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Julien Guerin (Ceremade & Mokaplan) interviendra sur Analysis of the sliced Wasserstein distances : comparison and barycenter properties.
Abstract :
Nearly twenty years ago, Marc Bernot, motivated by problems arising in image processing, introduced the so-called sliced Wasserstein distance (SW_p). It is defined as the average, over all directions in R^d, of the one-dimensional Wasserstein distances between the projections of the measures onto the given direction. Since Wasserstein distances are particularly easy to compute in one dimension, Bernot’s idea has enjoyed considerable success in applied domains such as machine learning and image processing. Somewhat paradoxically, it is only very recently that these distances have been the object of mathematically rigorous analysis. We will begin by addressing questions related to the optimal comparison between the sliced distance and the classical Wasserstein distance following the work of G. Carlier, A. Figalli, Q. Mérigot and Y. Wang. In a second part, we will discuss the notion of barycenters in the sense of SW2, already widely used in the literature much like classical Wasserstein barycenters, and we will state their existence, uniqueness, and quantitative stability properties.
Louis Tocquec (LMO & Mokaplan) interviendra sur On convergence rates of regularized unbalanced optimal transport: the discrete case.
Abstract :
Unbalanced optimal transport (UOT) is a natural extension of optimal transport (OT) allowing comparison between measures of different masses. It arises naturally in machine learning by offering a robustness against outliers. The aim of this work is to provide convergence rates of the regularized transport plans and potentials towards their original solution when both measures are weighted sums of Dirac masses.
Johannes Hertrich (Ceremade & Mokaplan) interviendra sur On the Relation between Rectified Flows and Optimal Transport.
Abstract :
We investigate the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than those previously acknowledged. In particular, we present several counter-examples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.
