A Toolbox for Refined Information-Theoretic Analyses with Applications

TL;DR

This paper presents a comprehensive toolbox of mathematical techniques for refined information-theoretic analysis, including generalizations of the method of types, saddle-point integration, and advanced inequalities, with diverse applications.

Contribution

It introduces new analytical tools and methods for more precise information-theoretic evaluations, extending existing techniques to Gaussian and exponential family settings.

Findings

Best known random-coding exponents for various problems

Refined bounds using Laplace and saddle-point methods

Enhanced inequalities for expectation evaluations

Abstract

This monograph offers a toolbox of mathematical techniques, which have been effective and widely applicable in information-theoretic analysis. The first tool is a generalization of the method of types to Gaussian settings, and then to general exponential families. The second tool is Laplace and saddle-point integration, which allow to refine the results of the method of types, and are capable of obtaining more precise results. The third is the type class enumeration method, a principled method to evaluate the exact random-coding exponent of coded systems, which results in the best known exponent in various problem settings. The fourth subset of tools aimed at evaluating the expectation of non-linear functions of random variables, either via integral representations, or by a refinement of Jensen's inequality via change-of-measure, by complementing Jensen's inequality with a reversed inequality, or by a class of generalized Jensen's inequalities that are applicable for functions beyond convex/concave. Various application examples of all these tools are provided along this monograph.