# Gaussian Process on the Product of Directional Manifolds

arXiv: https://arxiv.org/abs/2303.06799 · HTML: https://mlsanity.com/paper/2303.06799/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 34 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 7 of 7)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Modular Deep Reinforcement Learning for Continuous Motion Planning with Temporal Logic](https://mlsanity.com/paper/2102.12855/) (2021)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Directional Statistics-based Deep Metric Learning for Image Classification and Retrieval](https://mlsanity.com/paper/1802.09662/) (2018)
- [Unscented Orientation Estimation Based on the Bingham Distribution](https://mlsanity.com/paper/1311.5796/) (2013)

## Built on by (most cited first, 2 of 2)

- [Memory as Resonance: A Biomimetic Architecture for Infinite Context Memory on Ergodic Phonetic Manifolds](https://mlsanity.com/paper/2512.20245/) (2025)
- [Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable](https://mlsanity.com/paper/2411.18989/) (2024)

## Abstract (arXiv)

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.