Generalized Convolution Spectral Mixture for Multi-Task Gaussian Processes

Multi-task Gaussian processes (MTGPs) are a powerful approach for modeling structured dependencies among multiple tasks. Researchers on MTGPs have contributed to enhance this approach in various ways. Current MTGP methods, however, cannot model nonlinear task correlations in a general way. In this paper we address this problem. We focus on spectral mixture (SM) based kernels and propose an enhancement of this type of kernels, called multi-task generalized convolution spectral mixture (MT-GCSM) kernel. The MT-GCSM kernel can model nonlinear task correlations and mixtures dependency, including time and phase delay, not only between different tasks but also within a task at the spectral mixture level. Each task in MT-GCSM has its own generalized convolution spectral mixture kernel (GCSM) with a different number of convolution structures and all spectral mixtures from different tasks are dependent. Furthermore, the proposed kernel uses inner and outer full cross convolution between base spectral mixtures, so that the base spectral mixtures in the tasks are not necessarily aligned. Extensive experiments on synthetic and real-life datasets illustrate the difference between MT-GCSM and other kernels as well as the practical effectiveness of MT-GCSM.