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Gaussian Process on the Product of Directional Manifolds

Ziyu Cao, Kailai Li

TL;DR

This work develops Gaussian processes for inputs on the hypertorus by introducing the hypertoroidal von Mises (HvM) kernel, which explicitly models correlations across the three circular components of $\mathbb{T}^3$. It provides closed-form derivatives for hyperparameter optimization and demonstrates superior performance in a data-driven, ranging-based tracking setting using GP-based particle filtering. The approach outperforms kernels based on kernel products and parametric models, highlighting the importance of manifold-adaptive kernels in directional data. This method has potential for real-time applications and can be extended to higher-dimensional directional spaces with further theoretical guarantees.

Abstract

We present a principled study on defining Gaussian processes (GPs) with inputs on the product of directional manifolds. A circular kernel is first presented according to the von Mises distribution. Based thereon, the hypertoroidal von Mises (HvM) kernel is proposed to establish GPs on hypertori with consideration of correlated circular components. The proposed HvM kernel is demonstrated with multi-output GP regression for learning vector-valued functions on hypertori using the intrinsic coregionalization model. Analytic derivatives for hyperparameter optimization are provided for runtime-critical applications. For evaluation, we synthesize a ranging-based sensor network and employ the HvM-based GPs for data-driven recursive localization. Numerical results show that the HvM-based GP achieves superior tracking accuracy compared to parametric model and GPs of conventional kernel designs.

Gaussian Process on the Product of Directional Manifolds

TL;DR

This work develops Gaussian processes for inputs on the hypertorus by introducing the hypertoroidal von Mises (HvM) kernel, which explicitly models correlations across the three circular components of . It provides closed-form derivatives for hyperparameter optimization and demonstrates superior performance in a data-driven, ranging-based tracking setting using GP-based particle filtering. The approach outperforms kernels based on kernel products and parametric models, highlighting the importance of manifold-adaptive kernels in directional data. This method has potential for real-time applications and can be extended to higher-dimensional directional spaces with further theoretical guarantees.

Abstract

We present a principled study on defining Gaussian processes (GPs) with inputs on the product of directional manifolds. A circular kernel is first presented according to the von Mises distribution. Based thereon, the hypertoroidal von Mises (HvM) kernel is proposed to establish GPs on hypertori with consideration of correlated circular components. The proposed HvM kernel is demonstrated with multi-output GP regression for learning vector-valued functions on hypertori using the intrinsic coregionalization model. Analytic derivatives for hyperparameter optimization are provided for runtime-critical applications. For evaluation, we synthesize a ranging-based sensor network and employ the HvM-based GPs for data-driven recursive localization. Numerical results show that the HvM-based GP achieves superior tracking accuracy compared to parametric model and GPs of conventional kernel designs.
Paper Structure (15 sections, 18 equations, 5 figures)

This paper contains 15 sections, 18 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Gaussian Process Regression and Inference
  4. Gaussian Process Modeling
  5. Multi-Output Gaussian Processes
  6. Gaussian Processes on the Hypertorus
  7. A Circular Kernel Based on the von Mises Distribution
  8. The Hypertoroidal von Mises Kernel
  9. Hyperparameter Optimization
  10. Evaluation
  11. Scenario Setup
  12. Multi-Output GPs on Hypertorus
  13. GP-Based Particle Filtering
  14. Evaluation and Results
  15. Conclusions

Figures (5)

  • Figure 1: GP regression using different kernels on the circular function in Case \ref{['case1']}. Given the same training set (green dots), the vM kernel produces a geometry-adaptive posterior (with blue curves and gray areas denoting means and variances, respectively), whereas the SE kernel fails.
  • Figure 2: Proposed HvM kernels (normalized function values) configured in Case \ref{['case2']} with mappings to tori.
  • Figure 3: Considered trajectories (red) and estimates (blue) given by GP-based particle filtering using the proposed HvM kernel. Noise level is set at $\xi=0.01$ (corresponding to sequences of S1). Green dots denote reference points.
  • Figure 4: Results of particle filtering using parametric and different GP-based reweighting schemes in Sec. \ref{['subsec:res']}. RMSEs of the APEs are plotted for the trajectories ${\texttt{T1}}-{\texttt{T3}}$ using boxchart of default setting in Matlab. The proposed HvM-based GP (blue) reweighting consistently enables superior tracking accuracy compared to those based on PvM (red), and PPRD (lilac) kernels constructed via kernel product, whereas the parametric model (black) produces inferior performance with a considerable margin. The PSE kernel (green) leads to tracking failures in most sequences due to its nonperiodic definition.
  • Figure 5: Range measurements (red) and predictions (means and variances plotted by blue curves and gray areas, respectively) w.r.t. ground truth over T2 of noise level S2. The reference point on the top in Fig. \ref{['fig:exa']} is selected for demonstration.