Symmetry-Breaking Descent for Invariant Cost Functionals

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.