Fréchet Regression on the Bures-Wasserstein Manifold

Duc Toan Nguyen, Cesar A Uribe

Track: tiny paper (up to 4 pages)

Keywords: Fréchet regression, extrapolation, Bures-Wasserstein manifold, Riemannian optimization

TL;DR: We establish existence guarantees and scalable projection-free algorithms for Fréchet regression (conditional Barycenters) on the SPD manifold under the Bures–Wasserstein metric.

Abstract:

Fréchet 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. In this work, we study the existence and computation of conditional barycenters, specifically in the space of positive-definite matrices with the Bures-Wasserstein metric. We provide a sufficient condition for the existence of a minimizer of the conditional barycenter problem that characterizes the regression range of extrapolation. Moreover, we further characterize the optimization landscape, proving that under this condition, the objective is free of local maxima. Additionally, we develop a projection-free and provably correct algorithm for the approximate computation of first-order stationary points. Finally, we provide a stochastic reformulation that enables the use of off-the-shelf stochastic Riemannian optimization methods for large-scale setups.
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.

Anonymization: This submission has been anonymized for double-blind review via the removal of identifying information such as names, affiliations, and identifying URLs.

Submission Number: 38