A new converse in rate-distortion theory

Tan

IEEE Trans. Inform. Theory

2017

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.

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Section: Existing Resultsmentioning

confidence: 99%

Section: Existing Resultsmentioning

confidence: 99%

Section: A Main Contributionsmentioning

confidence: 99%

“…We remark that for a successively refinable discrete memoryless source-distortion measure triplet, (39) is replaced by [14,Property 1].…”

Section: A Tilted Information Densitymentioning

confidence: 99%

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Second-Order and Moderate Deviation Asymptotics for Successive Refinement

Tan

IEEE Trans. Inform. Theory

2017

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“…For the Kaspi problem, we first present the properties of optimal test channels and a parametric representation for the ratedistortion function. Second, we generalize the notion of the distortion-tilted information density for the lossy source coding problem [12] to the Kaspi problem and derive a non-asymptotic converse bound. The non-asymptotic converse bound holds for abstract sources, not restricted to DMSes.…”

Section: B Main Contributionsmentioning

confidence: 99%

Non-Asymptotic Converse Bounds and Refined Asymptotics for Two Source Coding Problems

IEEE Trans. Inform. Theory

2019

In this paper, we revisit two multi-terminal lossy source coding problems: the lossy source coding problem with side information available at the encoder and one of the two decoders, which we term as the Kaspi problem (Kaspi, 1994), and the multiple description coding problem with one semi-deterministic distortion measure, which we refer to as the Fu-Yeung problem (Fu and Yeung, 2002). For the Kaspi problem, we first present the properties of optimal test channels. Subsequently, we generalize the notion of the distortion-tilted information density for the lossy source coding problem to the Kaspi problem and prove a nonasymptotic converse bound using the properties of optimal test channels and the well-defined distortion-tilted information density. Finally, for discrete memoryless sources, we derive refined asymptotics which includes the second-order, large and moderate deviations asymptotics. In the converse proof of second-order asymptotics, we apply the Berry-Esseen theorem to the derived non-asymptotic converse bound. The achievability proof follows by first proving a type-covering lemma tailored to the Kaspi problem, then properly Taylor expanding the well-defined distortion-tilted information densities and finally applying the Berry-Esseen theorem. We then generalize the methods used in the Kaspi problem to the Fu-Yeung problem. As a result, we obtain the properties of optimal test channels for the minimum sum-rate function, a non-asymptotic converse bound and refined asymptotics for discrete memoryless sources. Since the successive refinement problem is a special case of the Fu-Yeung problem, as a byproduct, we obtain a non-asymptotic converse bound for the successive refinement problem, which is a strict generalization of the non-asymptotic converse bound for successively refinable sources (Zhou, Tan and Motani, 2017). Index TermsLossy source coding, multiple description coding problem, Non-asymptotic converse bound, Second-order asymptotics, Large Deviations, Moderate DeviationsThe authors are with the .sg. Part of this paper will be presented at Globecom 2017 [1], [2]. i.e., the second-order (cf. [13], [34], [35]), the large deviations (cf. [36], [37]) and the moderate deviations (cf. [38], [39])asymptotics. The converse proof of the second-order asymptotics follows by applying the Berry-Esseen theorem to the derived non-asymptotic converse bound and the achievability part relies on a type-covering lemma tailored to the Kaspi problem, proper uses of the Taylor's expansions, and the properties of the distortion-tilted information density. Since the Kaspi problem is a generalization of the lossy source coding problem and the lossy source coding problem with encoder and decoder side information, our second-order asymptotics for the Kaspi problem is a generalization of [13] and [40] for DMSes. We illustrate this point via numerical examples.We then extend the methods used in the Kaspi problem to the Fu-Yeung problem. First, we present the optimal test channels for the minimum sum-rate function. Subsequently,...

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Kaspi Problem Revisited: Non-Asymptotic Converse Bound and Second-Order Asymptotics

GLOBECOM 2017 - 2017 IEEE Global Communications Conference

2017