  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.
