On the Rate-Distortion-Perception Function for Gaussian Processes

TL;DR

This paper explores the rate-distortion-perception function for Gaussian processes, deriving analytical bounds and characterizing optimal reconstructions as Gaussian processes with shared eigenvectors, extending classical results to infinite-dimensional settings.

Contribution

It introduces an analytical tight upper bound for the RDPF of Gaussian processes and characterizes the optimal reconstruction process as a Gaussian process with shared eigenvectors.

Findings

Derived a tight upper bound for the RDPF of Gaussian processes.

Characterized the optimal reconstruction as a Gaussian process with shared eigenvectors.

Provided bounds on rate and distortion for stationary GPs as the interval length tends to infinity.

Abstract

In this paper, we investigate the rate-distortion-perception function (RDPF) of a source modeled by a Gaussian Process (GP) on 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, characterized by a covariance operator sharing the same set of eigenvectors of the source covariance operator. Similarly to the classical rate-distortion function, this allows us to formulate the RDPF problem in terms of the Karhunen-Lo\`eve transform coefficients of the involved GPs. Leveraging the similarities with the finite-dimensional Gaussian RDPF, we formulate an analytical tight upper bound for the RDPF for GPs, which recovers the optimal solution in the "perfect realism" regime. Lastly, in the case where the source is a stationary GP and $\Omega$ is the interval $[0, T]$ equipped with the Lebesgue measure, we derive an upper bound on the rate and the distortion for a fixed perceptual level and $T \to \infty$ as a function of the spectral density of the source process.