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

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.

💡 Research Summary

This paper studies the joint lossy compression of an N‑dimensional Gaussian source when each component is subject to its own distortion constraint. The classical lower bound on the rate‑distortion function (RDF) derived from the Hadamard inequality,
(R_{\text{low}}=\frac12\log\frac{\det K}{\det E}) (where (K) is the source covariance and (E=\operatorname{diag}(e_1,\dots,e_N)) contains the individual distortion limits), becomes exact only when the semidefinite condition (SDC) (K\succeq E) holds. Existing literature largely ignores the case where the SDC fails and provides no quantitative insight into how source correlations improve compression when the condition is satisfied.

The authors first reformulate the RDF as a convex “max‑det” problem and apply KKT conditions to characterize the optimal distortion matrix (D^\star) and the optimal reconstruction covariance (K_{\hat x^\star}=K-D^\star). Theorem 1 shows that if the SDC is satisfied and inactive ((K-D^\star\succ0)), then the Lagrange multiplier associated with the semidefinite constraint vanishes, forcing (D^\star=E) and making every individual distortion constraint tight. Conversely, when the SDC is violated, the optimal distortion matrix cannot be diagonal; instead (\det(K-D^\star)=0), meaning the optimal reconstruction is rank‑deficient. Hence, failure of the SDC inevitably leads to a reduction in the number of independent dimensions needed for reconstruction.

Theorem 2 provides two complementary upper bounds on the rank of the optimal reconstruction covariance:
\