Gradient Gibbs measures with non-convex potentials and the universality class of the Gaussian Free Field

What the math gives to ML

The paper's transferable asset is a constructive robustness principle: a Gibbs model with a strongly outward-drifting but locally non-convex potential can be represented as a mixture of strictly convex gradient models, retaining dimension-free Poincare control and Gaussian-free-field-like covariance decay. This suggests replacing isotropic optimizer noise or parameter regularization by a graph-structured mixture of convex Langevin dynamics, where each sampled component has stable curvature even though the aggregate process is non-convex. The most practical first test is a layerwise parameter graph with edge-difference energies, using randomly selected convex component potentials as a preconditioned Langevin optimizer and checking whether it improves stability or sample efficiency at equal compute.