Structured low-rank matrix learning: algorithms and applications

TL;DR

This paper introduces a novel factorization method for low-rank matrix learning constrained within a linear subspace, utilizing Riemannian optimization techniques to improve computational efficiency across various applications.

Contribution

It proposes a new factorization decoupling low-rank and structural constraints and formulates the problem on the Riemannian spectrahedron manifold for efficient optimization.

Findings

Effective in matrix completion tasks

Demonstrates robustness in noisy settings

Applicable to multi-task learning

Abstract

We consider the problem of learning a low-rank matrix, constrained to lie in a linear subspace, and introduce a novel factorization for modeling such matrices. A salient feature of the proposed factorization scheme is it decouples the low-rank and the structural constraints onto separate factors. We formulate the optimization problem on the Riemannian spectrahedron manifold, where the Riemannian framework allows to develop computationally efficient conjugate gradient and trust-region algorithms. Experiments on problems such as standard/robust/non-negative matrix completion, Hankel matrix learning and multi-task learning demonstrate the efficacy of our approach. A shorter version of this work has been published in ICML'18.