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

In the successive refinement problem, a fixed-length sequence emitted from an information source is encoded into two codewords by two encoders in order to give two reconstructions of the sequence. One of two reconstructions is obtained by one of two codewords, and the other reconstruction is obtained by all two codewords. For this coding problem, we give non-asymptotic inner and outer bounds on pairs of numbers of codewords of two encoders such that each probability that a distortion exceeds a given distortion level is less than a given probability level. We also give a general formula for the rate-distortion region for general sources, where the rate-distortion region is the set of rate pairs of two encoders such that each maximum value of possible distortions is less than a given distortion level.

[1]  Hirosuke Yamamoto.  Source coding theory for a triangular communication system , 1996, IEEE Trans. Inf. Theory.

[2]  Tomohiko Uyematsu,et al.  Revisiting the rate-distortion theory using smooth max Rényi divergence , 2014, 2014 IEEE Information Theory Workshop (ITW 2014).

[3]  Bixio Rimoldi,et al.  Successive refinement of information: characterization of the achievable rates , 1994, IEEE Trans. Inf. Theory.

[4]  Tomohiko Uyematsu,et al.  New Non-Asymptotic Bounds on Numbers of Codewords for the Fixed-Length Lossy Compression , 2016, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[5]  Ertem Tuncel,et al.  Successive Refinement of Abstract Sources , 2019, IEEE Transactions on Information Theory.

[6]  Naqueeb Ahmad Warsi,et al.  One-shot bounds for various information theoretic problems using smooth min and max Rényi divergences , 2013, 2013 IEEE Information Theory Workshop (ITW).

[7]  Hiroki Koga,et al.  Information-Spectrum Methods in Information Theory , 2002 .

[8]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[9]  Sergio Verdú,et al.  Fixed-Length Lossy Compression in the Finite Blocklength Regime , 2011, IEEE Transactions on Information Theory.

[10]  Tsachy Weissman,et al.  Strong Successive Refinability and Rate-Distortion-Complexity Tradeoff , 2015, IEEE Transactions on Information Theory.

[11]  Uyematsu Tomohiko,et al.  A General Formula of the Achievable Rate Region for the Successive Refinement Problem , 2016 .

[12]  Ertem Tuncel,et al.  The rate-distortion function for successive refinement of abstract sources , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[13]  Michelle Effros.  Distortion-rate bounds for fixed- and variable-rate multiresolution source codes , 1999, IEEE Trans. Inf. Theory.

[14]  Mehul Motani,et al.  Second-Order and Moderate Deviations Asymptotics for Successive Refinement , 2016, IEEE Transactions on Information Theory.

[15]  A. Kanlis,et al.  Error exponents for successive refinement by partitioning , 1995, Proceedings of 1995 IEEE International Symposium on Information Theory.