On the Rate-Distortion-Perception Function for Gaussian Processes

In this paper, we investigate the rate-distortion-perception function (RDPF) of a source modeled as a Gaussian Process (GP) over a measure space $\Omega$, under mean squared error (MSE) distortion and squared Wasserstein-2 perception metrics. First, we show that the optimal reconstruction process is itself a GP, whose covariance operator shares the same set of eigenvectors as the source's covariance operator. This structural property, akin to the classical rate-distortion function (RDF), allows us to reformulate the RDPF problem in terms of the Karhunen-Loève (KL) transform coefficients of the involved GPs. Leveraging the similarities with the finite-dimensional Gaussian RDPF, we derive a tight analytical upper bound on the RDPF for GPs, which recovers the optimal solution in the “perfect realism” regime. Finally, for stationary GPs over the interval $[0, T]$ with Lebesgue measure, we derive an upper bound on the rate and distortion for a fixed perceptual level and $T \rightarrow \infty$ as a function of the spectral density of the source process. We complement our theoretical findings with relevant simulation studies.