arXiv subjects

Emile Pierret

Publications and source records attributed to Emile Pierret.

5 recordsLinked to original sources

arXiv2026

Tessellations of Semi-Discrete Flow Matching

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

cs.LG↗

arXiv2025

Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions

Used as priors for Bayesian inverse problems, diffusion models have recently attracted considerable attention in the literature. Their flexibility and high variance enable them to generate multiple solutions for a given task, such as inpainting, super-resolution, and deblurring. However, there is still a lack of understanding about how accurately these conditional diffusion algorithms perform conditional sampling. In this article, we investigate the errors induced by these models when applied to a Gaussian data distribution for which the score function is exactly known. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems via conditional sampling and the practical distributions generated by conditional diffusion models. This is done by characterizing all the involved iterative Gaussian processes and by computing the exact Wasserstein distance between the distributions of the diffusion model samplers and the ideal conditional distribution associated with the inverse problem. Our findings allow for the comparison of two major algorithms from the literature, Deep Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Models ($\Pi$GDM), and the introduction of the new paradigm Conditional Gaussian Diffusion Models (CGDM) that is shown to be more accurate for Gaussian data distributions.

cs.LG↗

arXiv2024

Stochastic super-resolution for Gaussian microtextures

Super-Resolution (SR) is the problem that consists in reconstructing images that have been degraded by a zoom-out operator. This is an ill-posed problem that does not have a unique solution, and numerical approaches rely on a prior on high-resolution images. While optimization-based methods are generally deterministic, with the rise of image generative models more and more interest has been given to stochastic SR, that is, sampling among all possible SR images associated with a given low-resolution input. In this paper, we construct an efficient, stable and provably exact sampler for the stochastic SR of Gaussian microtextures. Even though our approach is limited regarding the scope of images it encompasses, our algorithm is competitive with deep learning state-of-the-art methods both in terms of perceptual metric and execution time when applied to microtextures. The framework of Gaussian microtextures also allows us to rigorously discuss the limitations of various reconstruction metrics to evaluate the efficiency of SR routines.

eess.IV↗

arXiv2024

Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors

Diffusion or score-based models recently showed high performance in image generation. They rely on a forward and a backward stochastic differential equations (SDE). The sampling of a data distribution is achieved by numerically solving the backward SDE or its associated flow ODE. Studying the convergence of these models necessitates to control four different types of error: the initialization error, the truncation error, the discretization error and the score approximation. In this paper, we theoretically study the behavior of diffusion models and their numerical implementation when the data distribution is Gaussian. Our first contribution is to derive the analytical solutions of the backward SDE and the probability flow ODE and to prove that these solutions and their discretizations are all Gaussian processes. Our second contribution is to compute the exact Wasserstein errors between the target and the numerically sampled distributions for any numerical scheme. This allows us to monitor convergence directly in the data space, while experimental works limit their empirical analysis to Inception features. An implementation of our code is available online.

cs.LG↗

arXiv2023

Stochastic Super-Resolution For Gaussian Textures

Super-resolution (SR) is an ill-posed inverse problem which consists in proposing high-resolution images consistent with a given low-resolution one. While most SR algorithms are deterministic, stochastic SR deals with designing a stochastic sampler generating any realistic SR solution. The goal of this paper is to show that stochastic SR is a well-posed and solvable problem when restricting to Gaussian stationary textures. Using Gaussian conditional sampling and exploiting the stationarity assumption, we propose an efficient algorithm based on fast Fourier transform. We also demonstrate the practical relevance of the approach for SR with a reference image. Although limited to stationary microtextures, our approach compares favorably in terms of speed and visual quality to some state of the art methods designed for a larger class of images.

eess.IV↗