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Colloquium: Daniel Lacker (Columbia)
18 Nov @ 3:30 pm - 4:30 pm
Colloquium: Daniel Lacker (Columbia)
18 Nov @ 3:30 pm – 4:30 pmTitle: Projected Langevin dynamics and gradient flows for optimal transport and variational inference
Abstract:
The classical Langevin diffusion provides a natural algorithm for sampling from a given (continuous) distribution, which can be viewed as the unique minimizer of an energy functional over the space of probability measures. More recently, many problems in machine learning and statistics require sampling from distributions which are characterized as minimizers of similar energy functionals over a smaller, constrained set of probability measures. Key examples are (regularized) optimal transport, for which the constraint set consists of couplings of some given marginals, as well as variational inference in Bayesian statistics for which a variety of constraint sets are used, the most popular two being Gaussians and product measures. This talk will discuss new stochastic dynamics arising as variants of the Langevin dynamics which are well-suited for these kinds of constrained sampling problems. These dynamics can be viewed as gradient flows on submanifolds of the Wasserstein space of probability measures (although this talk assumes no prior knowledge of gradient flow theory). Based in part on joint work with Giovanni Conforti and Soumik Pal.