Efficient Approximation of Channel Capacities

TL;DR

This paper introduces an efficient iterative method for approximating the capacity of discrete memoryless channels, including those with continuous inputs and countable outputs, providing explicit bounds and complexity analysis.

Contribution

It presents a novel convex duality-based iterative approach for capacity approximation with explicit bounds and applies it to Poisson channels with peak-power constraints.

Findings

The method achieves an $O(M^2 N \, \sqrt{\log N}/\varepsilon)$ complexity for capacity estimation.

Explicit upper and lower bounds for channel capacity are derived.

The approach extends to channels with continuous inputs and countable outputs, including Poisson channels.

Abstract

We propose an iterative method for approximately computing the capacity of discrete memoryless channels, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. The presented method requires $O(M^2 N \sqrt{\log N}/\varepsilon)$ to provide an estimate of the capacity to within $\varepsilon$, where $N$ and $M$ denote the input and output alphabet size; a single iteration has a complexity $O(M N)$. We also show how to approximately compute the capacity of memoryless channels having a bounded continuous input alphabet and a countable output alphabet under some mild assumptions on the decay rate of the channel's tail. It is shown that discrete-time Poisson channels fall into this problem class. As an example, we compute sharp upper and lower bounds for the capacity of a discrete-time Poisson channel with a peak-power input constraint.