Capacity of Noisy Permutation Channels

TL;DR

This paper determines the capacity of noisy permutation channels, showing it depends on the rank of the channel matrix and extends results to various channel types, including erasure and Z-channels.

Contribution

It establishes the exact capacity formula for a class of noisy permutation channels and extends the analysis to multiple channel types, providing new theoretical bounds.

Findings

Capacity equals half the rank of the channel matrix minus one for positive channels.

Derived sharp entropy bounds for sampled vectors through DMCs.

Extended capacity results to erasure and Z-channels.

Abstract

We establish the capacity of a class of communication channels introduced in [1]. The $n$-letter input from a finite alphabet is passed through a discrete memoryless channel $P_{Z|X}$ and then the output $n$-letter sequence is uniformly permuted. We show that the maximal communication rate (normalized by $\log n$) equals $1/2 (rank(P_{Z|X})-1)$ whenever $P_{Z|X}$ is strictly positive. This is done by establishing a converse bound matching the achievability of [1]. The two main ingredients of our proof are (1) a sharp bound on the entropy of a uniformly sampled vector from a type class and observed through a DMC; and (2) the covering $\epsilon$-net of a probability simplex with Kullback-Leibler divergence as a metric. In addition to strictly positive DMC we also find the noisy permutation capacity for $q$-ary erasure channels, the Z-channel and others.