Novel Compressible Adaptive Spectral Mixture Kernels for Gaussian Processes with Sparse Time and Phase Delay Structures

Spectral mixture (SM) kernels comprise a powerful class of kernels for Gaussian processes (GPs) capable of discovering structurally complex patterns and modeling negative covariances. Being a linear superposition of quasi-periodical kernel components, the state-of-the-art SM kernel does not consider component compression and dependency structures between components. In this paper, we investigate the benefits of component compression and modeling of both time and phase delay structures between basis components in the SM kernel. By verifying the presence of dependencies between function components using Gaussian conditionals and posterior covariance, we first propose a new SM kernel variant with a time and phase delay dependency structure (SMD) and then provide a structure adaptation (SA) algorithm for the SMD. The SMD kernel is constructed in two steps: first, time delay and phase delay are incorporated into each basis component; next, cross-convolution between a basis component and the reversed complex conjugate of another basis component is performed, which yields a complex-valued and positive definite kernel incorporating dependency structures between basis components. The model compression and dependency sparsity of the SMD kernel can be obtained by using automatic pruning in SA. We perform a thorough comparative experimental analysis of the SMD on both synthetic and real-life datasets. The results corroborate the efficacy of the dependency structure and SA in the SMD.