Variational filtering and smoothing with low-rank transports

Abstract

We use transport maps between probability distributions to devise algorithms for nonlinear Bayesian filtering and smoothing in high-dimensional state-space models. In particular, we consider a semi-parametric characterization of transports between a tractable reference distribution (e.g., a standard Gaussian) and target distributions represented by certain modifications of the lag--1 smoothing distributions. These transports can be combined to characterize the full smoothing distribution, or even the parameter posterior, of a state-space model, producing an inherently sequential smoothing algorithm, with a computational cost that is constant in time. These transports are identified through the solution of unconstrained variational problems. In order to circumvent the usual bottlenecks of representing high-dimensional transports, we exploit the fact that for sufficiently short lags, many dynamical systems of practical interest (e.g., Lorenz systems) may exhibit updates that act only on lower dimensional subspaces. This allows the construction of transports that are nearly linear with respect to most of their variables, and that encode nonlinearities only in the interaction of a handful of important directions. The methodology will be showcased on chaotic dynamical systems.

Contact information

Dr. Daniele Bigoni Massachusetts Institute of Technology