Coding on countably infinite alphabets

TL;DR

This paper develops universal lossless coding strategies for sources over countably infinite alphabets, establishing bounds on regret and redundancy, and proposing adaptive coding techniques for specific source classes.

Contribution

It extends universal coding theory to infinite alphabets by deriving bounds and proposing adaptive coding methods for algebraically vanishing envelope classes.

Findings

Upper bounds on minimax regret for infinite alphabet sources

Lower bounds on minimax redundancy using finite alphabet coders

Effective adaptive coding techniques for algebraically vanishing envelopes

Abstract

This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper-bounds on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend ourknowledge concerning universal coding to contexts where the key tools from parametric inference