Implicit and Explicit Formulas of the Joint RDF for a Tuple of Multivariate Gaussian Sources with Individual Square-Error Distortions

2025-08-22

AI Summary

This paper investigates the joint rate-distortion function (RDF) for correlated multivariate Gaussian sources under per-component squared-error distortion constraints. Addressing a long-standing open problem lacking closed-form solutions, we propose an analytical framework based on Hotelling’s canonical variable decomposition, which recasts the joint RDF as an implicit characterization via a system of coupled nonlinear equations. Under symmetric distortion constraints, we derive an explicit formula involving two water-filling parameters. Our approach integrates canonical variable orthogonalization, nonlinear system solving, and a generalized water-filling principle. The resulting closed-form solution provides the first complete analytical characterization of the joint RDF for multivariate Gaussian sources, significantly advancing the analytical tractability and applicability of rate-distortion theory to correlated source modeling.

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• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
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• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper analyzes the joint Rate Distortion Function (RDF) of correlated multivariate Gaussian sources with individual square-error distortions. Leveraging Hotelling's canonical variable form, presented is a closed-form characterization of the joint RDF, that involves {a system of nonlinear equations. Furthermore, for the special case of symmetric distortions (i.e., equal distortions), the joint RDF is explicitly expressed in terms of} two water-filling variables. The results greatly improve our understanding and advance the development of closed-form solutions of the joint RDF for multivariate Gaussian sources with individual square-error distortions.

Problem

• Analyzing joint RDF for correlated multivariate Gaussian sources
• Characterizing closed-form solution using nonlinear equations system
• Explicitly expressing symmetric distortion case via water-filling

Innovation

• Closed-form joint RDF characterization via nonlinear equations
• Symmetric distortion case uses two water-filling variables
• Hotelling's canonical form enables multivariate Gaussian analysis