# Fr'echet Regression on the Bures-Wasserstein Manifold

arXiv: https://arxiv.org/abs/2604.03566 · HTML: https://mlsanity.com/paper/2604.03566/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 61 references that are in the map.
- First posted on arXiv 4 Apr 2026.

## Builds on (most cited first, 28 of 28)

- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Learning Multilingual Word Embeddings in Latent Metric Space: A Geometric Approach](https://mlsanity.com/paper/1808.08773/) (2018)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Network Regression with Graph Laplacians](https://mlsanity.com/paper/2109.02981/) (2021)
- [Faster First-Order Methods for Stochastic Non-Convex Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1811.08109/) (2018)
- [Efficient Output Kernel Learning for Multiple Tasks](https://mlsanity.com/paper/1511.05706/) (2015)
- [Manifold valued data analysis of samples of networks, with applications in corpus linguistics](https://mlsanity.com/paper/1902.08290/) (2019)
- [Conditional Wasserstein Barycenters and Interpolation/Extrapolation of Distributions](https://mlsanity.com/paper/2107.09218/) (2021)
- [Bures-Wasserstein Means of Graphs](https://mlsanity.com/paper/2305.19738/) (2023)
- [Bures-Wasserstein Barycenters and Low-Rank Matrix Recovery](https://mlsanity.com/paper/2210.14671/) (2022)
- [R-SPIDER: A Fast Riemannian Stochastic Optimization Algorithm with Curvature Independent Rate](https://mlsanity.com/paper/1811.04194/) (2018)
- [Non-parametric regression for networks](https://mlsanity.com/paper/2010.00050/) (2020)
- [Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting](https://mlsanity.com/paper/1810.05231/) (2018)
- [Geodesic Optimal Transport Regression](https://mlsanity.com/paper/2312.15376/) (2023)
- [Wasserstein F-tests for Fr'echet regression on Bures-Wasserstein manifolds](https://mlsanity.com/paper/2404.03878/) (2024)
- [Barycentric coordinate neighbourhoods in Riemannian manifolds](https://mlsanity.com/paper/1606.01585/) (2016)
- [Metric extrapolation in the Wasserstein space](https://mlsanity.com/paper/2407.10516/) (2024)
- [Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size](https://mlsanity.com/paper/2501.18164/) (2025)
- [Wasserstein Transfer Learning](https://mlsanity.com/paper/2505.17404/) (2025)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Neural Local Wasserstein Regression](https://mlsanity.com/paper/2511.10824/) (2025)
- [An Optimal Transport Approach for Network Regression](https://mlsanity.com/paper/2406.12204/) (2024)
- [Generalized Wasserstein Barycenters](https://mlsanity.com/paper/2411.06838/) (2024)

## Abstract (arXiv)

Fr'echet 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.