Second-Order Converses via Reverse Hypercontractivity

Abstract

A strong converse shows that no procedure can beat the asymptotic (as blocklength $n\to \infty$) fundamental limit of a given information-theoretic problem for any fixed error probability. A second-order converse strengthens this conclusion by showing that the asymptotic fundamental limit cannot be exceeded by more than $O(\frac{1}{\sqrt{n}})$. While strong converses are achieved in a broad range of information-theoretic problems by virtue of the "blowing-up method"---a powerful methodology due to Ahlswede, Gács and Körner (1976) based on concentration of measure---this method is fundamentally unable to attain second-order converses and is restricted to finite-alphabet settings. Capitalizing on reverse hypercontractivity of Markov semigroups and functional inequalities, this paper develops the "smoothing-out" method, an alternative to the blowing-up approach that does not rely on finite alphabets and that leads to second-order converses in a variety of information-theoretic problems that were out of reach of previous methods.

61 pages, 3 figures, short version presented at ISIT 2017

References in corpus (2)

• Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance
• Information-Theoretic Perspectives on Brascamp-Lieb Inequality and Its Reverse

Cited by in corpus (2)

• Strong converse bounds in quantum network information theory: distributed hypothesis testing and source coding
• On Non-Interactive Simulation of Binary Random Variables