On the sum-rate loss of quadratic Gaussian multiterminal source coding

This work studies the sum-rate loss of quadratic Gaussian multiterminal source coding, i.e., the difference between the minimum sum-rates of distributed encoding and joint encoding (both with joint decoding) of correlated Gaussian sources subject to MSE distortion constraints on individual sources. It is shown that under the non-degraded assumption, i.e., all target distortions are simultaneously achievable by a Berger-Tung scheme, the supremum of the sum-rate loss of distributed encoding over joint encoding of L jointly Gaussian sources increases almost linearly in the number of sources L, with an asymptotic slope of 0.1083 b/s per source as L goes to infinity. This result is obtained even though we currently do not have the full knowledge of the minimum sum-rate for the distributed encoding case. The main idea is to upper-bound the minimum sum-rate of multiterminal source coding by that achieved by parallel Gaussian test channels while lower-bounding the minimum sum-rate of joint encoding by a reverse water-filling solution to a relaxed joint encoding problem of the same set of Gaussian sources with a sum-distortion constraint (that equals the sum of the individual target distortions). We show that under the non-degraded assumption, the supremum difference between the upper bound for distributed encoding and the lower bound for joint encoding is achieved in the bi-eigen equal-variance with equal distortion case, in which both bounds are known to be tight.