We consider the problem of finding the smallest or largest entry of a tensor of order $N$ that is specified via its rank decomposition. Stated in a different way, we are given $N$ sets of $R$-dimensional vectors and we wish to select one vector from each set such that the sum of the Hadamard product of the selected vectors is minimized or maximized. This is a fundamental tensor problem with numerous applications in embedding similarity search, recommender systems, graph mining, multivariate probability, and statistics. We show that this discrete optimization problem is NP-hard for any tensor rank higher than one, but also provide an equivalent continuous problem reformulation which is amenable to disciplined non-convex optimization. We propose a suite of gradient-based approximation algorithms whose performance in preliminary experiments appears to be promising.

翻译：我们考虑从秩分解形式给定的 $N$ 阶张量中寻找其最小或最大元素的问题。换而言之，给定 $N$ 组 $R$ 维向量，我们需从每组中选取一个向量，使得所选向量的哈达玛积之和最小化或最大化。这是一个基础的张量问题，在嵌入相似性搜索、推荐系统、图挖掘、多元概率与统计学中具有广泛应用。我们证明该离散优化问题在张量秩大于 1 时是 NP 难的，同时给出一个等价的连续问题重构形式，该形式适用于有纪律的非凸优化。我们提出一系列基于梯度的近似算法，初步实验结果表明其性能具有潜力。