On a Neural Implementation of Brenier's Polar Factorization
AuthorsNina Vesseron, Marco Cuturi Cameto
AuthorsNina Vesseron, Marco Cuturi Cameto
In 1991, Brenier proved a theorem that generalizes the polar decomposition for square matrices -- factored as PSD unitary -- to any vector field . The theorem, known as the polar factorization theorem, states that any field can be recovered as the composition of the gradient of a convex function with a measure-preserving map , namely . We propose a practical implementation of this far-reaching theoretical result, and explore possible uses within machine learning. The theorem is closely related to optimal transport (OT) theory, and we borrow from recent advances in the field of neural optimal transport to parameterize the potential as an input convex neural network. The map can be either evaluated pointwise using , the convex conjugate of , through the identity , or learned as an auxiliary network. Because is, in general, not injective, we consider the additional task of estimating the ill-posed inverse map that can approximate the pre-image measure using a stochastic generator. We illustrate possible applications of Brenier's polar factorization to non-convex optimization problems, as well as sampling of densities that are not log-concave.
December 3, 2024research area Methods and Algorithmsconference NeurIPS
Given a source and a target probability measure supported on , the Monge problem aims for the most efficient way to map one distribution to the other. This efficiency is quantified by defining a cost function between source and target data. Such a cost is often set by default in the machine learning literature to the squared-Euclidean distance, . The benefits of using elastic costs, defined...
June 20, 2023research area Methods and Algorithmsconference ICML
Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the "best" map to morph a continuous measure in into another must be the gradient of a convex function. To exploit...