Proliferation of non-linear excitations in the piecewise-linear perceptron

TL;DR

This paper studies the local minima of a non-convex spherical perceptron with a piecewise linear cost function, revealing complex non-linear excitations and pseudogaps that relate to jammed sphere packings, extending understanding of such landscapes.

Contribution

It demonstrates the emergence of multiple non-linear excitations and pseudogaps in the perceptron energy landscape, generalizing previous linear models to more complex non-linear cases.

Findings

Local minima are critical and marginally stable.

Four pseudogaps and power-law distributions are identified.

Local minima are approximately isostatic in the non-convex phase.

Abstract

We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.