In this paper we introduce and study the Maximum-Average Subtensor ($p$-MAS) problem, in which one wants to find a subtensor of size $k$ of a given random tensor of size $N$, both of order $p$, with maximum sum of entries. We are motivated by recent work on the matrix case of the problem in which several equilibrium and non-equilibrium properties have been characterized analytically in the asymptotic regime $1 \ll k \ll N$, and a puzzling phenomenon was observed involving the coexistence of a clustered equilibrium phase and an efficient algorithm which produces submatrices in this phase. Here we extend previous results on equilibrium and algorithmic properties for the matrix case to the tensor case. We show that the tensor case has a similar equilibrium phase diagram as the matrix case, and an overall similar phenomenology for the considered algorithms. Additionally, we consider out-of-equilibrium landscape properties using Overlap Gap Properties and Franz-Parisi analysis, and discuss the implications or lack-thereof for average-case algorithmic hardness.

翻译：本文提出并研究了最大平均子张量（p-MAS）问题，其目标是在给定大小为N的随机张量（两者均为p阶）中寻找大小为k的子张量，使得其元素和达到最大。该研究受到近期矩阵版本问题工作的启发，其中在渐近区域1≪k≪N内，多个平衡与非平衡特性已通过解析方法得到表征，并观察到一个涉及聚类平衡相与在该相中生成子矩阵的高效算法共存的现象。本文将先前关于矩阵版本的平衡特性和算法性质的结果扩展至张量情形。研究表明，张量情形具有与矩阵情形相似的平衡相图，且所考察算法呈现整体相似的现象学特征。此外，本文通过重叠间隙性质和Franz-Parisi分析探讨了非平衡景观特性，并讨论了其对平均情形算法复杂性的影响或缺失意义。