Rate Distortion With Side-Information at Many Decoders

Abstract

We present a new inner bound for the rate region of the t-stage successive-refinement problem with side-information. We also present a new upper bound for the rate-distortion function for lossysource coding with multiple decoders and side-information. Characterising this rate-distortion function is a long-standing open problem, and it is widely believed that the tightest upper bound is provided by Theorem 2 of Heegard and Berger's paper "Rate Distortion when Side Information may be Absent," IEEE Trans. Inform. Theory, 1985. We give a counterexample to Heegard and Berger's result. with side-information (fig. 4). A brief history of the literature on these problems is as follows. 1 Source q(x, y)Destination Fig. 1. The Wyner-Ziv Problem: (X, Y) = (X1, Y1), (X2, Y2), . . ., (Xn, Yn) is an iid random sequence emitted by a source q(x, y) = Pr[X = x, Y = y]. The encoder maps X to an index M , which belongs to a finite set M , at a rate r. Using M and Y, the decoder is required to generate a replicaX =X1,X2, . . . ,Xn of X to within an average distortion d, according to (1).The rate-distortion function R(d) is defined as the smallest rate for which such a reconstruction is possible. A single-letter expression for this function was first given in [1, Thm. 1].October 24, 2018 DRAFT A. The Wyner-Ziv Problem with t-DecodersSuppose that the side-information Y in Figure 1 is unreliable in the sense that it may or may not be available to the decoder. If the encoder does not know a priori when Y is available, thenWyner and Ziv's coding argument for (2) fails, and a more sophisticated argument is required to exploit Y. This observation inspired Kaspi [4] in 1980 (published by Wyner on behalf of Kaspi in 1994) as well as Heegard and Berger [5] in 1985 to independently study the problem shown in fig. 2 -the Kaspi/Heegard-Berger problem. As with the Wyner-Ziv problem, the objective of this problem is to characterise the corresponding rate-distortion function R (d 1 , d 2 ). That is, to find the smallest rate such that decoders 1 and 2 can produce replicasX 1 andX 2 of X to within average distortions d 1 and d 2 , respectively. To this end, Heegard and Berger [5, Thm. 1] showed that 1where the minimization is taken over all choices of two auxiliary random variables, U and W , that are jointly distributed with (X, Y ) and which satisfy the following two properties: (1) (U, W ) is conditionally independent of Y given X; and (2) there exist functionsX 1 (W ) and X 2 (Y, U, W ) with Eδ(X,X 1 (W )) ≤ d 1 and Eδ(X,X 2 (Y, U, W )) ≤ d 2 , respectively. The Kaspi/Heegard-Berger problem in Figure 2 was further generalised by Heegard and Berger in [5, Sec. VII] to the problem shown in Figure 3. There are t-decoders, each with different side-information, and the objective is to characterise the corresponding rate-distortion function R(d). Unfortunately, this function has eluded characterisation for all but a few special cases. For example, Heegard and Berger [5, Thm. 3] have characterised R(d) for stochastically degraded side-information 2 ;...

Vector Gaussian rate-distortion with variable side information

“…4 The distortion constraints in (20), (22), and (18) also imply that K X|Y 1 and K X|Y 2 are positive definite matrices. 5 Note that D 1 and D 2 are positive definite since D …”

“…For three separate special cases, we show that at least one of the lower bounds matches the best-known achievable rate [1], [5], thereby determining the optimal rate. The four lower bounds are all obtained using variations on the following argument.…”

“…), then the rate distortion function is known [1], [5]. Hence a natural way to obtain a lower bound to R(D) is to create degraded problems by providing extra side information to one decoder or the other.…”

Source coding with conditionally less noisy side information

“…The best achievability result (upper bound to R * ) can be distilled from [1], [2], [7] and is summarised next.…”

“…The described problem is a special case of the rate-distortion functions in [1], [2]. Single-letter expressions for R * are known in the following three special cases: (i) equal source components X = Y [8]; (ii) complementary side information U = Y and V = X [3]; and (iii) degraded side information (X, Y ) − −U − −V [1].…”