LP Bounds for Rate-Distortion With Variable Side Information

Abstract

We consider a rate-distortion problem with side information at multiple decoders. Several upper and lower bounds have been proposed for this general problem or special cases of it. We provide an upper bound for general instances of this problem, which takes the form of a linear program, by utilizing random binning and simultaneous decoding techniques [1] and compare it with the existing bounds. We also provide a lower bound for the general problem, which was inspired by a linear-programming lower bound for index coding, and show that it subsumes most of the lower bounds in literature. Using these upper and lower bounds, we explicitly characterize the rate-distortion function of a problem that can be seen as a Gaussian analogue of the "odd-cycle" index coding problem. I. INTRODUCTIONWe consider the rate-distortion tradeoff for a canonical problem in source coding: an encoder with access to a source of interest broadcasts a single message to multiple decoders, each endowed with side information about the source. Each decoder then wants to reproduce the source subject to a distortion constraint. This is essentially the multiple-decoder extension of the Wyner-Ziv [2] problem, sometimes referred to as the Heegard-Berger [3] problem.Even for the two-decoder case, the complete characterization of the rate-distortion function is a long-standing open problem. However, the rate-distortion function has been determined in several special cases, including when the side information at the various decoders can be ordered according to stochastic degradedness [3], when there are two decoders whose side information is "mismatch degraded" [4], and when there are two decoders and the side information at decoder 2 is "conditionally less noisy" than the side information at decoder 1 and decoder 1 seeks to losslessly reproduce a deterministic function of the source [5]. Also, instead of imposing some degraded structure on the side information, one can consider degraded reconstruction sets at the two decoders in which one component of the source is reconstructed at both decoders with vanishing block error probability and the other component of the source is only reconstructed at a single decoder [6]. Various vector Gaussian instances of the problem are solved [7], [8]. Several instances of the index coding problem, which is an important special case, have also been solved (e.g., [9]-[11]).Upper and lower bounds on the rate-distortion function in the general case are also available. Existing achievable schemes proceed by crafting separate messages for different subsets of decoders, which are encoded and decoded in a fixed order using random binning [3], [4], [12]. Our first contribution is to show how such schemes can be improved using simultaneous decoding [1], in which each decoder decodes all of its messages at once instead of sequentially. The resulting achievable bound involves optimizing over auxiliary random variables and, for each choice of such variables, solving a linear program (LP). Prior to this work, the best achievable bou...