On Universal D-Semifaithful Coding for Memoryless Sources with Infinite Alphabets
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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 (∞-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 ∞-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 ∞-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 Λ. 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 Λ, and the second is a vanishing information radius (I-radius) condition for Λ reminiscent of the condition known for lossless universal source coding.
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