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A New Algorithm for Computing $α$-Capacity

Akira Kamatsuka, Koki Kazama, Takahiro Yoshida

TL;DR

The paper tackles computing the $α$-capacity for $α>1$, which is linked to decoding exponents in channel coding. It introduces a novel alternating-optimization algorithm based on a variational characterization of the Augustin–Csiszár mutual information to directly compute $Cα^{C}$ for $α∈(1,∞)$. The work compares the convergence of the new method against established AO schemes (Arimoto and Jitsumatsu–Oohama) via numerical experiments, highlighting dependence on $α$ and initialization. This contributes a practical tool for evaluating $α$-capacities and decoding exponents in discrete channels, with potential extensions to global convergence analyses and rate studies.

Abstract

The problem of computing $α$-capacity for $α>1$ is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu--Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the $α$-capacity for $α>1$ based on a variational characterization of the Augustin--Csisz{á}r mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.

A New Algorithm for Computing $α$-Capacity

TL;DR

The paper tackles computing the -capacity for , which is linked to decoding exponents in channel coding. It introduces a novel alternating-optimization algorithm based on a variational characterization of the Augustin–Csiszár mutual information to directly compute for . The work compares the convergence of the new method against established AO schemes (Arimoto and Jitsumatsu–Oohama) via numerical experiments, highlighting dependence on and initialization. This contributes a practical tool for evaluating -capacities and decoding exponents in discrete channels, with potential extensions to global convergence analyses and rate studies.

Abstract

The problem of computing -capacity for is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu--Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the -capacity for based on a variational characterization of the Augustin--Csisz{á}r mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.
Paper Structure (8 sections, 6 theorems, 13 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 8 sections, 6 theorems, 13 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\alpha$-MI and $\alpha$-Capacity
  4. Arimoto Algorithm for Computing the Sibson and Arimoto Capacity
  5. Jitsumatsu--Oohama Algorithm for Computing the Sibson Capacity
  6. Proposed Algorithm
  7. Numerical Example
  8. Conclusion

Key Result

Proposition 1

Let $\alpha\in (0, 1)\cup (1, \infty)$. Then,

Figures (2)

  • Figure 1: Relationship between the algorithms for computing $\alpha$-capacity
  • Figure 2: Transitions of approximate value of $\alpha$-capacity $F^{(k)}$ as $k$ increases for (a) $\alpha=1.03$, (b) $\alpha=1.5$, (c) $\alpha=2.0$, and (d) $\alpha=5.0$. Blue solid curve expresses Algorithm \ref{['alg:Arimoto']} (Arimoto algorithm), orange dashed curve expresses Algorithm \ref{['alg:JOA']} (Jitsumasu--Oohama algorithm), and green dash-dotted curve expresses Algorithm \ref{['alg:Csiszar']} (This work), respectively.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2: $\alpha$-capacity
  • Remark 1
  • Proposition 1: arimoto1977,370121,e22050526
  • Proposition 2: 1055640,BN01990060en,kamatsuka2024new
  • Proposition 3: 8889422
  • Corollary 1
  • Proposition 4: kamatsuka2024algorithms
  • Proposition 5
  • proof