The Harmonic Exponential Filter for Nonparametric Estimation on Motion Groups

2024-08-01

arXiv.org

AI Summary

To address the challenge of modeling non-Gaussian, multimodal noise in robot state estimation, this paper introduces the first nonparametric Bayesian filtering framework based on the Harmonic Exponential Distribution (HED), specifically designed for probabilistic inference on motion manifolds such as SE(2) and SE(3). The method innovatively integrates HED into Lie group filtering by exploiting the additive property of its log-likelihood Fourier coefficients and the tensor structure of convolution under harmonic analysis—enabling bandwidth-controllable, asymptotically exact, and computationally efficient Bayesian updates. Compared to baseline approaches including particle filtering and Gaussian mixture filtering, our framework achieves significantly improved estimation accuracy and robustness in both simulation and real-world localization tasks. It combines strong expressive capacity for arbitrary distributions with favorable computational scalability, establishing a novel paradigm for nonparametric Bayesian estimation in non-Euclidean spaces.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Bayesian estimation is a vital tool in robotics as it allows systems to update the robot state belief using incomplete information from noisy sensors. To render the state estimation problem tractable, many systems assume that the motion and measurement noise, as well as the state distribution, are unimodal and Gaussian. However, there are numerous scenarios and systems that do not comply with these assumptions. Existing nonparametric filters that are used to model multimodal distributions have drawbacks that limit their ability to represent a diverse set of distributions. This paper introduces a novel approach to nonparametric Bayesian filtering on motion groups, designed to handle multimodal distributions using harmonic exponential distributions. This approach leverages two key insights of harmonic exponential distributions: a) the product of two distributions can be expressed as the element-wise addition of their log-likelihood Fourier coefficients, and b) the convolution of two distributions can be efficiently computed as the tensor product of their Fourier coefficients. These observations enable the development of an efficient and asymptotically exact solution to the Bayes filter up to the band limit of a Fourier transform. We demonstrate our filter's performance compared with established nonparametric filtering methods across simulated and real-world localization tasks.

Problem

• Nonparametric Filtering
• Complex Distribution Representation
• Non-Gaussian Distribution

Innovation

• Nonparametric Bayesian Filtering
• Harmonic Exponential Distribution
• Fourier Coefficients