非负张量完备化的最优化方法
Nonnegative Tensor Completion via Integer Optimization
摘要：
与矩阵完备化不同，张量完备化没有已知的算法来实现信息论样本复杂度。本文对非负张量完备化的特殊情况提出了一种新的算法。我们证明了我们的算法收敛在一个线性（数值公差）的预言步骤数，同时实现信息理论的速度。我们的方法是使用一个特定的0-1多面体的规范定义一个新的规范的非负张量;整数线性规划，反过来，可以用来解决这个多面体的线性分离问题。我们结合这一见解与弗兰克-沃尔夫算法的一个变种，构建我们的数值算法，我们证明了它的有效性和可扩展性，通过计算实验使用笔记本电脑上的张量高达一亿个条目。
英文摘要：
Unlike matrix completion, tensor completion does not have an algorithm that is known to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative tensors. We prove that our algorithm converges in a linear (in numerical tolerance) number of oracle steps, while achieving the information-theoretic rate. Our approach is to define a new norm for nonnegative tensors using the gauge of a particular 0-1 polytope; integer linear programming can, in turn, be used to solve linear separation problems over this polytope. We combine this insight with a variant of the Frank-Wolfe algorithm to construct our numerical algorithm, and we demonstrate its effectiveness and scalability through computational experiments using a laptop on tensors with up to one-hundred million entries.
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