Improved Bounds on Lossless Source Coding and Guessing Moments via R\'enyi Measures

TL;DR

This paper improves bounds on guessing moments and source coding measures using Rényi entropy, especially in the non-asymptotic regime, and relates these to error probabilities and codeword length distributions.

Contribution

It introduces significantly tighter non-asymptotic bounds on guessing moments and source coding measures using Rényi entropy, extending Arikan's asymptotic bounds.

Findings

Enhanced non-asymptotic bounds on guessing moments.

Derived bounds on the cumulant generating function of codeword lengths.

Established relationships between guessing moments and error probabilities.

Abstract

This paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikan's bounds, they are expressed in terms of the Arimoto-R\'enyi conditional entropy. Although Arikan's bounds are asymptotically tight, the improvement of the bounds in this paper is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probability are also established, characterizing the exact locus of their attainable values. The bounds on optimal guessing moments serve to improve non-asymptotic bounds on the cumulant generating function of the codeword lengths for fixed-to-variable optimal lossless source coding without prefix constraints. Non-asymptotic bounds on the reliability function of discrete memoryless sources are derived as well. Relying on these techniques, lower bounds on the cumulant generating function of the codeword lengths are derived, by means of the smooth R\'enyi entropy, for source codes that allow decoding errors.