Author:
Erba Vittorio,Krzakala Florent,Pérez Ortiz Rodrigo,Zdeborová Lenka
Abstract
Abstract
We study the maximum-average submatrix problem, wherein given an N × N matrix J, one needs to find the k × k submatrix with the largest average number of entries. We investigate 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 magnetisation. We analytically characterise the phase diagram of the model as a function of the submatrix average and the size of the submatrix k in the limit
N
→
∞
. 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 (1-RSB) and full-step RSB. In the limit of m → 0, we find a simpler phase diagram featuring a frozen 1-RSB phase, where the Gibbs measure comprises exponentially many pure states each with zero entropy. We discover an interesting phenomenon, reminiscent of the phenomenology of the binary perceptron: there are efficient algorithms that provably work in the frozen 1-RSB phase.
Subject
Statistics, Probability and Uncertainty,Statistics and Probability,Statistical and Nonlinear Physics