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

Jul 16, 2026

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This paper studies Gaussian-quadratic lossy compression for vector sources of arbitrary length under individual distortion constraints. A Hadamard-inequality-based lower bound on the rate-distortion function (RDF) is derived and shown to be tight exactly when a semidefinite condition (SDC) holds; otherwise the bound becomes loose. The paper also develops the missing quantitative relationship between source correlations and the RDF. New results are provided for different covariance matrices and distortion constraints, including spectral characterization of optimal reconstructions and complete RDFs across regions for a two-type correlation model under two-type distortion constraints, with optimal rate allocations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages60

Reading Time2 h 31 min

Abstract

Introduction

Distortion constraints and RDF trade-off

Sum distortion criterion

Individual distortion criteria

arXiv :2607 .09545v 1 [ cs .IT] 10 Jul 2026 On the Gaussian-Quadratic Rate-Distortion Function for Vector Sources with Individual Distortion Constraints Shuao Chen, Junyuan Gao, Yuxuan Shi, Yongpeng Wu, Giuseppe Caire, H. Vincent Poor, and Wenjun Zhang 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 twotype 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 S. Chen, Y. Wu, and W. Zhang are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mails: {shuao.chen, yongpeng.wu, [zhangwenjun](zhangwenjun}@sjtu.edu.cn)[}](zhangwenjun}@sjtu.edu.cn)[@sjtu.edu.cn](zhangwenjun}@sjtu.edu.cn)) (Corresponding author: Yongpeng Wu). J. Gao is with the Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong SAR, China (e-mail: [junyuan.gao@polyu.edu.hk](junyuan.gao@polyu.edu.hk)). Y. Shi is with the Department of Networked Intelligence, Pengcheng Laboratory, Shenzhen 410083, China (e-mail: [shiyx01@pcl.ac.cn](shiyx01@pcl.ac.cn)). G. Caire is with the Communications and Information Theory Group, Technische Universitt Berlin, Berlin 10587, Germany (e-mail: [caire@tu-berlin.de](caire@tu-berlin.de)) . H. V. Poor is with the Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544, USA (e-mail: [poor@princeton.edu](poor@princeton.edu)). Parts of this work were presented at the 2025 IEEE Global Communications Conference [1] . 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. Index Terms Distortion criterion, Hadamard inequality, lossy compression, rate-distortion function, vector source. I. INTRODUCTION The rapid growth of artificial intelligence (AI), Internet of Things (IoT), and edge computing has led to the generation of vast datasets containing components with heterogeneous significance, in which mission-critical elements coexist with non-essential data [2], [3] . The next generation of communication systems is expected to confront unprecedented challenges in handling the massive creation, transmission, and processing of data [4], [5] . Given the potentially stringent requirements for processing latency, transmission efficiency, and storage limitations, effective compression of these heterogeneous data streams will be indispensable in enhancing overall system efficiency. From an

FAQs

What distortion model is analyzed in this paper?

The paper analyzes Gaussian-quadratic lossy compression under individual distortion constraints, where each source component has its own allowable distortion level.

When is the Hadamard-inequality-based RDF lower bound tight?

The bound becomes tight if and only if the semidefinite condition (SDC) holds; if the SDC fails, the bound is not tight and becomes loose.

How do source correlations affect the rate-distortion function in this work?

The paper provides new theoretical results that explicitly incorporate source correlations into the RDF when the SDC holds, and further examines how correlations determine the optimal distortion (rate) allocations across regions.

Jul 16, 2026

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