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Optimum Achievable Rates in Two Random Number Generation Problems with f-Divergences Using Smooth Rényi Entropy.

Ryo Nomura1, Hideki Yagi2

  • 1Center for Data Science, Waseda University, Tokyo 169-8050, Japan.

Entropy (Basel, Switzerland)
|September 27, 2024
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Summary
This summary is machine-generated.

This study analyzes random number generation rates using smooth Rényi entropy for f-divergences. It reveals a duality between source resolvability and intrinsic randomness, extending previous findings.

Keywords:
Hellinger distanceKullback–Leibler divergencef-divergenceintrinsic randomnessrandom number generationsmooth Rényi entropysource resolvabilityvariational distance

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Area of Science:

  • Information Theory
  • Probability Theory
  • Random Number Generation

Background:

  • Two key problems in random number generation for general sources are source resolvability and intrinsic randomness.
  • Optimum achievable rates are typically characterized by information spectrum and smooth Rényi entropy.
  • Recent work characterized rates for f-divergences using information spectrum, but smooth Rényi entropy remains underexplored.

Purpose of the Study:

  • To analyze optimum achievable rates for random number generation problems with respect to f-divergences using smooth Rényi entropy.
  • To extend the class of f-divergences considered in these problems.
  • To investigate first-order and second-order rates, as well as optimistic achievable rates.

Main Methods:

  • Derivation of general formulas for first-order optimum achievable rates with respect to f-divergences.
  • Relaxation of conditions on f-divergence to generalize the formulas.
  • Particularization of general formulas to specific functions f.

Main Results:

  • General formulas for first-order optimum achievable rates with respect to f-divergences are derived.
  • The derived formulas are generalized by relaxing conditions on f-divergence.
  • A duality between resolvability and intrinsic randomness is revealed in terms of smooth Rényi entropy.

Conclusions:

  • The study provides a unified framework for analyzing random number generation rates using smooth Rényi entropy and f-divergences.
  • The findings extend existing results and offer new insights into the relationship between source resolvability and intrinsic randomness.
  • The derived general formulas simplify the calculation of optimum achievable rates for various important measures.