On Universal D-Semifaithful Coding for Memoryless Sources With Infinite Alphabets

DOI10.1109/TIT.2021.3134891zbMATH Open1497.94051arXiv2107.05082OpenAlexW3180765777MaRDI QIDQ5088421

Jorge F. Silva, Pablo Piantanida

Publication date: 13 July 2022

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: The problem of variable length and fixed-distortion universal source coding (or D-semifaithful source coding) for stationary and memoryless sources on countably infinite alphabets ($infty$-alphabets) is addressed in this paper. The main results of this work offer a set of sufficient conditions (from weaker to stronger) to obtain weak minimax universality, strong minimax universality, and corresponding achievable rates of convergences for the worse-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on $infty$-alphabets. An important implication of these results is that universal D-semifaithful source coding is not feasible for the complete family of stationary and memoryless sources on $infty$-alphabets. To demonstrate this infeasibility, a sufficient condition for the impossibility is presented for the envelope family. Interestingly, it matches the well-known impossibility condition in the context of lossless (variable-length) universal source coding. More generally, this work offers a simple description of what is needed to achieve universal D-semifaithful coding for a family of distributions $Lambda$. This reduces to finding a collection of quantizations of the product space at different block-lengths -- reflecting the fixed distortion restriction -- that satisfy two asymptotic requirements: the first is a universal quantization condition with respect to $Lambda$, and the second is a vanishing information radius (I-radius) condition for $Lambda$ reminiscent of the condition known for lossless universal source coding.

Full work available at URL: https://arxiv.org/abs/2107.05082

zbMATH Keywords

countably infinite alphabetsstationary and memoryless sourcesvariable length and fixed-distortion universal source coding

Mathematics Subject Classification ID

Signal theory (characterization, reconstruction, filtering, etc.) (94A12) Measures of information, entropy (94A17) Source coding (94A29)

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