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Joint Lossy Compression for a Vector Gaussian Source under Individual Distortion Criteria

Shuao Chen, Junyuan Gao, Yuxuan Shi, Yongpeng Wu, Giuseppe Caire, H. Vincent Poor, Wenjun Zhang

TL;DR

The paper tackles joint lossy compression of a vector Gaussian source under per-component distortion constraints, examining both cases where the semidefinite condition (SDC) holds and where it does not. It develops a backward test-channel framework and leverages KKT conditions to characterize the optimal reconstruction, establishing upper bounds on the reconstruction dimension and showing that lower-dimensional reconstructions emerge when the SDC fails. The work then specializes to a scalable two-type correlation (2TC) covariance model, deriving a closed-form SDC region and RDF that explicitly incorporate source correlations, and proving that the probability of SDC satisfaction decays exponentially with source length. Through both analytic results and simulations, the paper quantifies how correlations reduce the rate under fixed distortions and provides practical insights into exploiting correlation structure to improve compression efficiency.

Abstract

This paper investigates the joint compression problem of a vector Gaussian source, where an individual distortion constraint is imposed on each source component. It is known that the rate-distortion function (RDF) is lower-bounded by the rate derived from the Hadamard inequality, which becomes exact when the semidefinite condition (SDC) holds. However, existing works often overlook the case where the SDC is not satisfied. Moreover, even when the SDC holds, a quantitative characterization of how correlations enable more efficient compression is lacking. In this work, we refine the results when the SDC is satisfied and derive new theoretical results when the SDC is not satisfied, thereby establishing theoretical limits for practical source compression with correlations. Specifically, we examine the properties of optimal source reconstruction and provide upper bounds on its dimension, showing that lower-dimensional reconstructions are essential for efficient compression when the SDC does not hold. Within a scalable two-type correlation (2TC) covariance framework, we prove that the probability of satisfying the SDC decays exponentially with source length, emphasizing the importance of exploring theoretical limits when the SDC is not met. Additional, we determine the component-wise correlations that a vector source should possess to achieve the Hadamard compression rate, revealing the trade-off between distortion constraints and correlations. More importantly, by deriving an explicit RDF with correlations incorporated, we quantitatively characterize the gain in compression efficiency achieved by fully leveraging source correlations.

Joint Lossy Compression for a Vector Gaussian Source under Individual Distortion Criteria

TL;DR

The paper tackles joint lossy compression of a vector Gaussian source under per-component distortion constraints, examining both cases where the semidefinite condition (SDC) holds and where it does not. It develops a backward test-channel framework and leverages KKT conditions to characterize the optimal reconstruction, establishing upper bounds on the reconstruction dimension and showing that lower-dimensional reconstructions emerge when the SDC fails. The work then specializes to a scalable two-type correlation (2TC) covariance model, deriving a closed-form SDC region and RDF that explicitly incorporate source correlations, and proving that the probability of SDC satisfaction decays exponentially with source length. Through both analytic results and simulations, the paper quantifies how correlations reduce the rate under fixed distortions and provides practical insights into exploiting correlation structure to improve compression efficiency.

Abstract

This paper investigates the joint compression problem of a vector Gaussian source, where an individual distortion constraint is imposed on each source component. It is known that the rate-distortion function (RDF) is lower-bounded by the rate derived from the Hadamard inequality, which becomes exact when the semidefinite condition (SDC) holds. However, existing works often overlook the case where the SDC is not satisfied. Moreover, even when the SDC holds, a quantitative characterization of how correlations enable more efficient compression is lacking. In this work, we refine the results when the SDC is satisfied and derive new theoretical results when the SDC is not satisfied, thereby establishing theoretical limits for practical source compression with correlations. Specifically, we examine the properties of optimal source reconstruction and provide upper bounds on its dimension, showing that lower-dimensional reconstructions are essential for efficient compression when the SDC does not hold. Within a scalable two-type correlation (2TC) covariance framework, we prove that the probability of satisfying the SDC decays exponentially with source length, emphasizing the importance of exploring theoretical limits when the SDC is not met. Additional, we determine the component-wise correlations that a vector source should possess to achieve the Hadamard compression rate, revealing the trade-off between distortion constraints and correlations. More importantly, by deriving an explicit RDF with correlations incorporated, we quantitatively characterize the gain in compression efficiency achieved by fully leveraging source correlations.
Paper Structure (12 sections, 7 theorems, 31 equations, 3 figures)

This paper contains 12 sections, 7 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Setup
  4. Definition of RDF
  5. Source Compression with Arbitrary Covariance
  6. Source Compression with 2TC Covariance
  7. 2TC Framework for Covariance Modeling
  8. Characterization of SDC Region
  9. A Distortion-Constraint Perspective
  10. A Source-Correlation Perspective
  11. Simulation Results
  12. Conclusion

Key Result

Lemma 1

RDF in Eq. eq:vector_rdf_definition is given by the solution to where ${\sf d} = \mathsf{diag}(\mathsf{D})$ is the vector of diagonal elements of $\mathsf{D}$, and ${\sf e} = [e_1, \cdots, e_N]^{\sf T}$ is the normalized distortion constraint vector. ${\sf d} \le {\sf e}$ means $d_i \le e_i$ for all $i \in [N]$.

Figures (3)

  • Figure 1: Compression rate per component in Eqs. \ref{['eq:rd sdc iso']} and \ref{['eq:alliso rate per asym']} versus source length under identical distortion constraint $e=0.25$ in Theorems \ref{['thm:RE_psd_conditions']}.
  • Figure 2: The probability of SDC satisfaction versus the source length in Theorem \ref{['thm:sdc exponential asymp']} for $\rho_1=0.45$ and all distortion constraints $e_i \sim \mathcal{U}[0,1]$.
  • Figure 3: Evaluation of $\rho_0^m$ versus source length in Theorem \ref{['thm:max rho0']} with fixed $e_2 = 0.1$.

Theorems & Definitions (10)

  • Remark 1
  • Definition 1
  • Lemma 1: Formulation of RDF under the individual distortion criteria, xiao2005compression
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Theorem 5