on that, an efficient cyclic reconstruction algorithm is further proposed to perform forward and backward integration in the dual space of image-space signal and k-space measurement. We compared our method with state-of-the-art learning-based baselines on the fastMRI database of both single-coil knee and multi-coil brain MRIs. The results show that our proposed unsupervised method can significantly outperform existing unsupervised approaches, and achieve performance comparable to most supervised end-to-end and prior learning baselines trained on fully sampled MRI, while offering greater efficiency than the compared generative model-based approaches.

Recurrent Memory for Online Interdomain Gaussian Processes

We propose a novel online Gaussian process (GP) model that is capable of capturing long-term memory in sequential data in an online regression setting. Our model, Online HiPPO Sparse Variational Gaussian Process Regression (OHSGPR), leverages the HiPPO (High-order Polynomial Projection Operators) framework, which is popularized in the RNN domain due to its long-range memory modeling capabilities. We interpret the HiPPO time-varying orthogonal projections as inducing variables with time-dependent orthogonal polynomial basis functions, which allows the SGPR inducing points to memorize the process history. We show that the HiPPO framework fits naturally into the interdomain GP framework and demonstrate that the kernel matrices can also be updated online in a recurrence form based on the ODE evolution of HiPPO. We evaluate our method on time series regression tasks, showing that it outperforms the existing online GP method in terms of predictive performance and computational efficiency

Neural Flow Samplers with Shortcut Models

Sampling from unnormalized densities is a fundamental task across various domains. Flow-based samplers generate samples by learning a velocity field that satisfies the continuity equation, but this requires estimating the intractable time derivative of the partition function. While importance sampling provides an approximation, it suffers from high variance. To mitigate this, we introduce a velocity-driven Sequential Monte Carlo method combined with control variates to reduce variance. Additionally, we incorporate a shortcut model to improve efficiency by minimizing the number of sampling steps. Empirical results on both synthetic datasets and $n$-body system targets validate the effectiveness of our approach.

Energy-Based Modelling for Discrete and Mixed Data via Heat Equations on Structured Spaces

Energy-based models (EBMs) offer a flexible framework for probabilistic modelling across various data domains. However, training EBMs on data in discrete or mixed state spaces poses significant challenges due to the lack of robust and fast sampling methods. In this work, we propose to train discrete EBMs with Energy Discrepancy, a loss function which only requires the evaluation of the energy function at data points and their perturbed counterparts, thus eliminating the need for Markov chain Monte Carlo. We introduce perturbations of the data distribution by simulating a diffusion process on the discrete state space endowed with a graph structure. This allows us to inform the choice of perturbation from the structure of the modelled discrete variable, while the continuous time parameter enables fine-grained control of the perturbation. Empirically, we demonstrate the efficacy of the proposed approaches in a wide range of applications, including the estimation of discrete densities with non-binary vocabulary and binary image modelling. Finally, we train EBMs on tabular data sets with applications in synthetic data generation and calibrated classification.

Your Image is Secretly the Last Frame of a Pseudo Video

Diffusion models, which can be viewed as a special case of hierarchical variational autoencoders (HVAEs), have shown profound success in generating photo-realistic images. In contrast, standard HVAEs often produce images of inferior quality compared to diffusion models. In this paper, we hypothesize that the success of diffusion models can be partly attributed to the additional self-supervision information for their intermediate latent states provided by corrupted images, which along with the original image form a pseudo video. Based on this hypothesis, we explore the possibility of improving other types of generative models with such pseudo videos. Specifically, we first extend a given image generative model to their video generative model counterpart, and then train the video generative model on pseudo videos constructed by applying data augmentation to the original images. Furthermore, we analyze the potential issues of first-order Markov data augmentation methods, which are typically used in diffusion models, and propose to use more expressive data augmentation to construct more useful information in pseudo videos. Our empirical results on the CIFAR10 and CelebA datasets demonstrate that improved image generation quality can be achieved with additional self-supervised information from pseudo videos.

Mutual Information Multinomial Estimation

Estimating mutual information (MI) is a fundamental yet challenging task in data science and machine learning. This work proposes a new estimator for mutual information. Our main discovery is that a preliminary estimate of the data distribution can dramatically help estimate. This preliminary estimate serves as a bridge between the joint and the marginal distribution, and by comparing with this bridge distribution we can easily obtain the true difference between the joint distributions and the marginal distributions. Experiments on diverse tasks including non-Gaussian synthetic problems with known ground-truth and real-world applications demonstrate the advantages of our method.

Identifying Nonstationary Causal Structures with High-Order Markov Switching Models

Causal discovery in time series is a rapidly evolving field with a wide variety of applications in other areas such as climate science and neuroscience. Traditional approaches assume a stationary causal graph, which can be adapted to nonstationary time series with time-dependent effects or heterogeneous noise. In this work we address nonstationarity via regime-dependent causal structures. We first establish identifiability for high-order Markov Switching Models, which provide the foundations for identifiable regime-dependent causal discovery. Our empirical studies demonstrate the scalability of our proposed approach for high-order regime-dependent structure estimation, and we illustrate its applicability on brain activity data.

Diffusion Model With Optimal Covariance Matching

The probabilistic diffusion model has become highly effective across various domains. Typically, sampling from a diffusion model involves using a denoising distribution characterized by a Gaussian with a learned mean and either fixed or learned covariances. In this paper, we leverage the recently proposed full covariance moment matching technique and introduce a novel method for learning covariances. Unlike traditional data-driven covariance approximation approaches, our method involves directly regressing the optimal analytic covariance using a new, unbiased objective named Optimal Covariance Matching (OCM). This approach can significantly reduce the approximation error in covariance prediction. We demonstrate how our method can substantially enhance the sampling efficiency of both Markovian (DDPM) and non-Markovian (DDIM) diffusion model families.

C-TPT: Calibrated Test-Time Prompt Tuning for Vision-Language Models via Text Feature Dispersion

In deep learning, test-time adaptation has gained attention as a method for model fine-tuning without the need for labeled data. A prime exemplification is the recently proposed test-time prompt tuning for large-scale vision-language models such as CLIP. Unfortunately, these prompts have been mainly developed to improve accuracy, overlooking the importance of calibration, which is a crucial aspect for quantifying prediction uncertainty. However, traditional calibration methods rely on substantial amounts of labeled data, making them impractical for test-time scenarios. To this end, this paper explores calibration during test-time prompt tuning by leveraging the inherent properties of CLIP. Through a series of observations, we find that the prompt choice significantly affects the calibration in CLIP, where the prompts leading to higher text feature dispersion result in better-calibrated predictions. Introducing the Average Text Feature Dispersion (ATFD), we establish its relationship with calibration error and present a novel method, Calibrated Test-time Prompt Tuning (C-TPT), for optimizing prompts during test-time with enhanced calibration. Through extensive experiments on different CLIP architectures and datasets, we show that C-TPT can effectively improve the calibration of test-time prompt tuning without needing labeled data. The code is publicly accessible at https://github.com/hee-suk-yoon/C-TPT.