The maximum-average subtensor problem: equilibrium and out-of-equilibrium properties

TL;DR

This paper extends the analysis of the maximum-average submatrix problem from matrices to tensors, revealing similar equilibrium phases and exploring out-of-equilibrium properties to understand algorithmic hardness.

Contribution

It generalizes previous matrix results to tensors, characterizing equilibrium phases and analyzing landscape properties affecting algorithmic complexity.

Findings

Tensor case has similar equilibrium phase diagram as matrix case

Algorithms exhibit similar phenomenology in tensor setting

Out-of-equilibrium analysis suggests implications for algorithmic hardness

Abstract

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.