Riemannian adaptive stochastic gradient algorithms on matrix manifolds

TL;DR

This paper introduces new Riemannian stochastic gradient algorithms tailored for matrix manifolds, leveraging subspace structures to improve optimization, with proven convergence and demonstrated effectiveness in machine learning tasks.

Contribution

It develops novel adaptive stochastic gradient algorithms on Riemannian matrix manifolds that preserve subspace structures, with theoretical convergence guarantees.

Findings

Algorithms are provably convergent with rate O(log(T)/sqrt(T)).

Experiments show improved performance on various applications.

Preserving subspace structures enhances optimization on matrix manifolds.

Abstract

Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the intrinsic non-linear structure of the underlying manifold and the absence of a canonical coordinate system. In machine learning applications, however, most manifolds of interest are represented as matrices with notions of row and column subspaces. In addition, the implicit manifold-related constraints may also lie on such subspaces. For example, the Grassmann manifold is the set of column subspaces. To this end, such a rich structure should not be lost by transforming matrices to just a stack of vectors while developing optimization algorithms on manifolds. We propose novel stochastic gradient algorithms for problems on Riemannian matrix manifolds by adapting the row and column subspaces of gradients. Our algorithms are provably convergent and they achieve the convergence rate of order $\mathcal{O}(\log (T)/\sqrt{T})$, where $T$ is the number of iterations. Our experiments illustrate the efficacy of the proposed algorithms on several applications.