Gaussian Process on the Product of Directional Manifolds

We introduce a principled study on establishing 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 so-called hypertoroidal von Mises (HvM) kernel is proposed to establish GPs on hypertori with consideration of correlational circular components. The proposed HvM kernel is demonstrated with multi-output GP regression for learning vector-valued functions defined on hypertori using the intrinsic coregionalization model. Analytical derivatives in 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. The numerical results show that the HvM-based GP achieves superior tracking accuracy compared to parametric model and GPs based on conventional kernel designs.