Rate region of the vector Gaussian one-helper source-coding problem

Abstract

We determine the rate region of the vector Gaussian one-helper source-coding problem under a covariance matrix distortion constraint. The rate region is achieved by a simple scheme that separates the lossy vector quantization from the lossless spatial compression. The converse is established by extending and combining three analysis techniques that have been employed in the past to obtain partial results for the problem.

Rate Region of the Vector Gaussian One-Helper Source-Coding Problem

“…This addresses the first issue. One can verify that if X is a vector and Y is a scalar, then the second and third issues do not arise, and hence distortion projection together with Oohama's converse arguments is sufficient to solve the problem [7]. Liu and Viswanath [4] showed that the channel enhancement technique of Weingarten et al [3] is sufficient to solve the helper problem in the vector case, thereby addressing the third issue.…”

“…Source enhancement and Oohama's converse technique are lifted directly from [1] and [6]. The distortion projection, on the other hand, requires an extension beyond what was needed in the scalar helper case [7]. This extension requires us to first establish several properties of the optimal Gaussian solution to the problem, to which we turn next.…”

“…In prior work [7], we showed that it is possible to reduce the general case to this one by projecting the source and the distortion constraint in the directions in which the distortion constraint is met with equality for the candidate optimal scheme. We call this process distortion projection.…”

“…The case in which Y is a scalar and X is a vector was solved by the authors [7]. The proof did not use enhancement, but it did require a novel technique that we call distortion projection.…”

Source Coding in Networks With Covariance Distortion Constraints

“…The Gaussian one-helper problem with the covariance matrix distortion constraint was studied in [13], [14]. Here in addition to a main encoder which observes the direct source and sends a message to the decoder with rate R 1 , there is a helper with noisy observations, which sends a message with rate R 2 to the decoder.…”