Understanding Smoothness of Vector Gaussian Processes on Product Spaces

Vector Gaussian processes are becoming increasingly important in machine learning and statistics, with applications to many branches of applied sciences. Recent efforts have allowed to understand smoothness in scalar Gaussian processes defined over manifolds as well as over product spaces involving manifolds. Under assumptions of Gaussianity and mean-square continuity, the smoothness of a zero-mean scalar process is in one-to-one correspondence with the smoothness of the covariance kernel. Unfortunately, such a result is not available for vector-valued random fields, as the way each component in the covariance kernel contributes to the smoothness of the vector field is unclear. This paper challenges the problem of quantifying smoothness of matrix-valued continuous kernels that are associated with mean-square continuous vector Gaussian processes defined over non-Euclidean product manifolds. After noting that a constructive RKHS approach is unsuitable for this specific task, we proceed through the analysis of spectral properties. Specifically, we find a spectral representation to quantify smoothness through Sobolev spaces that are adapted to certain measure spaces of product measures obtained through the tensor product of Haar measures with multivariate Gaussian measures. Our results allow to measure smoothness in a simple way, and open for the study of foundational properties of certain machine learning techniques over product spaces.