An Extended Frank-Wolfe Method with Application to Low-Rank Matrix Completion

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. We present computational guarantees for the method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply our method to the low-rank matrix completion problem, where low-dimensional faces correspond to low-rank solutions. We present computational results for large-scale low-rank matrix completion problems that demonstrate significant speed-ups in computing low-rank near-optimal solutions on both artificial and real datasets. This is joint work with Paul Grigas (ORC) and Rahul Mazumder (Sloan).