Random Subspace Descent for Probability Flows

Wasserstein gradient flows connect sampling and generative modeling with optimization over probability measures. Kernel-based flows such as MMD descent, Sobolev descent, and Laplacian Adjusted Wasserstein Gradient Descent (LAWGD) yield interacting particle algorithms, but their descent directions can be expensive to compute. Random features and slicing reduce this cost, but many slicing algorithms ignore the geometry of the flow and need not decrease the original objective. We approximate the flows by solving their defining variational problem over a random span of ridge potentials. This preserves the original objective and geometry, reduces the update to a Gram solve, and yields an exact decomposition of captured and lost dissipation. Particle experiments demonstrate the advantage of projection over naive slicing.