Bounds on the benefits of interaction in distributed source coding for function computation

In this paper, we study a setting in which two terminals A and B respectively observe, or measure, two memoryless, possibly statistically dependent, sources X and Y; and they interact bidirectionally in the aim of computing, at terminal B, a function fB(X;Y) of the two sources. Essentially, we establish upper bounds on the maximum gain that can be brought up by the interaction, in terms of minimum sum rate improvement for a given average distortion. In particular, we show that this gain is bounded by the redundancy of the one-message minimal rate for computing the function fB(X;Y) in the case in which Terminal A does not know the side information Y over the one-message minimal rate for computing the same function with the same tolerance but with Terminal A informed about the side information Y. That is, the redundancy of the one-message Wyner-Ziv rate-distortion function for function computation over the one-message conditional rate-distortion function for function computation. In the special case of lossy source reproduction, the bound reduces to the rate loss of the Wyner-Ziv problem as studied by Zamir in the case of a difference distortion measure. In the case of lossy function computation, we use this bound to establish an alternate bound that is generally easier to compute. Furthermore, we also apply the results to some important special cases, thus allowing us to gain some fundamental insights on the benefits of the interaction for both lossless and lossy function computations in these cases.