🤖 AI Summary
To address the degeneracy of particle weights in standard Sequential Importance Resampling (SIR) particle filters—leading to failure in high-dimensional or degenerate-likelihood nonlinear filtering scenarios—this paper proposes a weight-free filtering framework grounded in Brenier optimal transport. By parameterizing the Brenier map via deep neural networks, the method directly transports the prior state distribution to the posterior distribution without explicit likelihood modeling, thereby eliminating resampling and circumventing weight degeneracy. This constitutes the first systematic integration of Brenier transport theory into nonlinear filtering. The approach naturally accommodates multimodal and non-Gaussian posteriors and enables differentiable, end-to-end learning through stochastic optimization and Monte Carlo sampling. Experiments demonstrate substantial improvements in sample efficiency and posterior approximation accuracy under high-dimensional and degenerate-likelihood conditions, outperforming both SIR particle filters and ensemble Kalman filters.
📝 Abstract
This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance resampling (SIR) particle filters suffer from fundamental limitations, in scenarios involving degenerate likelihoods or high-dimensional states, due to the weight degeneracy issue. In this paper, we explore an alternative method, which is based on estimating the Brenier optimal transport (OT) map from the current prior distribution of the state to the posterior distribution at the next time step. Unlike SIR particle filters, the OT formulation does not require the analytical form of the likelihood. Moreover, it allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare the OT method to the SIR particle filter and the ensemble Kalman filter, evaluating the performance in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.
