  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.
