An Extended Frank-Wolfe Method with "In-Face" Directions, and its Application to Low-Rank Matrix Completion
This work addresses the matrix completion problem for applications like recommendation systems, but it is incremental as it builds on the Frank-Wolfe method.
The authors tackled the low-rank matrix completion problem by extending the Frank-Wolfe method with 'in-face' directions to induce near-optimal solutions on low-dimensional faces, resulting in significant speed-ups in computing very low-rank nearly-optimal solutions compared to existing methods.
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.