Generalized Gaussian Multiterminal Source Coding: The Symmetric Case

Consider a generalized multiterminal source coding system, where <inline-formula> <tex-math notation="LaTeX">$\binom{\ell }{ m}$ </tex-math></inline-formula> encoders, each observing a distinct size-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> subset of <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$\ell \geq 2$ </tex-math></inline-formula>) zero-mean unit-variance exchangeable Gaussian sources with correlation coefficient <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula>, compress their observations in such a way that a joint decoder can reconstruct the sources within a prescribed mean squared error distortion based on the compressed data. The optimal rate-distortion performance of this system was previously known only for the two extreme cases <inline-formula> <tex-math notation="LaTeX">$m=\ell $ </tex-math></inline-formula> (the centralized case) and <inline-formula> <tex-math notation="LaTeX">$m=1$ </tex-math></inline-formula> (the distributed case), and except when <inline-formula> <tex-math notation="LaTeX">$\rho =0$ </tex-math></inline-formula>, the centralized system can achieve strictly lower compression rates than the distributed system under all non-trivial distortion constraints. Somewhat surprisingly, it is established in the present paper that the optimal rate-distortion performance of the afore-described generalized multiterminal source coding system with <inline-formula> <tex-math notation="LaTeX">$m\geq 2$ </tex-math></inline-formula> coincides with that of the centralized system for all distortions when <inline-formula> <tex-math notation="LaTeX">$\rho \leq 0$ </tex-math></inline-formula> and for distortions below an explicit positive threshold (depending on <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>) when <inline-formula> <tex-math notation="LaTeX">$\rho > 0$ </tex-math></inline-formula>. Moreover, when <inline-formula> <tex-math notation="LaTeX">$\rho > 0$ </tex-math></inline-formula>, the minimum achievable rate of generalized multiterminal source coding subject to an arbitrary positive distortion constraint <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> is shown to be within a finite gap (depending on <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>) from its centralized counterpart in the large <inline-formula> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> limit except for possibly the critical distortion <inline-formula> <tex-math notation="LaTeX">$d=1-\rho $ </tex-math></inline-formula>.