# On the Existence of the Augustin Mean

@article{Cheng2021OnTE,
title={On the Existence of the Augustin Mean},
author={Hao-Chung Cheng and Barış Nakiboğlu},
journal={2021 IEEE Information Theory Workshop (ITW)},
year={2021},
pages={1-6}
}
• Published 1 September 2021
• Mathematics
• 2021 IEEE Information Theory Workshop (ITW)
The existence of a unique Augustin mean and its invariance under the Augustin operator are established for arbitrary input distributions with finite Augustin information for channels with countably generated output $\sigma$-algebras. The existence is established by representing the conditional Rényi divergence as a lower semi-continuous and convex functional in an appropriately chosen uniformly convex space and then invoking the Banach-Saks property in conjunction with the lower semi-continuity…

## References

SHOWING 1-10 OF 35 REFERENCES
The Augustin Capacity and Center
For any channel, the existence of a unique Augustin mean is established for any positive order and probability mass function on the input set and the Augustin information is shown to be continuously differentiable in the order.
Properties of Scaled Noncommutative Rényi and Augustin Information
• Computer Science
2019 IEEE International Symposium on Information Theory (ISIT)
• 2019
This paper proves the uniform equicontinuity for all three quantum versions of Rényi information, hence it yields the joint continuity of these quantities in the orders and priors, and establishes the concavity in the region of s ∈ (−1, 0) for both Petz’s and the sandwiched versions.
A Simple Derivation of the Refined Sphere Packing Bound Under Certain Symmetry Hypotheses.
The trade-off between type I and type II error probabilities in the hypothesis testing problem with (possibly non-stationary) independent samples is determined up to some multiplicative constants, assuming that the probabilities of both types of error are decaying exponentially with the number of samples, using the Berry-Esseen theorem.
The Rényi Capacity and Center
• B. Nakiboğlu
• Computer Science
IEEE Transactions on Information Theory
• 2019
The van Erven– Harremoës conjecture is proved for any positive order and for any set of probability measures on a given measurable space and a generalization of it is established for the constrained variant of the problem.
Rényi Divergence and Kullback-Leibler Divergence
• Computer Science
IEEE Transactions on Information Theory
• 2014
The most important properties of Rényi divergence and Kullback- Leibler divergence are reviewed, including convexity, continuity, limits of σ-algebras, and the relation of the special order 0 to the Gaussian dichotomy and contiguity.
Correlation detection and an operational interpretation of the Rényi mutual information
• Computer Science
2015 IEEE International Symposium on Information Theory (ISIT)
• 2015
This work shows that the Rényi mutual information attains operational significance in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternate hypothesis consists of all product states that share one marginal with thenull hypothesis.
Divergence Radii and the Strong Converse Exponent of Classical-Quantum Channel Coding With Constant Compositions
• Computer Science
IEEE Transactions on Information Theory
• 2021
It is shown that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényu divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding.
Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
• Computer Science
IEEE Transactions on Information Theory
• 2017
This paper first extends the sphere packing bound for classical-quantum channels to the case of constant-composition codes, and shows that the obtained result is related to a variation of the Lovász theta function studied by Marton.
Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback-Leibler
The details of this phenomenon are discussed, which suggests the question of whether auxiliary channels are used in the optimal way in the second approach and whether recent results on the exact strong-converse exponent in classical-quantum channel coding might play a role in the considered problem.
Error estimates for low rate codes
The authors of [2] regret that their proof of the lower bound cannot be extended to infinite alphabet channels or nonstationary channels because of the use of “fixed composition codes” (while the proofs of the upper estimate can be easily transferred to those channels).