Compress and Estimate in Multiterminal Source Coding

We consider a multiterminal remote source coding problem in which a source sequence is estimated from the output of multiple source encoders, each having access only to a noisy observation of the source realization. Each remote encoder compresses its noisy observation sequence so as to minimize a local distortion measure which depends only on the distribution of its observed sequence, and is otherwise independent from the distribution of the underlying source. The latter is finally estimated at a central location from the output of each of the remote encoders. This source compression and estimation scenario leads to an achievable scheme for the remote multiterminal source coding problem which we term the “compress-and-estimate” (CE) scheme. For the case of an i.i.d source observed through multiple memoryless channels, we derive a single-letter expression for performance in the CE scheme, which we refer to as the CE distortion-rate function (CE-DRF). We prove that the CE-DRF can be achieved by estimating the source realization from the output of any set of encoders, as long as each encoder attains its local rate-distortion functions. Moreover we show that, for large enough blocklength, the distortion in estimating a finite sub-block of the source from the output of such encoders, averaged over all sub-blocks, does not exceeds the CE-DRF. Finally, we derive closed-form expressions for the CE-DRF in the case of a Gaussian source observed through multiple AWGN channels under quadratic distortion, and for the case of a binary source observed through multiple biflip channels under Hamming distortion. By comparing the resulting expressions to known optimal source coding performance, we highlight the performance lost due to a sub-optimal source code design which can be though of as arising from the lack of statistical knowledge on the relationship between source and observation at the encoders.