Regularized Kernel and Neural Sobolev Descent: Dynamic MMD Transport

We introduce Regularized Kernel and Neural Sobolev Descent for transporting a source distribution to a target distribution along smooth paths of minimum kinetic energy (defined by the Sobolev discrepancy), related to dynamic optimal transport. In the kernel version, we give a simple algorithm to perform the descent along gradients of the Sobolev critic, and show that it converges asymptotically to the target distribution in the MMD sense. In the neural version, we parametrize the Sobolev critic with a neural network with input gradient norm constrained in expectation. We show in theory and experiments that regularization has an important role in favoring smooth transitions between distributions, avoiding large discrete jumps. Our analysis could provide a new perspective on the impact of critic updates (early stopping) on the paths to equilibrium in the GAN setting.