# Symmetry Breaking in Symmetric Tensor Decomposition

arXiv: https://arxiv.org/abs/2103.06234 · HTML: https://mlsanity.com/paper/2103.06234/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 10 Mar 2021.

## Builds on (most cited first, 6 of 6)

- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions](https://mlsanity.com/paper/1804.02686/) (2018)
- [Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry](https://mlsanity.com/paper/2008.01805/) (2020)
- [Subspace power method for symmetric tensor decomposition](https://mlsanity.com/paper/1912.04007/) (2019)
- [On the Principle of Least Symmetry Breaking in Shallow ReLU Models](https://mlsanity.com/paper/1912.11939/) (2019)
- [Symmetry & critical points for a model shallow neural network](https://mlsanity.com/paper/2003.10576/) (2020)

## Built on by (most cited first, 10 of 10)

- [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)
- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [Annihilation of Spurious Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2210.06088/) (2022)
- [Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of Sn](https://mlsanity.com/paper/2107.02422/) (2021)
- [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)
- [Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU](https://mlsanity.com/paper/2510.25060/) (2025)
- [Ubiquitous Symmetry at Critical Points Across Diverse Optimization Landscapes](https://mlsanity.com/paper/2506.01959/) (2025)
- [Eigenvalues of the Hessian in Deep Learning: The Origin of Symmetry and Its Breaking](https://mlsanity.com/paper/2610.09919/) (2026)

## Abstract (arXiv)

In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of the loss function and show that critical points detected by standard gradient based methods are symmetry breaking with respect to the target tensor. The phenomena, seen for different choices of target tensors and norms, make possible the use of recently developed analytic and algebraic tools for studying nonconvex optimization landscapes exhibiting symmetry breaking phenomena of similar nature.