  We propose a graph spectrum-based Gaussian process for prediction of signals
defined on nodes of the graph. The model is designed to capture various graph
signal structures through a highly adaptive kernel that incorporates a flexible
polynomial function in the graph spectral domain. Unlike most existing
approaches, we propose to learn such a spectral kernel, where the polynomial
setup enables learning without the need for eigen-decomposition of the graph
Laplacian. In addition, this kernel has the interpretability of graph filtering
achieved by a bespoke maximum likelihood learning algorithm that enforces the
positivity of the spectrum. We demonstrate the interpretability of the model in
synthetic experiments from which we show the various ground truth spectral
filters can be accurately recovered, and the adaptability translates to
superior performances in the prediction of real-world graph data of various
characteristics.
