An Extended Frank-Wolfe Method with "In-Face" Directions, and its Application to Low-Rank Matrix Completion

TL;DR

This paper introduces an extended Frank-Wolfe algorithm with 'in-face' directions that efficiently produces low-rank solutions for matrix completion, offering faster convergence to nearly-optimal low-rank matrices.

Contribution

The paper develops a novel 'in-face' direction approach within the Frank-Wolfe framework, enhancing low-rank solution generation for matrix completion problems.

Findings

Significant speed-ups in computing low-rank solutions.

Effective in producing nearly-optimal low-rank matrices.

Demonstrated on both artificial and real datasets.

Abstract

Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face" directions at each iteration, as well as through new choice rules for selecting between in-face and ``regular" Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away-steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological approach at producing nearly-optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speed-ups in computing very low-rank nearly-optimal solutions as compared to either the Frank-Wolfe method or its traditional away-step variant.