On common information and the encoding of sources that are not successively refinable
This paper focuses on a new framework for scalable coding of information based on principles derived from common information of two dependent random variables. In the conventional successive refinement setting, the encoder generates two layers of information called the base layer and the enhancement layer. The first decoder, which receives only the base layer, produces a coarse reconstruction of the source, whereas the second decoder, which receives both the layers, uses the enhancement layer to refine the information further leading to a finer reconstruction. It is popularly known that asymptotic rate-distortion optimality at both the decoders is possible if and only if the source-distortion pair is successively refinable. However when the source is not successively refinable under the given distortion metric, it is impossible to achieve rate-distortion optimality at both the layers simultaneously. For this reason, most practical system designers resort to storing two individual representations of the source leading to significant overhead in transmission/storage costs. Inspired by the breadth of applications, in this paper, we propose a new framework for scalable coding wherein a subset of the bits sent to the first decoder is not sent to the second decoder. That is, the encoder generates one common bit stream which is routed to both the decoders, but unlike the conventional successive refinement setting, both the decoders receive an additional individual bitstream. By relating the proposed framework with the problem of common information of two dependent random variables, we derive a single letter characterization for the minimum sum rate achievable for the proposed setting when the two decoders are constrained to receive information at their respective rate-distortion functions. We show using a simple example that the proposed framework provides a strictly better asymptotic sum rate as opposed to the conventional scalable coding setup when the source-distortion pair is not successively refinable.