Fréchet Regression on the Bures-Wasserstein Manifold

2026-04-03

AI Summary

This work addresses the challenge of computing conditional Fréchet means—i.e., Fréchet regression—on the Bures-Wasserstein manifold, which entails non-convex optimization lacking both theoretical guarantees and efficient algorithms. Focusing on the space of positive definite matrices, the authors establish sufficient conditions for the existence of conditional barycenters and prove that the associated objective function possesses no local maxima. Building on these theoretical insights, they propose a projection-free first-order Riemannian optimization algorithm with provable convergence and further extend it to a stochastic Riemannian framework suitable for large-scale settings. Empirical validation on real biological networks and large synthetic diffusion tensor imaging datasets demonstrates the method’s effectiveness and scalability.

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.

Problem

• Fréchet regression
• conditional barycenters
• Bures-Wasserstein manifold
• non-convex optimization
• positive-definite matrices

Innovation

• Fréchet regression
• Bures-Wasserstein manifold
• conditional barycenter
• Riemannian optimization
• positive-definite matrices