Frozen $1$-RSB structure of the symmetric Ising perceptron

TL;DR

This paper proves that the symmetric Ising perceptron has a frozen 1-RSB structure with solutions forming isolated clusters, confirming a physics conjecture and explaining the model's solution landscape.

Contribution

It rigorously establishes the frozen 1-RSB structure of the symmetric Ising perceptron, including solution isolation and linear Hamming distance, using a novel comparison and concentration approach.

Findings

Solutions form isolated clusters with vanishing entropy density

Typical solutions are isolated with high probability

Hamming distance between solutions is linear in dimension

Abstract

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.