Seminar: Score matching and simulation of sub-Riemannian Brownian bridges
Erlend Grong, Associate Professor @ Department of Mathematics, University of Bergen
Abstract:
We will start by giving an introduction to score matching.
Estimating a probability density p(x) on R^d arriving as a result a Markov chain or a Stochastic Differential Equation (SDE) can usually be done by simulating samples and taking averages.
However, if we need good estimates for derivatives of p(x), we will need many many more samples to get good estimates, and we struggle in region where the probability p(x) has low values, and hence few samples.
Score matching is a process of estimating the score, i.e., the function s(x) = dlog p(x), without relying on numerical approximations of derivatives. It is a technique of how a neural network can learn the score from a given set of samples.
We will describe this techniques and their applications to diffusion networks.
We will also show the application of score matching for bridge process from solutions of SDEs, i.e., solutions that have been conditioned on hitting a given point at a given final time. I will then describe our own research into the special case where the SDEs have a sub-Riemannian infinitesimal generator. The latter is a joint work with Stefan Sommer (Copenhagen) and Karen Habermann (Warwick).