秩一布尔张量分解与多线性多面体
Rank-one Boolean tensor factorization and the multilinear polytope
AI总结：
本文研究秩一布尔张量分解这一NP难问题，利用多线性多面体的面结构提出新的线性规划松弛，并给出确定性及高概率恢复保证，证明特定面可显著提升恢复性能。
摘要：
我们考虑一个NP难问题：在给定二进制张量的条件下，寻找与之最接近的秩一二进制张量，并将其称为秩一布尔张量分解（BTF）问题。该优化问题可用于从含噪观测中恢复一个被植入的秩一张量。我们将秩一BTF形式化为在一个高度结构化的多线性集合上最小化线性函数的问题。借助我们先前关于多线性多面体面结构的结果，我们为秩一BTF提出了新的线性规划松弛。随后，我们建立了确定性充分条件，在这些条件下，我们提出的线性规划能够恢复被植入的秩一张量。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。我们的理论结果以及数值模拟表明，多线性多面体的某些面显著改善了秩一BTF线性规划松弛的恢复性能。
英文摘要：
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.
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