Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition) (1212.4663v8)

Abstract: During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory, information theory, theoretical computer science, and learning theory. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors. The first part of the monograph introduces classical concentration inequalities for martingales, as well as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication. The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities, which underlie the so-called functional approach to concentration of measure, and then from a complementary information-theoretic viewpoint based on transportation-cost inequalities and probability in metric spaces. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, we discuss several applications of the entropy method to problems in communications and coding, including strong converses, empirical distributions of good channel codes, and an information-theoretic converse for concentration of measure.

• The paper comprehensively covers classical and modern concentration inequalities, detailing derivations using the entropy method and tensorized approaches.
• It demonstrates significant applications of these inequalities in communications and coding, including performance analysis of LDPC codes and implications for 5G systems.
• The work provides practical insights for designing efficient coding schemes and suggests future theoretical extensions for dependent variables and new distributions.

Overview of the Paper

This academic monograph dives deep into the topic of concentration inequalities, particularly their derivation using modern mathematical tools and their connections to information theory and coding. The document is divided into thoughtful sections, each dedicated to a facet of concentration inequalities, elucidating both classical results and novel advancements.

Key Contributions

• Classical and Modern Inequalities: The monograph explores classical concentration inequalities for martingales, tracing their evolution and refinements over time. Core results such as the Azuma–Hoeffding and McDiarmid inequalities are examined in depth, providing a foundation for understanding the power and versatility of these techniques.
• Entropy Method: A substantial portion of the monograph is dedicated to the entropy method, showcasing its utility in deriving concentration inequalities. The entropy method's basis lies in its ability to relate concentration inequalities to logarithmic Sobolev inequalities and information-theoretic aspects.
• Applications: The document culminates in various applications of the theoretical constructs to real-world problems. These include implications in communications and coding, such as the performance analysis of binary linear block codes, expansion properties of bipartite graphs, and properties of LDPC codes in noisy channels.
• Tensorized Approaches: A critical aspect discussed is the tensorization of entropic measures, allowing n-dimensional problems to be decomposed into more tractable one-dimensional problems. This technique is pivotal in applying concentration inequalities to functions of independent random variables.

Strong Numerical Results and Bold Claims

The authors provide strong numerical examples to illustrate the tightness of various bounds obtained from improved versions of classical inequalities. These examples are meticulously chosen to highlight the effectiveness of refined inequalities over their classical counterparts in settings like OFDM signals and random regular bipartite graphs.

Moreover, the authors make a compelling case for the broader applicability of these refined information-theoretic inequalities in new domains beyond traditional settings, pressing the bounds of current theoretical knowledge.

Practical and Theoretical Implications

From a practical standpoint, the concentration results have profound implications for designing low-complexity, efficient coding schemes, especially in scenarios where transmission reliability is critical, such as 5G communications. The enhanced understanding of LDPC codes' performance under iterative decoding schemes is particularly noteworthy.

Theoretically, the monograph raises intriguing possibilities for the future development of AI systems, where concentration inequalities could better quantify uncertainties posed by model predictions and decisions.

Future Directions

The monograph suggests a promising future where concentration inequalities derived through the entropy method might be extended to handle more general cases of dependent random variables, leading to tighter and more general bounds applicable in broader contexts. The exploration of logarithmic Sobolev inequalities in new distributional contexts offers fertile ground for future research.

Conclusion

Overall, this work is not merely a survey but a comprehensive theoretical exposition enriched with new research results derived by the authors. The profound insights into concentration inequalities and their applications exemplify how mathematical theory can inform and improve practices in communications and information theories, underscoring the significance of these results in the broader computational and applied mathematics landscape.