Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities (1504.02608v1)

Abstract: This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.

• The paper extends classical hypothesis testing and channel coding by using second-order asymptotic expansions that incorporate dispersion metrics for non-vanishing error regimes.
• It offers comprehensive insights into joint source-channel coding and multi-terminal networks, identifying scenarios where traditional separation theorems incur second-order penalties.
• The work provides practical guidance for designing adaptive coding schemes that bridge theoretical limits with the operational constraints of realistic communication systems.

Review of "Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities"

This monograph by Vincent Y.F. Tan examines the asymptotic behavior of various information-theoretic quantities under the setting where error probabilities are not vanishing. The focus is on extending classical results in the domain of information theory, which traditionally centered around the vanishing error probability regime, to settings where the error probability remains non-zero as the blocklength tends to infinity. This allows for a more nuanced understanding of the operational limits in practical communication scenarios where non-zero errors are tolerated.

Key Contributions

1. Hypothesis Testing: The monograph begins by revisiting Strassen's classical results concerning binary hypothesis testing in a non-vanishing error regime. Tan extends these findings to provide asymptotic expressions for the hypothesis testing divergence and the information spectrum divergence. The results emphasize the significance of second-order terms, particularly the role of dispersion, which captures the rate at which the probabilities of type I and type II errors converge.
2. Point-to-Point Communication: The work explores the point-to-point communication models, particularly lossless and lossy source coding, as well as channel coding. It provides second-order asymptotic results for these problems by leveraging recent advancements in non-asymptotic information theory.
3. Source-Channel Coding: Tan discusses the interplay between source and channel coding in the finite blocklength regime. The monograph explores the celebrated separation theorem and presents scenarios where joint source-channel coding can be optimal, as well as those where separation incurs a penalty in the second-order sense.
4. Network Information Theory: The exploration extends to multi-terminal setups, such as the Gaussian interference channel in very strong interference regimes and Slepian-Wolf coding, highlighting the second-order behaviors. The focus is on characterizing rate regions with non-vanishing errors, using tools such as Gaussian approximations and information spectrum methods.
5. Channels with State: The text also addresses channels with random states, both when the state information is available at the encoder and/or decoder and when it is not. Notably, the writing on dirty paper problem is revisited with new insights into the second-order asymptotics.

Implications and Future Directions

The implications of this work are manifold for theoretical and practical aspects of communication systems. The refined asymptotic analyses bridge the gap between theory and applications by considering realistic scenarios where error probabilities are non-zero. For practitioners, the insights into the achievable second-order terms in finite blocklength regimes can guide the design of more efficient coding schemes that better meet operational constraints.

For future developments in AI and communication technologies, this work suggests avenues for research into adaptive coding strategies that dynamically adjust to real-world conditions. Moreover, it invites exploration into other multi-terminal communication scenarios untouched by this monograph, such as those involving feedback and more complex network configurations.

In essence, Tan's monograph provides a comprehensive treatment of information theory in the non-vanishing error regime, setting a foundation for further exploration into practical communication limits in the finite blocklength context. It serves as a crucial reference for researchers aiming to develop models and techniques that align more closely with the constraints and requirements found in contemporary communication systems.