The Augustin Capacity and Center

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-Harremoës 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 Rényi-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.

Properties of Scaled Noncommutative Rényi and Augustin Information

“…Augustin [30] and Csiszár [31] proposed another generalization 3 of Shannon's mutual information, which is termed as the order α Augustin information for a prior P [30,31,34,35]:…”

“…When maximizing over all priors P , both Rényi information and Augustin information equal the order α Rényi capacity [36,31,37,35]:…”

“…The properties of the auxiliary functions can also be derived via those of the Rényi and Augustin information. We refer the readers to the review literature by Verdú [74,33], Dalai [34], and the excellent expositions by Nakiboglu [37,35] from a measure-theoretic aspect. In classical information theory, the (channel) output space consists of probability measures.…”

The Sphere Packing Bound for DSPCs With Feedback à la Augustin

“…Augustin's SPB given in [14,Thm 36.6] applies to the Poisson channels, but not to various Gaussian channels analyzed in [2], [15], [16] because the quadratic cost function is not bounded. In [17], we have proved a SPB for codes on the cost constrained memoryless channels -without assuming the cost function to be bounded-using the constrained Augustin capacity and center analyzed in [14, §34] and [18]. The SPB given in [17,Thm.…”

The Augustin center and the sphere packing bound for memoryless channels

“…Theorem 2, presented in §3, establishes the SPB for a framework that subsumes all of the models considered in [2]- [26] by employing [27], which analyzes Augustin's information measures. Our use of [27] and Augustin's information measures is similar to the use of [28] and Rényi 's information measures in [26]. For the product channels, [26,Thm.…”