Universal Weak Variable-Length Source Coding on Countable Infinite Alphabets

TL;DR

This paper introduces the concept of almost lossless source coding for countably infinite alphabets, showing it is feasible under certain conditions and can outperform lossless coding in convergence rates.

Contribution

It defines almost lossless source coding for infinite alphabets, characterizes achievable rates, and establishes conditions for universal coding with vanishing distortion.

Findings

Shannon entropy characterizes the minimum achievable rate.

Almost lossless universal coding is feasible for finite-entropy stationary sources.

Faster convergence rates for redundancy are possible with almost lossless coding.

Abstract

Motivated from the fact that universal source coding on countably infinite alphabets is not feasible, this work introduces the notion of almost lossless source coding. Analog to the weak variable-length source coding problem studied by Han (IEEE TIT, 2000, 46, 1217-1226), almost lossless source coding aims at relaxing the lossless block-wise assumption to allow an average per-letter distortion that vanishes asymptotically as the block-length tends to infinity. In this setup, we show on one hand that Shannon entropy characterizes the minimum achievable rate (similarly to the case of finite alphabet sources) while on the other that almost lossless universal source coding becomes feasible for the family of finite-entropy stationary memoryless sources with infinite alphabets. Furthermore, we study a stronger notion of almost lossless universality that demands uniform convergence of the average per-letter distortion to zero, where we establish a necessary and sufficient condition for the so-called family of envelope distributions to achieve it. Remarkably, this condition is the same necessary and sufficient condition needed for the existence of a strongly minimax (lossless) universal source code for the family of envelope distributions. Finally, we show that an almost lossless coding scheme offers faster rate of convergence for the (minimax) redundancy compared to the well-known information radius developed for the lossless case at the expense of tolerating a non-zero distortion that vanishes to zero as the block-length grows. This shows that even when lossless universality is feasible, an almost lossless scheme can offer different regimes on the rates of convergence of the (worst case) redundancy versus the (worst case) distortion.