  The main challenge with the tensor completion problem is a fundamental
tension between computation power and the information-theoretic sample
complexity rate. Past approaches either achieve the information-theoretic rate
but lack practical algorithms to compute the corresponding solution, or have
polynomial-time algorithms that require an exponentially-larger number of
samples for low estimation error. This paper develops a novel tensor completion
algorithm that resolves this tension by achieving both provable convergence (in
numerical tolerance) in a linear number of oracle steps and the
information-theoretic rate. Our approach formulates tensor completion as a
convex optimization problem constrained using a gauge-based tensor norm, which
is defined in a way that allows the use of integer linear optimization to solve
linear separation problems over the unit-ball in this new norm. Adaptations
based on this insight are incorporated into a Frank-Wolfe variant to build our
algorithm. We show our algorithm scales-well using numerical experiments on
tensors with up to ten million entries.
