Mathematics Colloquia and Seminars

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Description

Preconditioning can make probability flows converge much faster, but applying the preconditioner is often expensive. I'll start by discussing two ways to precondition a probability flow: mirror Langevin dynamics and Laplacian-adjusted Wasserstein gradient descent (LAWGD). Both converge quickly to the target distribution, and understanding why tells us what an approximation needs to preserve. Next, I'll describe randomized versions of each. Subspace Langevin approximates matrix-preconditioned Langevin, a special case of mirror Langevin, by updating only along randomly sampled low-dimensional subspaces. Subspace LAWGD instead approximates the transport potential in the span of random ridge functions and uses a geometry-aware projection, so descent still holds at the population level. In the last part, I'll show how to learn the inverse Langevin operator directly from samples, without access to the target density or its score. The common thread is the trade-off between convergence, approximation quality, and computational cost, which I'll illustrate with applications to sampling and generative modeling.