Tensor Completion via Integer Optimization
Abstract
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.
Cite
@article{arxiv.2402.05141,
title = {Tensor Completion via Integer Optimization},
author = {Xin Chen and Sukanya Kudva and Yongzheng Dai and Anil Aswani and Chen Chen},
journal= {arXiv preprint arXiv:2402.05141},
year = {2024}
}