Tensor Completion via Integer Optimization
This work addresses a fundamental computational bottleneck in tensor completion for applications like data imputation and machine learning, offering a practical solution with theoretical guarantees.
The paper tackles the tensor completion problem by developing a novel algorithm that achieves both provable convergence in linear oracle steps and the information-theoretic sample complexity rate, resolving the trade-off between computation and sample efficiency, with numerical experiments scaling to tensors of up to ten million entries.
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.