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.

翻译：我们考虑NP难问题：在给定布尔张量中寻找最接近的秩一布尔张量，即秩一布尔张量分解（BTF）问题。该优化问题可用于从含噪观测中恢复植入的秩一张量。我们将秩一BTF形式化为在高度结构化的多线性集上最小化线性函数的问题。基于先前关于多线性多面体面结构的研究结果，我们提出了秩一BTF的新型线性规划松弛方法。随后建立了确定性充分条件，在此条件下所提出的线性规划能够恢复植入的秩一张量。为分析这些确定性条件的有效性，我们考虑含噪张量的半随机模型，并获得了线性规划的高概率恢复保证。理论结果与数值模拟均表明，多线性多面体的某些面能显著改善秩一BTF线性规划松弛的恢复性能。