Second-Order and Moderate Deviation Asymptotics for Successive Refinement

Abstract

We derive the optimal second-order coding region and moderate deviations constant for successive refinement source coding with a joint excess-distortion probability constraint. We consider two scenarios: (i) a discrete memoryless source (DMS) and arbitrary distortion measures at the decoders and (ii) a Gaussian memoryless source (GMS) and quadratic distortion measures at the decoders. For a DMS with arbitrary distortion measures, we prove an achievable second-order coding region, using type covering lemmas by Kanlis and Narayan and by No, Ingber and Weissman. We prove the converse using the perturbation approach by Gu and Effros. When the DMS is successively refinable, the expressions for the second-order coding region and the moderate deviations constant are simplified and easily computable. For this case, we also obtain new insights on the second-order behavior compared to the scenario where separate excess-distortion proabilities are considered. For example, we describe a DMS, for which the optimal second-order region transitions from being characterizable by a bivariate Gaussian to a univariate Gaussian, as the distortion levels are varied. We then consider a GMS with quadratic distortion measures. To prove the direct part, we make use of the sphere covering theorem by Verger-Gaugry, together with appropriately-defined Gaussian type classes. To prove the converse, we generalize Kostina and Verdú's one-shot converse bound for point-to-point lossy source coding. We remark that this proof is applicable to general successively refinable sources. In the proofs of the moderate deviations results for both scenarios, we follow a strategy similar to that for the second-order asymptotics and use the moderate deviations principle.

The Dispersion of Mismatched Joint Source-Channel Coding for Arbitrary Sources and Additive Channels

“…We remark that the choice of M i for any i ∈ [1 : N ] is universal because it only depends on the quantization level Υ(i) (see (5)), which is fixed a priori, and the type of source codebook (see (10) and (12)). It does not depend on the source codebook realization.…”

“…To describe our NN-JSCC scheme, we resort to a framework that is ubiquitous in joint source-channel coding, e.g., [6], [7]. We define the notion of power types for positive reals similar to [10]. Let ξ be a positive number.…”

Exponential Strong Converse for Successive Refinement with Causal Decoder Side Information

“…Recall that we use T (see (8)) to denote the collection of random variables (X, Y k , S k ,X k ) and use t, T similarly to denote a realization of T and its alphabet, respectively. For any P T ∈ P sh (recall (55)), any (α k , β k ) ∈ [0, 1] 2k satisfying (15) and any λ ∈ R + , for any t ∈ T , definẽ…”

Non-Asymptotic Bounds and a General Formula for the Rate-Distortion Region of the Successive Refinement Problem

“…Recently, the asymptotic analysis of the second-order rates to the blocklength becomes an active target of the study. Especially, for the successive refinement problem, No et al [9] and Zhou et al [10] gave a lot of results to the set of second-order rates for discrete and Gaussian stationary memoryless sources. No et al [9] considered separate excess-distortion criteria such that a probability that a distortion exceeds a given distortion level is less than a given probability level separately for each reconstruction.…”

“…No et al [9] considered separate excess-distortion criteria such that a probability that a distortion exceeds a given distortion level is less than a given probability level separately for each reconstruction. On the other hand, Zhou et al [10] considered the joint excess-distortion criterion such that a probability that either of distortions exceeds a given distortion level is less than a given probability level. Although they also gave several nonasymptotic bounds on the set of pairs of rates, they mainly focus on the asymptotic behavior of the set.…”