The Augustin Capacity and Center

TL;DR

This paper introduces the Augustin capacity, center, and related quantities, establishing their existence, uniqueness, and properties for various channels, and connects them to Renyi-Gallager measures, with applications to Gaussian channels.

Contribution

It formally defines the Augustin capacity and center, proves their existence and uniqueness, and links them to Renyi-Gallager quantities, expanding the theoretical understanding of channel information measures.

Findings

Existence and uniqueness of Augustin mean and center for channels.

Continuous differentiability of Augustin information in the order.

Equality of Augustin-Legendre and Renyi-Gallager quantities.

Abstract

For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set. The Augustin mean is shown to be the unique fixed point of an operator defined in terms of the order and the input distribution. The Augustin information is shown to be continuously differentiable in the order. For any channel and convex constraint set with finite Augustin capacity, the existence of a unique Augustin center and the associated van Erven-Harremoes bound are established. The Augustin-Legendre (A-L) information, capacity, center, and radius are introduced and the latter three are proved to be equal to the corresponding Renyi-Gallager quantities. The equality of the A-L capacity to the A-L radius for arbitrary channels and the existence of a unique A-L center for channels with finite A-L capacity are established. For all interior points of the feasible set of cost constraints, the cost constrained Augustin capacity and center are expressed in terms of the A-L capacity and center. Certain shift invariant families of probabilities and certain Gaussian channels are analyzed as examples.