2025 Volume 79 Issue 1 Pages 35-76
In this paper, we prove that the convergence speed of certain quadratic recurrence formulas in the Arimoto–Blahut algorithm is of order O(1/N). The Arimoto–Blahut algorithm is a well-known algorithm in information theory for computing the capacity of a discrete memoryless channel. There have been many studies on exponential convergence, whereas we found previously (Nakagawa et al. IEEE Trans. Inform. Theory 67(10) (2021), 6810–6831) that there exists a channel for which the convergence is of order O(1/N). In that reference, the convergence of order O(1/N) was analyzed by using quadratic recurrence formulas consisting of the first- and second-order terms of the Taylor expansion of the defining function of the Arimoto–Blahut algorithm. However, there we assumed an infinite number of inequalities and the proof was given under this assumption. In this present paper, we prove the convergence of order O(1/N) by assuming only a finite number of inequalities. The important contribution of this paper is the novelty of the proof. A key idea of the proof is to examine a continuous-time function (i.e., a function defined over the non-negative real numbers) obtained by interpolating the discrete-time function (i.e., a sequence defined over the non-negative integers) of the quadratic recurrence formulas with line segments. The correctness of the proof is demonstrated by several numerical examples.
