# Explaining Near-Zero Hessian Eigenvalues Through Approximate Symmetries in Neural Networks

arXiv: https://arxiv.org/abs/2607.07845 · HTML: https://mlsanity.com/paper/2607.07845/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 61 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## Builds on (most cited first, 30 of 37)

- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced](https://mlsanity.com/paper/1806.00900/) (2018)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Revisiting Natural Gradient for Deep Networks](https://mlsanity.com/paper/1301.3584/) (2013)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [Finite Versus Infinite Neural Networks: an Empirical Study](https://mlsanity.com/paper/2007.15801/) (2020)
- [Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics](https://mlsanity.com/paper/2012.04728/) (2020)
- [Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra](https://mlsanity.com/paper/2008.11865/) (2020)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [Hessian-based Analysis of Large Batch Training and Robustness to Adversaries](https://mlsanity.com/paper/1802.08241/) (2018)
- [Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians](https://mlsanity.com/paper/1901.08244/) (2019)
- [Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape](https://mlsanity.com/paper/1907.02911/) (2019)
- [The jamming transition as a paradigm to understand the loss landscape of deep neural networks](https://mlsanity.com/paper/1809.09349/) (2018)
- [Analytic Insights into Structure and Rank of Neural Network Hessian Maps](https://mlsanity.com/paper/2106.16225/) (2021)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [Emergent properties of the local geometry of neural loss landscapes](https://mlsanity.com/paper/1910.05929/) (2019)
- [Abide by the Law and Follow the Flow: Conservation Laws for Gradient Flows](https://mlsanity.com/paper/2307.00144/) (2023)

## Abstract (arXiv)

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.