Extension of the Blahut-Arimoto algorithm for maximizing directed information

Abstract

Abstract-In this paper, we extend the Blahut-Arimoto algorithm for maximizing Massey's directed information. The algorithm can be used for estimating the capacity of channels with delayed feedback, where the feedback is a deterministic function of the output. In order to maximize the directed information, we apply the ideas from the regular Blahut-Arimoto algorithm, i.e., the alternating maximization procedure, to our new problem. We provide both upper and lower bound sequences that converge to the optimum global value. Our main insight in this paper is that in order to find the maximum of the directed information over a causal conditioning probability mass function, one can use a backward index time maximization combined with the alternating maximization procedure. We give a detailed description of the algorithm, showing its complexity and the memory needed, and present several numerical examples.

Capacity and Coding for the Ising Channel With Feedback

“…Recall that T γ,δ is the DP operator restricted to the policy γ, δ, hence it is the DP operator without the supremum. Using basic algebra and setting γ * , δ * as in (22), (23) with a as a variable, we obtain…”

“…In this section we verify that the function h * (z), as found in (30), and ρ * = 2H(a) 3+a , indeed satisfy the Bellman Equation, T h * (z) = h * (z) + ρ * . Furthermore, we show that the selection of δ * , γ * , as in ( 23), (22) maximizes T h * (z). Namely, we prove Theorem 1.…”

“…then ρ = ρ * . Using Lemma 4 we obtain that δ * and γ * , as defined in (22),(23), maximize T h * (z) when a ≈ 0.4503. In addition, we show that h * (z), which is defined in (30), satisfies the Bellman Equation,…”

“…We begin with proving the following lemma: Lemma 4. The policies γ * , δ * which are defined in (22), (23):…”

“…The capacity of the Ising channel with feedback was approximated numerically [22] using an extension of Blahut-Arimoto algorithm for directed information. Here, we present the explicit expression together with a simple capacity-achieving coding scheme.…”