doi: 10.17706/ijapm.2025.15.2.91-101
Symmetry-Breaking Lie Flows for Optimisation of Discontinuous Functionals
Mikhail Osipov
- Independent Researcher, Milan, Italy.
- Manuscript received July 9, 2025
- accepted August 20, 2025
- published September 23, 2025
Abstract
We address the problem of reducing a task cost functional <em>W(S)</em>, defined over Sobolev-class signals <em>S</em>, when <em>W</em> is invariant under a global symmetry group <em>G</em> ⊂ <em>Diff</em>(<em>M</em>) and only accessible as a black-box. Such settings arise in machine learning, imaging, and inverse problems, where performance metrics are non-differentiable and internal to pretrained models. We propose a variational method that leverages symmetry to construct explicit, symmetry-breaking deformations of the input. By minimizing an auxiliary energy, we obtain a gauge field whose induced deformation <em>h</em> = <em>A</em><sub><em>ϕ</em></sub>[<em>S</em>] lies generically transverse to the <em>G</em>-orbit. We show that—even for discontinuous <em>W</em>—a simple double-sign test on h descends to a strictly lower-cost region with positive probability, and almost surely under mild geometric conditions. This method requires no model gradients or labels and operates entirely at test time. It offers a principled mechanism for optimizing invariant cost functionals via Lie-algebraic flows, with applications to black-box systems and symmetry-constrained tasks.
Keywords
- Symmetry breaking
- non-smooth optimisation
- machine learning
How to Cite
Mikhail Osipov, "Symmetry-Breaking Lie Flows for Optimisation of Discontinuous Functionals," International Journal of Applied Physics and Mathematics, vol. 15, no. 2, pp. 91-101, 2025. https://doi.org/ 10.17706/ijapm.2025.15.2.91-101
Copyright & License
Copyright © 2025 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).