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：本文引入并研究了最大平均子张量（$p$-MAS）问题：对于给定的$p$阶随机张量（尺寸为$N$），寻找其$p$阶尺寸为$k$的子张量，使得其元素和达到最大。我们的研究动机源于近期对该问题矩阵情形（$p=2$）的工作，其中多个平衡与非平衡性质在渐近区域$1 \ll k \ll N$下得到了解析刻画，并观察到一个令人困惑的现象——聚类平衡相与能在该相中生成子矩阵的高效算法共存。本文将先前关于矩阵情形的平衡性质与算法性质的结果推广至张量情形。我们证明张量情形具有与矩阵情形相似的平衡相图，且所考察算法呈现出整体相似的现象学特征。此外，我们利用重叠间隙性质与Franz-Parisi分析考察了非平衡景观特性，并讨论了其对平均情形算法困难性的影响或无影响。