  In this paper, we study the computation of the rate-distortion-perception
function (RDPF) for a multivariate Gaussian source under mean squared error
(MSE) distortion and, respectively, Kullback-Leibler divergence, geometric
Jensen-Shannon divergence, squared Hellinger distance, and squared
Wasserstein-2 distance perception metrics. To this end, we first characterize
the analytical bounds of the scalar Gaussian RDPF for the aforementioned
divergence functions, also providing the RDPF-achieving forward "test-channel"
realization. Focusing on the multivariate case, we establish that, for
tensorizable distortion and perception metrics, the optimal solution resides on
the vector space spanned by the eigenvector of the source covariance matrix.
Consequently, the multivariate optimization problem can be expressed as a
function of the scalar Gaussian RDPFs of the source marginals, constrained by
global distortion and perception levels. Leveraging this characterization, we
design an alternating minimization scheme based on the block nonlinear
Gauss-Seidel method, which optimally solves the problem while identifying the
Gaussian RDPF-achieving realization. Furthermore, the associated algorithmic
embodiment is provided, as well as the convergence and the rate of convergence
characterization. Lastly, for the "perfect realism" regime, the analytical
solution for the multivariate Gaussian RDPF is obtained. We corroborate our
results with numerical simulations and draw connections to existing results.
