Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching

In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback--Leibler divergence, this is in general not hold true for the Wasserstein distance. In this paper, we introduce a conditional Wasserstein distance via a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. Interestingly, the dual formulation of the conditional Wasserstein-1 flow resembles losses in the conditional Wasserstein GAN literature in a quite natural way. We derive theoretical properties of the conditional Wasserstein distance, characterize the corresponding geodesics and velocity fields as well as the flow ODEs. Subsequently, we propose to approximate the velocity fields by relaxing the conditional Wasserstein distance. Based on this, we propose an extension of OT Flow Matching for solving Bayesian inverse problems and demonstrate its numerical advantages on an inverse problem and class-conditional image generation.

Mixed Noise and Posterior Estimation with Conditional DeepGEM

Motivated by indirect measurements and applications from nanometrology with a mixed noise model, we develop a novel algorithm for jointly estimating the posterior and the noise parameters in Bayesian inverse problems. We propose to solve the problem by an expectation maximization (EM) algorithm. Based on the current noise parameters, we learn in the E-step a conditional normalizing flow that approximates the posterior. In the M-step, we propose to find the noise parameter updates again by an EM algorithm, which has analytical formulas. We compare the training of the conditional normalizing flow with the forward and reverse KL, and show that our model is able to incorporate information from many measurements, unlike previous approaches.

Learning from small data sets: Patch-based regularizers in inverse problems for image reconstruction

The solution of inverse problems is of fundamental interest in medical and astronomical imaging, geophysics as well as engineering and life sciences. Recent advances were made by using methods from machine learning, in particular deep neural networks. Most of these methods require a huge amount of (paired) data and computer capacity to train the networks, which often may not be available. Our paper addresses the issue of learning from small data sets by taking patches of very few images into account. We focus on the combination of model-based and data-driven methods by approximating just the image prior, also known as regularizer in the variational model. We review two methodically different approaches, namely optimizing the maximum log-likelihood of the patch distribution, and penalizing Wasserstein-like discrepancies of whole empirical patch distributions. From the point of view of Bayesian inverse problems, we show how we can achieve uncertainty quantification by approximating the posterior using Langevin Monte Carlo methods. We demonstrate the power of the methods in computed tomography, image super-resolution, and inpainting. Indeed, the approach provides also high-quality results in zero-shot super-resolution, where only a low-resolution image is available. The paper is accompanied by a GitHub repository containing implementations of all methods as well as data examples so that the reader can get their own insight into the performance.

Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances

In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback Leibler divergence, it does not hold true for the Wasserstein distance. We will introduce a conditional Wasserstein distance with a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. By deriving its dual, we find a rigorous way to motivate the loss of conditional Wasserstein GANs. We outline conditions under which the vanilla and the conditional Wasserstein distance coincide. Furthermore, we will show numerical examples where training with the conditional Wasserstein distance yields favorable properties for posterior sampling.

Posterior Sampling Based on Gradient Flows of the MMD with Negative Distance Kernel

We propose conditional flows of the maximum mean discrepancy (MMD) with the negative distance kernel for posterior sampling and conditional generative modeling. This MMD, which is also known as energy distance, has several advantageous properties like efficient computation via slicing and sorting. We approximate the joint distribution of the ground truth and the observations using discrete Wasserstein gradient flows and establish an error bound for the posterior distributions. Further, we prove that our particle flow is indeed a Wasserstein gradient flow of an appropriate functional. The power of our method is demonstrated by numerical examples including conditional image generation and inverse problems like superresolution, inpainting and computed tomography in low-dose and limited-angle settings.

Generative Sliced MMD Flows with Riesz Kernels

Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels $K(x,y) = - \|x-y\|^r$, $r \in (0,2)$ have exceptional properties which allow for their efficient computation. First, the MMD of Riesz kernels coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for $r=1$, a simple sorting algorithm can be applied to reduce the complexity from $O(MN+N^2)$ to $O((M+N)\log(M+N))$ for two empirical measures with $M$ and $N$ support points. For the implementations we approximate the gradient of the sliced MMD by using only a finite number $P$ of slices. We show that the resulting error has complexity $O(\sqrt{d/P})$, where $d$ is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for large scale applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.

Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems

Conditional generative models became a very powerful tool to sample from Bayesian inverse problem posteriors. It is well-known in classical Bayesian literature that posterior measures are quite robust with respect to perturbations of both the prior measure and the negative log-likelihood, which includes perturbations of the observations. However, to the best of our knowledge, the robustness of conditional generative models with respect to perturbations of the observations has not been investigated yet. In this paper, we prove for the first time that appropriately learned conditional generative models provide robust results for single observations.

Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation

Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. Existing SBDMs are typically formulated in a finite-dimensional setting, where images are considered as tensors of a finite size. This papers develops SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently on multiple resolution levels. We thereby hope to obtain diffusion models that generalize across different resolution levels and improve the efficiency of the training process. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting. First, we modify the forward process to ensure that the latent distribution is well-defined in the infinite-dimensional setting using the notion of trace class operators. Second, we illustrate that approximating the score function with an operator network, in our case Fourier neural operators (FNOs), is beneficial for multilevel training. After deriving the forward and reverse process in the infinite-dimensional setting, we show their well-posedness, derive adequate discretizations, and investigate the role of the latent distributions. We provide first promising numerical results on two datasets, MNIST and material structures. In particular, we show that multilevel training is feasible within this framework.

PatchNR: Learning from Small Data by Patch Normalizing Flow Regularization

Learning neural networks using only a small amount of data is an important ongoing research topic with tremendous potential for applications. In this paper, we introduce a regularizer for the variational modeling of inverse problems in imaging based on normalizing flows. Our regularizer, called patchNR, involves a normalizing flow learned on patches of very few images. The subsequent reconstruction method is completely unsupervised and the same regularizer can be used for different forward operators acting on the same class of images. By investigating the distribution of patches versus those of the whole image class, we prove that our variational model is indeed a MAP approach. Our model can be generalized to conditional patchNRs, if additional supervised information is available. Numerical examples for low-dose CT, limited-angle CT and superresolution of material images demonstrate that our method provides high quality results among unsupervised methods, but requires only few data.

A Unified Approach to Variational Autoencoders and Stochastic Normalizing Flows via Markov Chains

Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. In this paper, we provide a unified framework to handle these approaches via Markov chains. Indeed, we consider stochastic normalizing flows as pair of Markov chains fulfilling some properties and show that many state-of-the-art models for data generation fit into this framework. The Markov chains point of view enables us to couple both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers and variational autoencoders in a mathematically sound way. Besides layers with densities as Langevin layers, diffusion layers or variational autoencoders, also layers having no densities as deterministic layers or Metropolis-Hasting layers can be handled. Hence our framework establishes a useful mathematical tool to combine the various approaches.