The simplest model of jamming

TL;DR

This paper investigates the perceptron model as a simple representation of jamming phenomena, revealing critical behaviors and universality classes in different optimization regimes, with implications for understanding complex systems.

Contribution

It characterizes the critical jamming phase in perceptrons, linking it to high-dimensional sphere packings and highlighting the role of replica symmetry breaking.

Findings

Jamming in convex regimes is hypostatic and non-critical.

Non-convex regimes exhibit isostatic and critical jamming.

Exponents match those in high-dimensional hard sphere systems.

Abstract

We study a well known machine learning model -the perceptron- as a simple model of jamming of hard objects. We exhibit two regimes: 1) a convex optimisation regime where jamming is hypostatic and non-critical. 2) a non convex optimisation regime where jamming is isostatic and critical. We characterise the critical jamming phase through exponents describing the distributions law of forces and gaps. Surprisingly we find that these exponents coincide with the corresponding ones recently computed in high dimensional hard spheres. In addition, modifying the perceptron to a random linear programming problem, we show that isostaticity is not a sufficient condition for singular force and gap distributions. For that, fragmentation of the space of solutions (replica symmetry breaking) appears to be a crucial ingredient. We hypothesise universality for a large class of non-convex constrained satisfaction problems with continuous variables.