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Are Guessing, Source Coding and Tasks Partitioning Birds of A Feather?

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Summary

This study reveals a strong connection between four information theory problems, linking them through a moment minimization problem solved using Renyi entropy. This unified framework establishes an equivalence, allowing solutions from one problem to solve others.

Keywords:
Kullback–Leibler divergenceRényi entropyShannon entropySundaresan’s divergenceguessingrelative α-entropysource codingtasks partitioning

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Area of Science:

  • Information Theory
  • Coding Theory
  • Statistical Inference

Background:

  • Four distinct information-theoretic problems are explored: Campbell source coding, Arikan guessing, Huleihel et al. memoryless guessing, and Bunte and Lapidoth tasks' partitioning.
  • These problems are analyzed within the independent and identically distributed (IID) lossless data compression framework.

Purpose of the Study:

  • To establish a mathematical relationship among the four specified information theoretic problems.
  • To introduce a general framework for analyzing mismatched versions of these problems.
  • To investigate a practically relevant variant of the Bunte-Lapidoth tasks partitioning problem.

Main Methods:

  • Mathematical formulation relating the problems via a general moment minimization problem.
  • Utilizing Renyi entropy to define the optimal solution for the minimization problem.
  • Developing a unified framework to derive asymptotic results for mismatched scenarios.
  • Analyzing a generalized variant of the Bunte-Lapidoth problem.

Main Results:

  • A close relationship is established among Campbell source coding, Arikan guessing, Huleihel et al. memoryless guessing, and Bunte and Lapidoth tasks' partitioning problems.
  • The optimal solution for the moment minimization problem is expressed in terms of Renyi entropy.
  • A unified framework is proposed for mismatched versions, yielding asymptotic results.
  • A practically appealing variant of the Bunte-Lapidoth problem is studied, generalizing Arıkan's guessing problem.

Conclusions:

  • An equivalence is demonstrated among all four problems under the unified framework.
  • Solutions for one problem can be asymptotically optimized for the others, simplifying analysis and application.