Filtering with State-Observation Examples via Kernel Monte Carlo Filter

TL;DR

This paper introduces Kernel Monte Carlo Filter, a nonparametric filtering method that uses kernel mean embeddings to perform state estimation from example-based observations, especially when the observation model is unknown.

Contribution

It proposes a novel filtering approach that leverages kernel mean embeddings and kernel Bayes' rule to handle cases with only example-based observation data, without explicit probabilistic models.

Findings

Kernel Monte Carlo Filter effectively estimates states from example-based observations.

Theoretical analysis shows sampling performance depends on effective sample size.

Resampling improves sampling performance by increasing effective sample size.

Abstract

This paper addresses the problem of filtering with a state-space model. Standard approaches for filtering assume that a probabilistic model for observations (i.e. the observation model) is given explicitly or at least parametrically. We consider a setting where this assumption is not satisfied; we assume that the knowledge of the observation model is only provided by examples of state-observation pairs. This setting is important and appears when state variables are defined as quantities that are very different from the observations. We propose Kernel Monte Carlo Filter, a novel filtering method that is focused on this setting. Our approach is based on the framework of kernel mean embeddings, which enables nonparametric posterior inference using the state-observation examples. The proposed method represents state distributions as weighted samples, propagates these samples by sampling, estimates the state posteriors by Kernel Bayes' Rule, and resamples by Kernel Herding. In particular, the sampling and resampling procedures are novel in being expressed using kernel mean embeddings, so we theoretically analyze their behaviors. We reveal the following properties, which are similar to those of corresponding procedures in particle methods: (1) the performance of sampling can degrade if the effective sample size of a weighted sample is small; (2) resampling improves the sampling performance by increasing the effective sample size. We first demonstrate these theoretical findings by synthetic experiments. Then we show the effectiveness of the proposed filter by artificial and real data experiments, which include vision-based mobile robot localization.