On the Gaussian-Quadratic Rate-Distortion Function for Vector Sources with Individual Distortion Constraints

Abstract

This paper investigates the Gaussian-quadratic lossy compression with arbitrary source length under individual distortion constraints. The rate-distortion function (RDF) is lower-bounded by a Hadamard inequality-based rate, which is tight if and only if the semidefinite condition (SDC) holds. Otherwise, this bound becomes loose, and analytical results are lacking. Moreover, the fundamental quantitative relationship between source correlations and the RDF remains incomplete. In this paper, we provide new theoretical results under different source covariance matrices and distortion constraints. First, under arbitrary covariance and distortion constraints, we obtain the spectral properties of the optimal source reconstruction achieving the RDF, and a stronger scalar inequality version of the SDC. We propose a class of source covariance matrices based on hierarchical correlations and show that studying the two-type correlation (2-TC) model is sufficient to establish the analytical foundation for the broader class. Under this covariance, we obtain the RDF with source correlations explicitly incorporated when the SDC holds, and analyze the SDC from the perspectives of distortion constraints and source correlations. Next, under the 2-TC covariance and two-type distortion (2-TD) constraints, we establish the complete RDFs over seven regions on a distortion plane, with the optimal distortion (rate) allocations determined in each region. It is revealed that the essence of pursuing the complete RDF lies in thoroughly analyzing the correlations between the optimal distortions. Finally, under isotropic correlation and identical constraints, we provide the per-component compression rate and show that exploiting correlations can significantly reduce compression costs.