Caffarelli Estimates under Lipschitz Perturbations

What the math gives to ML

The paper establishes a dimension-free stability principle for quadratic optimal transport: if a Gaussian target density is perturbed by a globally L-Lipschitz log-density term, its Brenier map remains globally Lipschitz even without convexity or semiconvexity of the perturbation. This can be transferred to latent-variable generators by parameterizing the generator as the gradient of a convex potential and controlling the Lipschitz constant of the target log-density perturbation. The resulting bound is conservative and grows exponentially in L squared, so its most practical use is as a stability certificate or adaptive Jacobian regularizer rather than as a hard architectural constraint.