Polar codes for distributed hierarchical source coding

Hierarchical source coding, also known as successive refinement of information, was introduced by Koshelev [1,2] and Equitz and Cover [3]. This problem is concerned with the construction of a source code for a discrete memoryless source X with respect to a given distortion measure d1 that can be further refined to represent the same source within another distortion measure d2 so that both representations approach the best possible compression rates for the given distortion values. This property is also termed divisibility of sources and it has an obvious interpretation in the context of ǫ-nets in metric spaces [4]. Koshelev found a sufficient condition for successive refinement in [1], showing that the source is divisible if the coarse descriptionX1 is independent of X given the fine descriptionX2, and Equitz and Cover showed that this condition is also necessary. If the Markov condition is not satisfied, then attaining the rate-distortion functions for both descriptions is impossible, and the excess rate needed to represent the source was quantified by Rimoldi [5] (see also Koshelev [1]). As observed in [3], successive refinement is a particular case of the multiple description problem for which the region of achievable rates was established by Ahlswede [6] and El Gamal and Cover [7]. A constructive scheme based on codes with low-density generator matrices together with message-passing encoding was presented by Zhang et al. [8]. A version of the successive refinement problem that incorporates side information used to represent the source was considered by Steinberg and Merhav in [9]. In this problem, the side information is expressed as a pair of random variables that form a Markov chain with the source random variables and are used at the initial stage of representing the source and at the refinement stage, respectively. Paper [9] found the minimum possible rates for reproducing the source at the given distortion levels in the presence of side information. The aim of this paper is to construct an explicit scheme for successive refinement for the aforementioned problems using polar codes. Polar codes were initially designed to support communication at rates approaching capacity of binary-input symmetric memoryless channels [10]. Subsequently they were shown to approach optimal performance for a number of information-theoretic problems with two or more users. A sampling of results includes lossless and lossy source coding problems [11, 12], multiple-access channels [13], the degraded wiretap channel [14], as well as a range of other problems that previously relied on random coding (see the recent preprint [15] for a more detailed overview of applications of polar codes). Recently Honda and Yamamoto [16] showed that it is possible to modify the construction of polar codes so that the coding scheme supports capacity-achieving communication for channels that are not necessarily symmetric. This result paves way for new applications of polar codes such as achieving optimal rates for broadcast channels [15]. In this paper we note that the asymmetric polar coding scheme can be also used for achieving rate-distortion functions in a range of problems of multiterminal source beginning with the basic successive