B0295
Title: Optimal transport based denoising
Authors: Nicolas Garcia Trillos - University of Wisconsin Madison (United States) [presenting]
Abstract: In the standard formulation of the classical denoising problem, one is given a probabilistic model of latent variables and observations, and the goal is to construct a map to recover latent variables from observations. While there are many classical approaches for building denoising estimators, including the posterior mean, these estimators are often unable to adapt to the geometric structure of the prior distribution of latent variables. A new perspective is taken on the denoising problem inspired by optimal transport (OT) theory. New estimands are proposed that are motivated by theoretical considerations, first assuming that the prior distribution is known. It is rigorously proven that, under general assumptions, these estimands are mathematically well-defined and are closely connected to solutions to Monge OT problems. After this, approaches are explored for recovering defined estimands in realistic settings and in particular prove that, when the likelihood model is an exponential family, and assuming additional identifiability of the model, the estimands can be recovered solely from the information of the marginal distribution of observations after solving a linear relaxation of the original problem that is reminiscent to standard multi-marginal OT. The family of OT-like relaxations is of interest in its own right and the denoising problem suggests alternative numerical methods inspired by the rich literature on computational OT.
