Sampling from Boltzmann densities with physics informed low-rank formats

Our method proposes the efficient generation of samples from an unnormalized Boltzmann density by solving the underlying continuity equation in the low-rank tensor train (TT) format. It is based on the annealing path commonly used in MCMC literature, which is given by the linear interpolation in the space of energies. Inspired by Sequential Monte Carlo, we alternate between deterministic time steps from the TT representation of the flow field and stochastic steps, which include Langevin and resampling steps. These adjust the relative weights of the different modes of the target distribution and anneal to the correct path distribution. We showcase the efficiency of our method on multiple numerical examples.

Optimizing Federated Learning by Entropy-Based Client Selection

Deep learning is an emerging field revolutionizing various industries, including natural language processing, computer vision, and many more. These domains typically require an extensive amount of data for optimal performance, potentially utilizing huge centralized data repositories. However, such centralization could raise privacy issues concerning the storage of sensitive data. To address this issue, federated learning was developed. It is a newly distributed learning technique that enables to collaboratively train a deep learning model on decentralized devices, referred to as clients, without compromising their data privacy. Traditional federated learning methods often suffer from severe performance degradation when the data distribution among clients differs significantly. This becomes especially problematic in the case of label distribution skew, where the distribution of labels varies across clients. To address this, a novel method called FedEntOpt is proposed. FedEntOpt is designed to mitigate performance issues caused by label distribution skew by maximizing the entropy of the global label distribution of the selected client subset in each federated learning round. This ensures that the aggregated model parameters from the clients were exhibited to data from all available labels, which improves the accuracy of the global model. Extensive experiments on several benchmark datasets show that the proposed method outperforms several state-of-the-art algorithms by up to 6% in classification accuracy, demonstrating robust and superior performance, particularly under low participation rates. In addition, it offers the flexibility to be combined with them, enhancing their performance by over 40%.

Conditional Generative Models for Contrast-Enhanced Synthesis of T1w and T1 Maps in Brain MRI

Contrast enhancement by Gadolinium-based contrast agents (GBCAs) is a vital tool for tumor diagnosis in neuroradiology. Based on brain MRI scans of glioblastoma before and after Gadolinium administration, we address enhancement prediction by neural networks with two new contributions. Firstly, we study the potential of generative models, more precisely conditional diffusion and flow matching, for uncertainty quantification in virtual enhancement. Secondly, we examine the performance of T1 scans from quantitive MRI versus T1-weighted scans. In contrast to T1-weighted scans, these scans have the advantage of a physically meaningful and thereby comparable voxel range. To compare network prediction performance of these two modalities with incompatible gray-value scales, we propose to evaluate segmentations of contrast-enhanced regions of interest using Dice and Jaccard scores. Across models, we observe better segmentations with T1 scans than with T1-weighted scans.

Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry

We deal with the task of sampling from an unnormalized Boltzmann density $\rho_D$ by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $\rho_Z$. First, we examine conditions under which Fisher-Rao flows are absolutely continuous in the Wasserstein geometry. Second, we address specific interpolations $f_t$ and the learning of the related density/velocity pairs $(\rho_t,v_t)$. It was numerically observed that the linear interpolation, which requires only a parametrization of the velocity field $v_t$, suffers from a "teleportation-of-mass" issue. Using tools from the Wasserstein geometry, we give an analytical example, where we can precisely measure the explosion of the velocity field. Inspired by M\'at\'e and Fleuret, who parametrize both $f_t$ and $v_t$, we propose an interpolation which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds to the Wasserstein gradient flow of the Kullback-Leibler divergence related to Langevin dynamics. We demonstrate by numerical examples that our model provides a well-behaved flow field which successfully solves the above sampling task.

PnP-Flow: Plug-and-Play Image Restoration with Flow Matching

In this paper, we introduce Plug-and-Play (PnP) Flow Matching, an algorithm for solving imaging inverse problems. PnP methods leverage the strength of pre-trained denoisers, often deep neural networks, by integrating them in optimization schemes. While they achieve state-of-the-art performance on various inverse problems in imaging, PnP approaches face inherent limitations on more generative tasks like inpainting. On the other hand, generative models such as Flow Matching pushed the boundary in image sampling yet lack a clear method for efficient use in image restoration. We propose to combine the PnP framework with Flow Matching (FM) by defining a time-dependent denoiser using a pre-trained FM model. Our algorithm alternates between gradient descent steps on the data-fidelity term, reprojections onto the learned FM path, and denoising. Notably, our method is computationally efficient and memory-friendly, as it avoids backpropagation through ODEs and trace computations. We evaluate its performance on denoising, super-resolution, deblurring, and inpainting tasks, demonstrating superior results compared to existing PnP algorithms and Flow Matching based state-of-the-art methods.

