Copula-based Estimation of Continuous Sources for a Class of Constrained Rate-Distortion-Functions

TL;DR

This work develops a copula-based framework to estimate the rate–distortion–perception tradeoff in the perfect realism regime (PR-RDPF) for multivariate continuous sources under a single-letter distortion constraint. By linking PR-RDPF with OC-RDF and entropic optimal transport (EOT) through an I-projection view on copula distributions, the authors derive a tractable relaxation that yields a computable lower bound whose accuracy improves as the number of moment constraints grows. They obtain a parametric closed-form for the relaxed projection, solve the resulting convex dual via a gradient-based algorithm with Monte Carlo gradient estimates, and establish a generalized Shannon lower bound under MSE distortion. Numerical experiments for scalar and vector sources demonstrate accurate PR-RDPF estimation, with Gaussian sources achieving the SLB and non-Gaussian cases exhibiting expected tightness behavior in high-resolution regimes. Overall, the framework provides a scalable, convergence-guaranteed method for computing PR-RDPF and related problems, enhancing practical rate-distortion-perception tradeoffs in continuous-data applications.

Abstract

We present a new method to estimate the rate-distortion-perception function in the perfect realism regime (PR-RDPF), for multivariate continuous sources subject to a single-letter average distortion constraint. The proposed approach is not only able to solve the specific problem but also two related problems: the entropic optimal transport (EOT) and the output-constrained rate-distortion function (OC-RDF), of which the PR-RDPF represents a special case. Using copula distributions, we show that the OC-RDF can be cast as an I-projection problem on a convex set, based on which we develop a parametric solution of the optimal projection proving that its parameters can be estimated, up to an arbitrary precision, via the solution of a convex program. Subsequently, we propose an iterative scheme via gradient methods to estimate the convex program. Lastly, we characterize a Shannon lower bound (SLB) for the PR-RDPF under a mean squared error (MSE) distortion constraint. We support our theoretical findings with numerical examples by assessing the estimation performance of our iterative scheme using the PR-RDPF with the obtained SLB for various sources.

Figures (2)

• Figure 1: PR-RDPF for various source distributions under (a) MSE distortion metric and (b) MAE distortion metric.
• Figure 2: PR-RDPF under MSE distortion metric for a (a) Gaussian, and (b) exponential bivariate source.