On a class of geodesically convex optimization problems solved via Euclidean MM methods

TL;DR

This paper introduces efficient Euclidean Majorization-Minorization algorithms for geodesically convex problems that are expressed as differences of Euclidean convex functions, applicable in statistics and machine learning.

Contribution

It develops algorithms exploiting g-convexity and split structures to ensure global optimality while avoiding expensive Riemannian computations.

Findings

Algorithms guarantee global optimality with iteration complexity bounds.

Bypass the need for exponential maps and parallel transport in Riemannian optimization.

Applied to matrix scaling, covariance estimation, and inequalities in machine learning.

Abstract

We study geodesically convex (g-convex) problems that can be written as a difference of Euclidean convex functions. This structure arises in several optimization problems in statistics and machine learning, e.g., for matrix scaling, M-estimators for covariances, and Brascamp-Lieb inequalities. Our work offers efficient algorithms that on the one hand exploit g-convexity to ensure global optimality along with guarantees on iteration complexity. On the other hand, the split structure permits us to develop Euclidean Majorization-Minorization algorithms that help us bypass the need to compute expensive Riemannian operations such as exponential maps and parallel transport. We illustrate our results by specializing them to a few concrete optimization problems that have been previously studied in the machine learning literature. Ultimately, we hope our work helps motivate the broader search for mixed Euclidean-Riemannian optimization algorithms