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

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 (1-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 worst-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on 1-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 1-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 ⇤. 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 ⇤, and the second is a vanishing information radius (I-radius) condition for ⇤ reminiscent of the condition known for lossless universal source coding.

Lossy Compression of Individual Sequences Revisited: Fundamental Limits of Finite-State Encoders

“…The absence of source knowledge imposes a cost in terms of enlarging the multiplicative constant associated with log n n . Yu and Speed [19] established weak universality, introducing a constant that grows with the cardinalities of the source and reconstruction alphabets [20]. Ornstein and Shields [17] delved into universal d-semifaithful coding for stationary and ergodic sources concerning the Hamming distortion measure, demonstrating convergence to the rate-distortion function with the probability one.…”

Lossy Compression of Individual Sequences Revisited: Fundamental Limits of Finite-State Encoders

“…The absence of source knowledge imposes a cost in terms of enlarging the multiplicative constant associated with log n n . Yu and Speed [19] established weak universality, introducing a constant that grows with the cardinalities of the source and reconstruction alphabets [20]. Ornstein and Shields [17] delved into universal d-semifaithful coding for stationary and ergodic sources concerning the Hamming distortion measure, demonstrating convergence to the rate-distortion function with probability one.…”

A Universal Random Coding Ensemble for Sample-Wise Lossy Compression

“…Not knowing the source conveys the price of enlarging the multiplicative constant in front of . Indeed, Yu and Speed [ 19 ] have established weak universality with a constant that grows with the cardinalities of the alphabets of the source and the reconstruction [ 20 ]. Ornstein and Shields [ 17 ] have considered universal d -semifaithful coding for stationary and ergodic sources with respect to the Hamming distortion measure, and established convergence with probability one to the rate–distortion function.…”