We develop a penalized likelihood inference approach for highly multivariate Gaussian stochastic processes. Since zeros in the inverse cross-spectral density encode process-level conditional independence, sparse modeling of this object provides a natural route to scalable inference in the highly multivariate case. We introduce a latent variable representation which, combined with random Fourier features, yields a stochastic approximation expectation--maximization algorithm for penalized maximum likelihood estimation of a sparse inverse cross-spectral density matrix under a graphical LASSO penalty. The augmentation is crucial because sparsity is present in the inverse spectral density of the latent variable, whereas the marginal inverse covariance remains dense in the presence of a spatial nugget. Our frequency domain formulation enables joint penalized likelihood estimation of process-specific spatial covariance parameters and sparse process-level conditional dependence without requiring a pre-specified reference set of locations. The method is demonstrated on both synthetic and the MERRA-2 aerosol reanalysis data sets, yielding benefit over existing alternatives, especially for prediction at unobserved locations under spatial misalignment.

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