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：我们考虑的问题是,要找到通过其等级分解指定的单价最低或最大的一元项。 以不同的方式,我们得到一亿元的立方矢量,我们希望从每组中选择一个矢量,这样选定的矢量的哈达马德产品总和可以最小化或最大化。 这是一个根本性的抗拉问题,在嵌入类似搜索、推荐系统、图解挖掘、多变概率和统计等诸多应用中,这是一个根本性的抗拉问题。 我们表明,对于任何超过一个的数级的超高级体,这种离散优化问题都很难解决,但也提供了类似的连续重订问题,这可以符合纪律的非凝固式优化。 我们建议了一套基于梯度的近似算法,其初步实验的性能似乎很有希望。</s>