$f$-divergence Inequalities (1508.00335v7)

Abstract: This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the best possible constant upper bounding a ratio of $f$-divergences. Another approach used for the derivation of bounds among $f$-divergences relies on moment inequalities and the logarithmic-convexity property, which results in tight bounds on the relative entropy and Bhattacharyya distance in terms of $\chi^2$ divergences. A rich variety of bounds are shown to hold under boundedness assumptions on the relative information. Special attention is devoted to the total variation distance and its relation to the relative information and relative entropy, including "reverse Pinsker inequalities," as well as on the $E_\gamma$ divergence, which generalizes the total variation distance. Pinsker's inequality is extended for this type of $f$-divergence, a result which leads to an inequality linking the relative entropy and relative information spectrum. Integral expressions of the R\'enyi divergence in terms of the relative information spectrum are derived, leading to bounds on the R\'enyi divergence in terms of either the variational distance or relative entropy.

• The paper introduces systematic derivations of f-divergence inequalities that extend classical Pinsker bounds using functional domination.
• It employs moment inequalities and logarithmic convexity to establish tighter bounds and optimal constants between divergence measures.
• The results have practical implications for hypothesis testing and information theory by improving numerical error estimates and convergence rates.

An Analysis of f-Divergence Inequalities

The paper "f-Divergence Inequalities" by Igal Sason and Sergio Verdu discusses the derivation of inequalities among f-divergence measures, which serve as metrics for dissimilarity between probability distributions. This work chiefly aims to provide systematic methodologies for deducing inequalities tied to diverse f-divergences. The discussion encompasses integral representations, extensions of well-known inequalities, and new mathematical solutions across various divergence measures.

Summary of Approaches and Results

The authors deploy several systematic methods for deriving f-divergence inequalities:

• Functional Domination: This method involves proving that one divergence measure bounds another in terms of a multiplicative constant. The authors present insights into conditions under which optimal constants for these inequalities can be identified, thus potentially improving on existing results.
• Moment Inequalities and Logarithmic Convexity: Through leveraging properties like moment inequalities, the paper establishes bounds on divergences, notably providing refinements to existing relationships such as those seen in classical Pinsker-type inequalities.
• Relative Information Boundedness: A novel approach analyzes scenarios where the relative information is bounded. By assuming such boundedness, the paper proposes tight bounds on ratios of different divergence measures, circumventing the need for strict dominations.

Noteworthy Contributions and Theoretical Implications

1. Pinsker's Inequality Extensions: The paper extensively discusses generalized and refined versions of Pinsker's inequality, a fundamental relationship connecting total variation distance and Kullback-Leibler divergence. Notably, it extends Pinsker’s inequality to broader contexts, including the total variation distance and relative information spectrum.
2. Local Behavior and Optimality: Through new analytic tools, the authors examine the local behavior of f-divergences. They determine that under a convergence of distributions, the relative behavior of divergences can often be ascertained with greater precision, leading to potential optimizations in numerical approaches.
3. Renyi Divergence: The paper explores Renyi divergence, proposing integral expressions defining it in terms of the relative information spectrum. These provide new insights into how Renyi divergence relates to other metrics under certain bounded conditions.
4. Numerical Tightness and Applications: Across the board, the authors rigorously validate their results' numerical tightness. By revealing bounds with optimized constants, they set the stage for various applications in fields requiring precise control over statistical differences, such as hypothesis testing, information theory, and data processing.

The theoretical exploration of these inequalities extends into practical domains, impacting the bounds used in error probabilities in binary hypothesis testing and the convergence rates in stochastic processes. Moreover, the general results about the relationship between Hellinger divergences and f-divergences suggest broader implications in measure concentration results.

Further Research and Applications

Given the robust nature of these approaches, future exploration may include:

• Complex Alphabet Spaces: While the current paper efficiently handles finite and some infinite alphabets, extending these methods to more complex or mixed spaces could yield significant new insights.
• Operational Interpretations in Network Information Theory: Extensions of these inequality results could enhance our understanding of multi-user information theory settings where joint distributions across networks are considered.
• Application to Quantum Information: Examining the applicability of these inequalities in quantum divergences could bridge quantum channel coding and classical information-theoretic results, leading to further interdisciplinary advancements.

Overall, Sason and Verdu's work presents a comprehensive and methodically founded exploration of f-divergence inequalities, contributing fundamentally to both the theory and application of statistical measures in information theory and related fields.