Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of S k

Yossi Arjevani*, Michael Field*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural networks. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on k symbols, S k ; in particular, the standard representation of S k . It is shown that spurious minima (non-global local minima) do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic S k -equivariant bifurcation. We describe minimal models for forced symmetry breaking that give a lower bound on the dynamic complexity involved in the creation of spurious minima when there is no symmetry. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterbach & Melbourne on the instability of solutions when there are quadratic equivariants.

Original languageAmerican English
Pages (from-to)2809-2857
Number of pages49
JournalNonlinearity
Volume35
Issue number6
DOIs
StatePublished - Jun 2022

Bibliographical note

Publisher Copyright:
© 2022 IOP Publishing Ltd & London Mathematical Society

Keywords

  • 37C80, 37G40, 34C23, 58K70
  • equivariant bifurcation
  • forced symmetry breaking
  • gradient dynamics
  • minimal models
  • spurious minima
  • symmetric group

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