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Fréchet Regression on the Bures-Wasserstein Manifold

Duc Toan Nguyen, César A. Uribe

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.

Fréchet Regression on the Bures-Wasserstein Manifold

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.

Paper Structure

This paper contains 20 sections, 14 theorems, 47 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related Work
  4. Fréchet Regression.
  5. Existence and uniqueness of Bures-Wasserstein conditional barycenters
  6. Preliminaries on the Bures-Wasserstein Geometry
  7. Existence of Bures-Wasserstein conditional barycenters
  8. Uniqueness of Bures-Wasserstein conditional barycenters
  9. Riemannian gradient-based methods
  10. Bures-Wasserstein Conditional Barycenter Gradient Method
  11. Pairwise Riemannian Stochastic Gradient Descent methods for large-scale Bures-Wasserstein conditional barycenters
  12. Numerical Analysis
  13. Network regression on an Ant Social Organization
  14. Simulated Diffusion Tensors Imaging (DTI)
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 3.2

Let $\Sigma_1,\ldots,\Sigma_n\in\mathbb{S}_{++}^d$ and $\lambda_1,\ldots,\lambda_n\in\mathbb R$ with $\sum_{k=1}^n \lambda_k=1$. If the Spectral Dominance of Positive Weights condition holds, i.e., then Problem eqn:frechet-obj admits a solution. $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: Ranges of interpolation (all positive weights) and extrapolation (possibly negative weights) for Fréchet regression with $X_i \in \mathbb{R}$ and $Y_i \in \Omega$, for $i \in \lbrace 1,...,8 \rbrace$
  • Figure 2: Interpolation (green) and extrapolation (red) between two (black) ellipsoids (3x3 SPD matrices) under BW metric
  • Figure 3: Results on the Ant social organization network dataset
  • Figure 4: Objective Values over 100 iterations of Algorithm \ref{['alg:rsgd']} from regression process of four tensors
  • Figure 5: Helix visualization ground-truth and predicted tensors
  • ...and 3 more figures

Theorems & Definitions (21)

  • Example 3.1
  • Theorem 3.2: Spectral Dominance of Positive Weights
  • Proposition 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Proposition 3.7: Unique existence of minimizer
  • Remark 3.8
  • Proposition 4.1
  • Lemma 4.2
  • ...and 11 more