Analysis of the Convergence Speed of the Arimoto-Blahut Algorithm by the Second Order Recurrence Formula

TL;DR

This paper analyzes the convergence speed of the Arimoto-Blahut algorithm, identifying conditions for exponential and slower convergence, and provides a comprehensive understanding of its behavior across different channel matrices.

Contribution

It introduces a novel analysis of slow convergence using second order recurrence formulas and clarifies conditions for different convergence rates.

Findings

Exponential convergence for many channel matrices

Slower than exponential convergence for some matrices

Numerical validation of theoretical convergence speeds

Abstract

In this paper, we investigate the convergence speed of the Arimoto-Blahut algorithm. For many channel matrices the convergence is exponential, but for some channel matrices it is slower than exponential. By analyzing the Taylor expansion of the defining function of the Arimoto-Blahut algorithm, we will make the conditions clear for the exponential or slower convergence. The analysis of the slow convergence is new in this paper. Based on the analysis, we will compare the convergence speed of the Arimoto-Blahut algorithm numerically with the values obtained in our theorems for several channel matrices. The purpose of this paper is a complete understanding of the convergence speed of the Arimoto-Blahut algorithm.