A1469
Title: Frechet regression on the Bures-Wasserstein manifold
Authors: Cesar Augusto Uribe Meneses - Rice University (United States) [presenting]
Duc Toan Nguyen - Rice University (United States)
Abstract: Frechet regression, or conditional barycenters, is a flexible framework for modeling relationships between covariates (usually Euclidean) and response variables on general metric spaces, e.g., probability distributions or positive definite matrices. However, in contrast to classical barycenter problems, computing conditional counterparts in many non-Euclidean spaces remains an open challenge, as they yield non-convex optimization problems with an affine structure. The existence and computation of conditional barycenters is studied, specifically in the space of positive-definite matrices with the Bures-Wasserstein metric. A sufficient condition for the existence of a minimizer of the conditional barycenter problem that characterizes the regression range of extrapolation is provided. Moreover, the optimization landscape is further characterized, proving that under this condition, the objective is free of local maxima. Additionally, a projection-free and provably correct algorithm for the approximate computation of first-order stationary points is developed. Finally, a stochastic reformulation that enables the use of off-the-shelf stochastic Riemannian optimization methods for large-scale setups is provided. Numerical experiments validate the performance of the proposed methods on regression problems of real-world biological networks and on large-scale synthetic Diffusion Tensor Imaging problems.
