# Symmetry-Breaking Descent for Invariant Cost Functionals

arXiv: https://arxiv.org/abs/2505.13578 · HTML: https://mlsanity.com/paper/2505.13578/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 19 May 2025.

## Builds on (most cited first, 11 of 11)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Mitigating Adversarial Effects Through Randomization](https://mlsanity.com/paper/1711.01991/) (2017)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Test-Time Training with Self-Supervision for Generalization under Distribution Shifts](https://mlsanity.com/paper/1909.13231/) (2019)
- [Symmetry and Generalisation in Machine Learning](https://mlsanity.com/paper/2501.03858/) (2025)
- [Dysarthria Normalization via Local Lie Group Transformations for Robust ASR](https://mlsanity.com/paper/2504.12279/) (2025)

## Abstract (arXiv)

We study the problem of reducing a task cost functional W : Hs(M) → ℝ, not assumed continuous or differentiable, defined over Sobolev-class signals S ∈ Hs(M) , in the presence of a global symmetry group G subset Diff(M). The group acts on signals by pullback, and the cost W is invariant under this action. Such scenarios arise in machine learning and related optimization tasks, where performance metrics may be discontinuous or model-internal. We propose a variational method that exploits the symmetry structure to construct explicit deformations of the input signal. A deformation control field  φ: M → mathbb Rd, obtained by minimizing an auxiliary energy functional, induces a flow that generically lies in the normal space (with respect to the L2 inner product) to the G-orbit of S, and hence is a natural candidate to cross the decision boundary of the G -invariant cost. We analyze two variants of the coupling term: (1) purely geometric, independent of W, and (2) weakly coupled to W. Under mild conditions, we show that symmetry-breaking deformations of the signal can reduce the cost. Our approach requires no gradient backpropagation or training labels and operates entirely at test time. It provides a principled tool for optimizing discontinuous invariant cost functionals via Lie-algebraic variational flows.