# Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of Sn

arXiv: https://arxiv.org/abs/2107.02422 · HTML: https://mlsanity.com/paper/2107.02422/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 69 references that are in the map.
- First posted on arXiv 6 Jul 2021.

## Builds on (most cited first, 4 of 4)

- [Spurious Local Minima are Common in Two-Layer ReLU Neural Networks](https://mlsanity.com/paper/1712.08968/) (2017)
- [Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry](https://mlsanity.com/paper/2008.01805/) (2020)
- [Symmetry & critical points for a model shallow neural network](https://mlsanity.com/paper/2003.10576/) (2020)
- [Symmetry Breaking in Symmetric Tensor Decomposition](https://mlsanity.com/paper/2103.06234/) (2021)

## Built on by (most cited first, 5 of 5)

- [Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks: A Tale of Symmetry II](https://mlsanity.com/paper/2107.10370/) (2021)
- [Annihilation of Spurious Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2210.06088/) (2022)
- [Symmetry & Critical Points for Symmetric Tensor Decomposition Problems](https://mlsanity.com/paper/2306.07886/) (2023)
- [Hidden Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2312.16819/) (2023)
- [Symmetry & Critical Points](https://mlsanity.com/paper/2408.14445/) (2024)

## Abstract (arXiv)

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 nets. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on n symbols, Sn; in particular, the standard representation of Sn. It is shown that spurious 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 Sn-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, Lauterback & Melbourne on the instability of solutions when there are quadratic equivariants.