# Asymptotically exact unweighted particle filter for manifold-valued hidden states and point process observations

## TL;DR

This paper introduces an asymptotically exact particle filter for manifold-valued hidden states with point process observations, utilizing intrinsic dynamics and PDE-based control terms to improve filtering accuracy.

## Contribution

It develops a novel filter (ppFPF) that extends feedback particle filtering to manifolds with point process data, using PDE solutions for control, ensuring intrinsic and accurate state estimation.

## Key findings

- The filter accurately updates particles on manifolds during observations.
- It leverages PDE solutions similar to weighted Poisson equations for control.
- The method is compatible with existing PDE approximation algorithms.

## Abstract

The filtering of a Markov diffusion process on a manifold from counting process observations leads to `large' changes in the conditional distribution upon an observed event, corresponding to a multiplication of the density by the intensity function of the observation process. If that distribution is represented by unweighted samples or particles, they need to be jointly transformed such that they sample from the modified distribution. In previous work, this transformation has been approximated by a translation of all the particles by a common vector. However, such an operation is ill-defined on a manifold, and on a vector space, a constant gain can lead to a wrong estimate of the uncertainty over the hidden state. Here, taking inspiration from the feedback particle filter (FPF), we derive an asymptotically exact filter (called ppFPF) for point process observations, whose particles evolve according to intrinsic (i.e. parametrization-invariant) dynamics that are composed of the dynamics of the hidden state plus additional control terms. While not sharing the gain-times-error structure of the FPF, the optimal control terms are expressed as solutions to partial differential equations analogous to the weighted Poisson equation for the gain of the FPF. The proposed filter can therefore make use of existing approximation algorithms for solutions of weighted Poisson equations.