Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rank-one Boolean tensor factorization and the multilinear polytope

Alberto Del Pia, Aida Khajavirad

TL;DR

This work tackles the NP-hard problem of rank-one Boolean tensor factorization by reframing it as a linear objective over a structured multilinear set and introducing strong LP relaxations based on the multilinear polytope. The authors derive a simple standard LP and two stronger relaxations—the flower LP and the complete LP—by augmenting the convex hull with additional inequalities that capture multilinear relationships. They establish deterministic and high-probability recovery guarantees under semi-random corruption, showing that stronger LPs yield higher recovery thresholds (for equal-density ground truths, p < 1/2 for the original problem, p < r_w /(1+2 r_w) for the flower LP, and p < r_w /(2(1+ r_w)) for the standard LP under certain regimes). Numerical experiments corroborate the theory, illustrating that the complete LP significantly outperforms the others, with the flower LP offering noticeable improvements over the standard LP. Overall, the paper advances both theory and practice by connecting ML/MAP recovery, robust LP relaxations, and polyhedral facets to reliable planted-tensor recovery in a semi-random setting.

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.

Rank-one Boolean tensor factorization and the multilinear polytope

TL;DR

This work tackles the NP-hard problem of rank-one Boolean tensor factorization by reframing it as a linear objective over a structured multilinear set and introducing strong LP relaxations based on the multilinear polytope. The authors derive a simple standard LP and two stronger relaxations—the flower LP and the complete LP—by augmenting the convex hull with additional inequalities that capture multilinear relationships. They establish deterministic and high-probability recovery guarantees under semi-random corruption, showing that stronger LPs yield higher recovery thresholds (for equal-density ground truths, p < 1/2 for the original problem, p < r_w /(1+2 r_w) for the flower LP, and p < r_w /(2(1+ r_w)) for the standard LP under certain regimes). Numerical experiments corroborate the theory, illustrating that the complete LP significantly outperforms the others, with the flower LP offering noticeable improvements over the standard LP. Overall, the paper advances both theory and practice by connecting ML/MAP recovery, robust LP relaxations, and polyhedral facets to reliable planted-tensor recovery in a semi-random setting.

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.
Paper Structure (35 sections, 20 theorems, 91 equations, 2 figures)

This paper contains 35 sections, 20 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Recovery of a planted model
  4. Fully random versus semi-random models
  5. Our contribution
  6. Outline.
  7. Optimization and recovery
  8. Maximum likelihood and maximum a posteriori estimators
  9. Robustness of optimization algorithms
  10. LP relaxations for rank-one BTF
  11. The multilinear polytope and new LP relaxations
  12. Statements of recovery guarantees
  13. Recovery conditions for rank-one BTF
  14. Recovery conditions for the standard LP
  15. Recovery conditions for the flower LP
  16. ...and 20 more sections

Key Result

proposition 1

Consider the fully-random corruption model with $p < \frac{1}{2}$. Then solving Problem pr comb is equivalent to finding the ML estimator. Furthermore, if we assume that all ground truth rank-one binary tensors are equally likely a priori, solving Problem pr comb is equivalent to finding the MAP est

Figures (2)

  • Figure 1: Comparison of the recovery threshold for two LP relaxations of Rank-one BTF under the semi-random corruption model as given by Corollary \ref{['corLP1']} and Theorem \ref{['th LP2']}.
  • Figure 2: The empirical rate of recovery for LP relaxations of Rank-one BTF under the fully-random corruption model. The black curves in Figures \ref{['fig2a']} and \ref{['fig2b']} correspond to the recovery guarantees given by Corollary \ref{['corLP1']} and Theorem \ref{['th LP2']}, respectively.

Theorems & Definitions (23)

  • proposition 1
  • proposition 2
  • proposition 3
  • theorem 1: Necessary conditions for recovery
  • theorem 2: Sufficient conditions for recovery
  • remark 1
  • theorem 3
  • corollary 1
  • remark 2
  • theorem 4
  • ...and 13 more