Coding on countably infinite alphabets

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 inferenceComment: 33 page

Coding on countably infinite alphabets

33 pagesInternational audienceThis 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 inferenc

Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory

Fano's inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano's inequality is generalized to a broad class of information measures, which contains those of Shannon and R\'{e}nyi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff's theorem proven by R\'{e}v\'{e}sz [Acta Math. Hungar. 1962, 3, 188{\textendash}198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon's and R\'{e}nyi's information measures, we also investigate the asymptotic behavior of the sequence of Shannon's and R\'{e}nyi's equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano's inequality.Comment: 44 pages, 3 figure

Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $\alpha$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundanc

Universal Densities Exist for Every Finite Reference Measure

As it is known, universal codes, which estimate the entropy rate consistently, exist for stationary ergodic sources over finite alphabets but not over countably infinite ones. We generalize universal coding as the problem of universal densities with respect to a fixed reference measure on a countably generated measurable space. We show that universal densities, which estimate the differential entropy rate consistently, exist for finite reference measures. Thus finite alphabets are not necessary in some sense. To exhibit a universal density, we adapt the non-parametric differential (NPD) entropy rate estimator by Feutrill and Roughan. Our modification is analogous to Ryabko's modification of prediction by partial matching (PPM) by Cleary and Witten. Whereas Ryabko considered a mixture over Markov orders, we consider a mixture over quantization levels. Moreover, we demonstrate that any universal density induces a strongly consistent Ces\`aro mean estimator of conditional density given an infinite past. This yields a universal predictor with the $0-1$ loss for a countable alphabet. Finally, we specialize universal densities to processes over natural numbers and on the real line. We derive sufficient conditions for consistent estimation of the entropy rate with respect to infinite reference measures in these domains.Comment: 28 pages, no figure

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019

Phase transitions for suspension flows

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note "Thermodynamic formalism for the positive geodesic flow on the modular surface" arXiv:1009.462