Rate-Distortion Function for a Heegard-Berger Problem With Two Sources and Degraded Reconstruction Sets

Abstract

In this work, we investigate an instance of the Heegard-Berger problem with two sources and arbitrarily correlated side information sequences at two decoders, in which the reconstruction sets at the decoders are degraded. Specifically, two sources are to be encoded in a manner that one of the two is reproduced losslessly by both decoders, and the other is reproduced to within some prescribed distortion level at one of the two decoders. We establish a single-letter characterization of the rate-distortion function for this model.The investigation of this result in some special cases also sheds light on the utility of joint compression of the two sources. Furthermore, we also generalize our result to the setting in which the source component that is to be recovered by both users is reconstructed in a lossy fashion, under the requirement that all terminals (i.e., the encoder and both decoders) can share an exact copy of the compressed version of this source component, i.e., a common encoder-decoders reconstruction constraint. For this model as well, we establish a single-letter characterization of the associated rate-distortion function.M. Benammar is with the Mathematics and Algorithmic Sciences Lab.,

Rate-Distortion Region of a Gray–Wyner Model with Side Information

“…As for the proof of achievability, it combines the optimal coding scheme of the Heegard–Berger problem with degraded reconstruction sets [ 9 ] and the double-binning based scheme of Shayevitz and Wigger (Theorem 2, [ 4 ]) for the Gray–Wyner problem with side information, and is outlined in the following.…”

“…The Gray–Wyner model with side information generalizes another long standing open source coding problem, the famous Heegard–Berger problem [ 2 ]. Full single-letter characterization of the optimal rate-distortion function of the Heegard–Berger problem is known only in few specific cases, the most important of which are the cases of : (i) stochastically degraded side information sequences [ 2 ] (see also [ 5 ]); (ii) Sgarro’s result [ 6 ] on the corresponding lossless problem; (iii) Gaussian sources with quadratic distortion measure [ 3 , 7 ]; (iv) some instances of conditionally less-noisy side information sequences [ 8 ]; and (v) the recently solved HB model with general side information sequences and degraded reconstruction sets [ 9 ], i.e., the model of Figure 2 with — in the lossless case, a few other optimal results were shown, such as for the so-called complementary delivery [ 10 ]. A lower bound for general instances of the rate distortion problem with side information at multiple decoders, which is inspired by a linear-programming lower bound for index coding, has been developed recently by Unal and Wagner in [ 11 ].…”

LP Bounds for Rate-Distortion With Variable Side Information

“…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].…”

“…Although X is a vector, we can view it as an ordered set which also induces an ordered set structure on the subsets. Hence, we can use the set notation whenever it is convenient 6. The statement of the result in[10] does not contain the (monotonicity) condition, although it is clear from the proof that it was intended to be included.…”

“…a ⊕ b denotes componentwise exclusive-OR operation where the shorter vector is zero padded as necessary 8. In a recent work of Benammar et al[6], the rate-distortion problem with two decoders having degraded reconstruction sets is considered and the corresponding rate-distortion function is characterized. The construction of auxiliary random variables in the converse result of Benammar et al[6] is specific to that problem setting and at this point it is unclear whether the LP lower bound subsumes this converse result.…”