Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

TL;DR

This paper proves the contiguity conjecture and the lognormal limit for the symmetric binary perceptron, confirming several longstanding conjectures and advancing understanding of its phase transitions and structure.

Contribution

It establishes the lognormal distribution of the partition function and proves key conjectures in the symmetric perceptron model unconditionally, using a novel dense graph conditioning method.

Findings

Partition function converges to a lognormal distribution.

Proves the contiguity conjecture between planted and unplanted models.

Establishes the sharp threshold and frozen 1-RSB conjectures.

Abstract

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-M\'ezard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognormal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.