Nonnegative Tensor Completion via Integer Optimization

TL;DR

This paper introduces a novel algorithm for nonnegative tensor completion that achieves the optimal information-theoretic sample complexity rate, utilizing integer optimization and a new tensor norm, with demonstrated scalability on large datasets.

Contribution

The paper presents a new tensor completion algorithm for nonnegative tensors that converges linearly and achieves the theoretical sample complexity rate, using integer optimization and a novel tensor norm.

Findings

Algorithm converges linearly in numerical tolerance.

Achieves the information-theoretic sample complexity rate.

Scalable to tensors with up to 100 million entries.

Abstract

Unlike matrix completion, tensor completion does not have an algorithm that is known to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative tensors. We prove that our algorithm converges in a linear (in numerical tolerance) number of oracle steps, while achieving the information-theoretic rate. Our approach is to define a new norm for nonnegative tensors using the gauge of a particular 0-1 polytope; integer linear programming can, in turn, be used to solve linear separation problems over this polytope. We combine this insight with a variant of the Frank-Wolfe algorithm to construct our numerical algorithm, and we demonstrate its effectiveness and scalability through computational experiments using a laptop on tensors with up to one-hundred million entries.