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Spectral mixture (SM) kernels comprise a powerful class of generalizedkernels for Gaussian processes (GPs) to describe complex patterns. This paperintroduces model compression and time- and phase (TP) modulated dependencystructures to the original (SM) kernel for improved generalization of GPs.Specifically, by adopting Bienaym\'es identity, we generalize the dependencystructure through cross-covariance between the SM components. Then, we proposea novel SM kernel with a dependency structure (SMD) by using cross-convolutionbetween the SM components. Furthermore, we ameliorate the expressiveness of thedependency structure by parameterizing it with time and phase delays. Thedependency structure has clear interpretations in terms of spectral density,covariance behavior, and sampling path. To enrich the SMD with effectivehyperparameter initialization, compressible SM kernel components, and sparsedependency structures, we introduce a novel structure adaptation (SA) algorithmin the end. A thorough comparative analysis of the SMD on both synthetic andreal-life applications corroborates its efficacy.

Compressible Spectral Mixture Kernels with Sparse Dependency Structures for Gaussian Processes