Kernel embedding of maps for Bayesian inference: The variational mapping particle filter

PULIDO M. AND P. J. VANLEEWEN

Munich

Simposio; 6th International Symposium of Data Assimilation.; 2018

In this work, a novel particle filter is introduced which aims to an efficient sampling of high-dimensional state spaces considering a limited number of particles. The filter is based on variational importance sampling. Particles are mapped from the proposal to the posterior density using the principles of optimal transport. The Kullback-Leibler divergence between the posterior density and the proposal divergence is optimized using  variational principles. A key ingredient of the mapping is that the transformations are embedded in a reproducing kernel Hilbert space which constrains the dimensions of the space for the optimal transport to the number of particles. Gradient information of the Kullback-Leibler divergence allows a quick convergence using well known gradient-based optimization algorithms from machine learning, adadelta and adam, which do not require cost function calculations. Evaluation of the method and comparison with a SIR filter is conducted as a proof-of-concept in the Lorenz-63 system. No resampling is required even for long recursive implementations. The number of effective particles remains close to the total number of particles in all the recursion. Hence, the mapping particle filter does not suffer from sample impoverishment.