Statistical mechanics of the maximum-average submatrix problem

TL;DR

This paper analyzes the maximum-average submatrix problem in large random matrices by mapping it to a spin-glass model, revealing a complex phase diagram with multiple symmetry-breaking phases and efficient algorithms in the frozen phase.

Contribution

It provides an analytical characterization of the phase diagram for the maximum-average submatrix problem using spin-glass theory, including novel insights into algorithmic performance in the frozen phase.

Findings

Rich phase diagram with multiple RSB phases

Existence of efficient algorithms in the frozen 1-RSB phase

Analytical mapping to a spin-glass model

Abstract

We study the maximum-average submatrix problem, in which given an $N \times N$ matrix $J$ one needs to find the $k \times k$ submatrix with the largest average of entries. We study the problem for random matrices $J$ whose entries are i.i.d. random variables by mapping it to a variant of the Sherrington-Kirkpatrick spin-glass model at fixed magnetization. We characterize analytically the phase diagram of the model as a function of the submatrix average and the size of the submatrix $k$ in the limit $N\to\infty$. We consider submatrices of size $k = m N$ with $0 < m < 1$. We find a rich phase diagram, including dynamical, static one-step replica symmetry breaking and full-step replica symmetry breaking. In the limit of $m \to 0$, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure is composed of exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there exist efficient algorithms that provably work in the frozen 1-RSB phase.