# Symmetry & Critical Points for Symmetric Tensor Decomposition Problems

arXiv: https://arxiv.org/abs/2306.07886 · HTML: https://mlsanity.com/paper/2306.07886/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 75 references that are in the map.
- First posted on arXiv 13 Jun 2023.

## Builds on (most cited first, 14 of 14)

- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Efficient approaches for escaping higher order saddle points in non-convex optimization](https://mlsanity.com/paper/1602.05908/) (2016)
- [Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry](https://mlsanity.com/paper/2008.01805/) (2020)
- [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)
- [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)
- [Symmetry Breaking in Symmetric Tensor Decomposition](https://mlsanity.com/paper/2103.06234/) (2021)
- [Annihilation of Spurious Minima in Two-Layer ReLU Networks](https://mlsanity.com/paper/2210.06088/) (2022)
- [A Robust Spectral Algorithm for Overcomplete Tensor Decomposition](https://mlsanity.com/paper/2203.02790/) (2022)
- [Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of Sn](https://mlsanity.com/paper/2107.02422/) (2021)
- [Complexity aspects of local minima and related notions](https://mlsanity.com/paper/2008.06148/) (2020)

## Built on by (most cited first, 3 of 3)

- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [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)

We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points represented by Puiseux series in the problem dimension, and so obtain precise analytic estimates on the objective function value and the Hessian spectrum. The results enable an analytic characterization of various obstructions to local optimization methods, revealing, in particular, a complex array of saddles and minima that differ in their symmetry, structure, and analytic properties. A notable phenomenon, observed for all critical points considered, concerns the index of the Hessian increasing with the objective function value.