Fréchet Regression on the Bures-Wasserstein Manifold

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.

Adjusted Shuffling SARAH: Advancing Complexity Analysis via Dynamic Gradient Weighting

In this paper, we propose Adjusted Shuffling SARAH, a novel algorithm that integrates shuffling techniques with the well-known variance-reduced algorithm SARAH while dynamically adjusting the stochastic gradient weights in each update to enhance exploration. Our method achieves the best-known gradient complexity for shuffling variance reduction methods in a strongly convex setting. This result applies to any shuffling technique, which narrows the gap in the complexity analysis of variance reduction methods between uniform sampling and shuffling data. Furthermore, we introduce Inexact Adjusted Reshuffling SARAH, an inexact variant of Adjusted Shuffling SARAH that eliminates the need for full-batch gradient computations. This algorithm retains the same linear convergence rate as Adjusted Shuffling SARAH while showing an advantage in total complexity when the sample size is very large.

Towards Tuning-Free Minimum-Volume Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) is a versatile and powerful tool for discovering latent structures in data matrices, with many variations proposed in the literature. Recently, Leplat et al.\@ (2019) introduced a minimum-volume NMF for the identifiable recovery of rank-deficient matrices in the presence of noise. The performance of their formulation, however, requires the selection of a tuning parameter whose optimal value depends on the unknown noise level. In this work, we propose an alternative formulation of minimum-volume NMF inspired by the square-root lasso and its tuning-free properties. Our formulation also requires the selection of a tuning parameter, but its optimal value does not depend on the noise level. To fit our NMF model, we propose a majorization-minimization (MM) algorithm that comes with global convergence guarantees. We show empirically that the optimal choice of our tuning parameter is insensitive to the noise level in the data.