T-FAKE: Synthesizing Thermal Images for Facial Landmarking

Facial analysis is a key component in a wide range of applications such as security, autonomous driving, entertainment, and healthcare. Despite the availability of various facial RGB datasets, the thermal modality, which plays a crucial role in life sciences, medicine, and biometrics, has been largely overlooked. To address this gap, we introduce the T-FAKE dataset, a new large-scale synthetic thermal dataset with sparse and dense landmarks. To facilitate the creation of the dataset, we propose a novel RGB2Thermal loss function, which enables the transfer of thermal style to RGB faces. By utilizing the Wasserstein distance between thermal and RGB patches and the statistical analysis of clinical temperature distributions on faces, we ensure that the generated thermal images closely resemble real samples. Using RGB2Thermal style transfer based on our RGB2Thermal loss function, we create the T-FAKE dataset, a large-scale synthetic thermal dataset of faces. Leveraging our novel T-FAKE dataset, probabilistic landmark prediction, and label adaptation networks, we demonstrate significant improvements in landmark detection methods on thermal images across different landmark conventions. Our models show excellent performance with both sparse 70-point landmarks and dense 478-point landmark annotations. Our code and models are available at https://github.com/phflot/tfake.

Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functions

We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the negative distance kernel $K(x,y) := -|x-y|$. In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone $\mathcal C(0,1) \subset L_2(0,1)$ of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on $L_2(0,1)$. Based on the construction of an appropriate counterpart of $\mathcal F_\nu$ on $L_2(0,1)$ and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures $\nu$, this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of $\mathcal C(0,1)$. For certain $\mathcal F_\nu$-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees.

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.

Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels

In this paper, we are concerned with estimating the joint probability of random variables $X$ and $Y$, given $N$ independent observation blocks $(\boldsymbol{x}^i,\boldsymbol{y}^i)$, $i=1,\ldots,N$, each of $M$ samples $(\boldsymbol{x}^i,\boldsymbol{y}^i) = \bigl((x^i_j, y^i_{\sigma^i(j)}) \bigr)_{j=1}^M$, where $\sigma^i$ denotes an unknown permutation of i.i.d. sampled pairs $(x^i_j,y_j^i)$, $j=1,\ldots,M$. This means that the internal ordering of the $M$ samples within an observation block is not known. We derive a maximum-likelihood inference functional, propose a computationally tractable approximation and analyze their properties. In particular, we prove a $\Gamma$-convergence result showing that we can recover the true density from empirical approximations as the number $N$ of blocks goes to infinity. Using entropic optimal transport kernels, we model a class of hypothesis spaces of density functions over which the inference functional can be minimized. This hypothesis class is particularly suited for approximate inference of transfer operators from data. We solve the resulting discrete minimization problem by a modification of the EMML algorithm to take addional transition probability constraints into account and prove the convergence of this algorithm. Proof-of-concept examples demonstrate the potential of our method.

Wasserstein Gradient Flows for Moreau Envelopes of f-Divergences in Reproducing Kernel Hilbert Spaces

Most commonly used $f$-divergences of measures, e.g., the Kullback-Leibler divergence, are subject to limitations regarding the support of the involved measures. A remedy consists of regularizing the $f$-divergence by a squared maximum mean discrepancy (MMD) associated with a characteristic kernel $K$. In this paper, we use the so-called kernel mean embedding to show that the corresponding regularization can be rewritten as the Moreau envelope of some function in the reproducing kernel Hilbert space associated with $K$. Then, we exploit well-known results on Moreau envelopes in Hilbert spaces to prove properties of the MMD-regularized $f$-divergences and, in particular, their gradients. Subsequently, we use our findings to analyze Wasserstein gradient flows of MMD-regularized $f$-divergences. Finally, we consider Wasserstein gradient flows starting from empirical measures and provide proof-of-the-concept numerical examples with Tsallis-$\alpha$ divergences.