Gaussian Process on the Product of Directional Manifolds

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.