关于具有个体失真约束的向量源的高斯 - 二次速率 - 失真函数
On the Gaussian-Quadratic Rate-Distortion Function for Vector Sources with Individual Distortion Constraints
AI总结：
研究具有个体失真约束的向量源的高斯 - 二次速率 - 失真函数，通过提出基于分层相关性的源协方差矩阵类等方法，在不同条件下得到最优源重构谱性质、完整RDF等结果，揭示相关性对降低压缩成本的作用。
AI中文摘要：
本文研究了在个体失真约束下任意源长度的高斯 - 二次有损压缩。速率 - 失真函数（RDF）由基于哈达玛不等式的速率下界界定，当且仅当半正定条件（SDC）成立时该界紧密，否则界变宽松且缺乏解析结果，源相关性与RDF的基本定量关系也不完整。本文在不同源协方差矩阵和失真约束下给出新理论结果。首先，在任意协方差和失真约束下，得到实现RDF的最优源重构的谱性质及更强的SDC标量不等式版本；提出基于分层相关性的源协方差矩阵类，表明研究两类型相关（2 - TC）模型足以建立更广泛类的解析基础，在此协方差下得到SDC成立时明确纳入源相关性的RDF，并从失真约束和源相关性角度分析SDC。其次，在2 - TC协方差和两类型失真（2 - TD）约束下，在失真平面上七个区域建立完整的RDF，确定每个区域的最优失真（速率）分配，揭示追求完整RDF的本质在于深入分析最优失真之间的相关性。最后，在各向同性相关和相同约束下，给出每个分量的压缩率，表明利用相关性可显著降低压缩成本。
英文摘要：
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.
