# Broken scale symmetries in undercomplete linear autoencoders

arXiv: https://arxiv.org/abs/2610.03640 · HTML: https://mlsanity.com/paper/2610.03640/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 20 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 16 of 16)

- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability](https://mlsanity.com/paper/2209.15594/) (2022)
- [What Happens after SGD Reaches Zero Loss? --A Mathematical Framework](https://mlsanity.com/paper/2110.06914/) (2021)
- [Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics](https://mlsanity.com/paper/2012.04728/) (2020)
- [Understanding Optimization in Deep Learning with Central Flows](https://mlsanity.com/paper/2410.24206/) (2024)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [Loss Landscapes of Regularized Linear Autoencoders](https://mlsanity.com/paper/1901.08168/) (2019)
- [Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks](https://mlsanity.com/paper/2306.04251/) (2023)
- [Edge of Stochastic Stability: Revisiting the Edge of Stability for SGD](https://mlsanity.com/paper/2412.20553/) (2024)
- [Neural Thermodynamics: Entropic Forces in Deep and Universal Representation Learning](https://mlsanity.com/paper/2505.12387/) (2025)
- [Noise Balance and Stationary Distribution of Stochastic Gradient Descent](https://mlsanity.com/paper/2308.06671/) (2023)
- [Stochastic Gradient Descent-Induced Drift of Representation in a Two-Layer Neural Network](https://mlsanity.com/paper/2302.02563/) (2023)
- [Contribution of task-irrelevant stimuli to drift of neural representations](https://mlsanity.com/paper/2510.21588/) (2025)
- [SGD at the Edge of Stability: Stochastic Stabilization with Large Learning Rates](https://mlsanity.com/paper/2606.30930/) (2026)

## Abstract (arXiv)

Neural network loss landscapes have many symmetries, which are preserved by gradient flow but broken by finite-stepsize stochastic gradient descent (SGD). A canonical example of such a symmetry is scale in homogeneous networks: one can scale up the parameters in one layer and down in the next without changing the network output. Previous work has documented cases in which SGD breaks this symmetry in favor of balancing gradient noise or minimizing fluctuations. Here, we show that the solution geometry of undercomplete linear autoencoders instead selects a preferred sign for scale drift: on the PCA solution manifold, SGD favors large decoder weights. This directed scale drift occurs on a slow timescale, and its dynamics admit an analytically-tractable effective description. However, it cannot continue indefinitely: increasing scale eventually drives the dynamics towards a finite-stepsize stability boundary. The resulting solutions are sharper than a balanced baseline in the sense of the maximum eigenvalue of the loss Hessian, but different sharpness measures can move in opposing directions. Thus, undercomplete autoencoders give a concrete illustration of how loss geometry can convert residual gradient noise into directed motion along a manifold of functionally-equivalent solutions.