We consider the NP-hard problem of approximating a tensor with binary entries by a rank-one tensor, referred to as rank-one Boolean tensor factorization problem. We formulate this problem, in an extended space of variables, 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 Boolean tensor factorization. To analyze the performance of the proposed linear programs, we consider a random corruption model for the input tensor. We first consider the original NP-hard problem and establish information theoretic limits under the random model. Next, we obtain sufficient conditions under which the proposed linear programming relaxations recover the ground truth with high probability. 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 Boolean tensor factorization.
翻译:我们考虑NP硬性问题,即用一等-一的 10 来接近带有二进制的单进器,称为级- 10, 称为 级- 单波级的 Excrimination 问题。 我们将这个问题在变数的扩大空间中表述为在高度结构化的多线性一组中最大限度地减少线性函数的问题。 我们利用我们以前关于多线性多面形的面部结构的结果, 提议对级- 一波列an Exor 参数化进行新的线性编程松动。 为了分析拟议的线性程序, 我们考虑对输入点进行随机的腐败模式。 我们首先考虑原始的 NP- 硬性问题, 并在随机模型中建立信息理论性限值。 其次, 我们获得了充分的条件, 拟议的线性编程松动后极有可能恢复地面的真相。 我们的理论结果以及数字模拟表明, 多线性多线性多线性多线性多功能化的有些方面极大地改进了级- 级- 线性编程放松编程性能的恢复特性。