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

by Antonio Sclocchi, Pierfrancesco Urbani

Submission summary

As Contributors: Antonio Sclocchi
Arxiv Link: https://arxiv.org/abs/2010.10253v2 (pdf)
Date accepted: 2020-12-29
Date submitted: 2020-12-18 01:50
Submitted by: Sclocchi, Antonio
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational

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.

Published as SciPost Phys. 10, 013 (2021)



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Resubmission 2010.10253v2 on 18 December 2020

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