Rank-one Boolean tensor factorization and the multilinear polytope

TL;DR

This paper addresses the NP-hard problem of rank-one Boolean tensor factorization by formulating it as a linear program over a multilinear polytope, providing new relaxations and recovery guarantees.

Contribution

It introduces novel linear programming relaxations for rank-one Boolean tensor factorization and establishes deterministic and probabilistic recovery conditions.

Findings

Linear programming relaxations can recover planted tensors under certain conditions.

Facets of the multilinear polytope enhance recovery performance.

Theoretical guarantees are supported by numerical simulations.

Abstract

We consider the NP-hard problem of finding the closest rank-one binary tensor to a given binary tensor, which we refer to as the rank-one Boolean tensor factorization (BTF) problem. This optimization problem can be used to recover a planted rank-one tensor from noisy observations. We formulate rank-one BTF as the problem of minimizing a linear function over a highly structured multilinear set. Leveraging on our prior results regarding the facial structure of multilinear polytopes, we propose novel linear programming relaxations for rank-one BTF. We then establish deterministic sufficient conditions under which our proposed linear programs recover a planted rank-one tensor. To analyze the effectiveness of these deterministic conditions, we consider a semi-random model for the noisy tensor, and obtain high probability recovery guarantees for the linear programs. Our theoretical results as well as numerical simulations indicate that certain facets of the multilinear polytope significantly improve the recovery properties of linear programming relaxations for rank-one BTF.