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Motivated principally by the low-rank matrix completion problem, we presentan extension of the Frank-Wolfe method that is designed to induce near-optimalsolutions on low-dimensional faces of the feasible region. This is accomplishedby a new approach to generating ``in-face" directions at each iteration, aswell as through new choice rules for selecting between in-face and ``regular"Frank-Wolfe steps. Our framework for generating in-face directions generalizesthe notion of away-steps introduced by Wolfe. In particular, the in-facedirections always keep the next iterate within the minimal face containing thecurrent iterate. We present computational guarantees for the new method thattrade off efficiency in computing near-optimal solutions with upper bounds onthe dimension of minimal faces of iterates. We apply the new method to thematrix completion problem, where low-dimensional faces correspond to low-rankmatrices. We present computational results that demonstrate the effectivenessof our methodological approach at producing nearly-optimal solutions of verylow rank. On both artificial and real datasets, we demonstrate significantspeed-ups in computing very low-rank nearly-optimal solutions as compared toeither the Frank-Wolfe method or its traditional away-step variant.

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