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. We show that this fundamental tensor problem is NP-hard for any tensor rank higher than one, and polynomial-time solvable in the rank-one case. We also propose a continuous relaxation and prove that it is tight for any rank. For low-enough ranks, the proposed continuous reformulation is amenable to low-complexity gradient-based optimization, and we propose a suite of gradient-based optimization algorithms drawing from projected gradient descent, Frank-Wolfe, or explicit parametrization of the relaxed constraints. We also show that our core results remain valid no matter what kind of polyadic tensor model is used to represent the tensor of interest, including Tucker, HOSVD/MLSVD, tensor train, or tensor ring. Next, we consider the class of problems that can be posed as special instances of the problem of interest. We show that this class includes the partition problem (and thus all NP-complete problems via polynomial-time transformation), integer least squares, integer linear programming, integer quadratic programming, sign retrieval (a special kind of mixed integer programming / restricted version of phase retrieval), and maximum likelihood decoding of parity check codes. We demonstrate promising experimental results on a number of hard problems, including state-of-art performance in decoding low density parity check codes and general parity check codes.

翻译：本文考虑通过秩分解指定的N阶张量的最小或最大元素查找问题。换言之，给定N组R维向量集合，需从每个集合中选取一个向量，使得所选向量哈达玛积之和最小化或最大化。我们证明：当张量秩大于1时，该基础张量问题为NP难问题；而秩为1时可在多项式时间内求解。同时提出连续松弛方法，并证明其对任意秩均具有紧致性。对于足够低的秩，所提连续重构模型可采用低复杂度梯度优化求解，我们据此设计了基于投影梯度下降、Frank-Wolfe算法及显式参数化松弛约束的梯度优化算法体系。研究进一步表明，无论采用何种多面张量模型（包括Tucker分解、HOSVD/MLSVD、张量列或张量环）表示目标张量，核心结论均保持成立。继而考虑可转化为本研究目标问题特例的问题类别，证明该类包含划分问题（因此可通过多项式时间变换包含所有NP完全问题）、整数最小二乘、整数线性规划、整数二次规划、符号检索（一种特殊混合整数规划/相位检索受限版本）及校验码最大似然译码。我们在多个困难问题上取得突破性实验结果，尤其在低密度校验码及通用校验码译码任务中达到当前最优性能。