Symmetry-Breaking Descent for Invariant Cost Functionals

We study the problem of reducing a task cost functional $W : H^s(M) \to \mathbb{R}$, not assumed continuous or differentiable, defined over Sobolev-class signals $S \in H^s(M)$, in the presence of a global symmetry group $G \subset \mathrm{Diff}(M)$. 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 $\phi: M \to \mathbb R^d$, obtained by minimizing an auxiliary energy functional, induces a flow that generically lies in the normal space (with respect to the $L^2$ 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.