Emile Pierret

Emile Pierret, Bruno Galerne

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.

Emile Pierret, Johannes Hertrich, Samuel Hurault et al.

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.

Emile Pierret, Bruno Galerne

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, several unresolved questions remain about how well they perform. In this article, we investigate the accuracy of these models when applied to a Gaussian data distribution for deblurring. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems and their resolution obtained using diffusion models by computing the exact Wasserstein distance between the distribution of the diffusion model sampler and the ideal distribution of solutions to the inverse problem. Our findings allow for the comparison of different algorithms from the literature.