Schrödinger bridges for Bayesian filtering - Enabling cost-effective filtering in nonlinear state space models

Abstract

While particle filters are able to approximate the filtering distribution in nonlinear and non-Gaussian scenarios, they suffer from an inherent flaw known as weight degeneracy. This leads to rough state estimates and wasted computational resources. This thesis seeks to mitigate weight degeneracy in particle filters by formulating the proposal as an optimal transport problem – the Schrödinger bridge. Theoretically, this allows for exact sampling from the target distribution, entirely bypassing the traditional weight update. Because the exact numerical solution to the Schrödinger bridge is computationally heavy, we approximate the optimal transport dynamics as a neural network using amortized learning. Our results demonstrate that using the exact Schrödinger bridge proposal completely eliminates the weight degeneracy, although the computations are slow and scale poorly. We also find that the neural networks currently follow the dynamic processes too poorly to bypass the weight updates entirely. However, when used as a proposal distribution to a marginal particle filter, the network provides a viable state estimation. In fact, it competes or bests commonly used filtering methods such as the bootstrap particle filter and the unscented Kalman filter in the nonlinear case, both with respect to the mean squared error to the true position and weight variance. These findings indicate the potential of using approximate Schrödinger bridges in particle filters. With further refinement to the network training, this integration has the potential to give rise to a highly efficient particle filter, free from weight degeneracy.