Non-Asymptotic Fundamental Limits of Guessing Subject to Distortion

TL;DR

This paper provides non-asymptotic bounds on the guessing problem with distortion, extending previous asymptotic results by using Rényi entropy measures and deriving a single-letter characterization for stationary memoryless sources.

Contribution

It introduces non-asymptotic bounds for guessing with distortion using Rényi entropy, and derives a single-letter asymptotic exponent for stationary memoryless sources.

Findings

Non-asymptotic achievability and converse bounds established.

Bounds are expressed via Rényi and Arimoto-Rényi conditional entropies.

Single-letter characterization of the asymptotic guessing exponent obtained.

Abstract

This paper investigates the problem of guessing subject to distortion, which was introduced by Arikan and Merhav. While the primary concern of the previous study was asymptotic analysis, our primary concern is non-asymptotic analysis. We prove non-asymptotic achievability and converse bounds of the moment of the number of guesses without side information (resp. with side information) by using a quantity based on the R\'enyi entropy (resp. the Arimoto-R\'enyi conditional entropy). Also, we introduce an error probability and show similar results. Further, from our bounds, we derive a single-letter characterization of the asymptotic exponent of guessing moment for a stationary memoryless source.