Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn $f$-Divergences: Integral Representations, Local Behavior, and Inequalities
167
0
0.0
(
0
)
Added by
Igal Sason
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Igal Sason
Ask ChatGPT about the research
No Arabic abstract
This paper is focused on $f$-divergences, consisting of three main contributions. The first one introduces integral representations of a general $f$-divergence by means of the relative information spectrum. The second part provides a new approach for the derivation of $f$-divergence inequalities, and it exemplifies their utility in the setup of Bayesian binary hypothesis testing. The last part of this paper further studies the local behavior of $f$-divergences.
rate research
Read More
This paper is focused on derivations of data-processing and majorization inequalities for $f$-divergences, and their applications in information theory and statistics. For the accessibility of the material, the main results are first introduced without proofs, followed by exemplifications of the theorems with further related analytical results, interpretations, and information-theoretic applications. One application refers to the performance analysis of list decoding with either fixed or variable list sizes; some earlier bounds on the list decoding error probability are reproduced in a unified way, and new bounds are obtained and exemplified numerically. Another application is related to a study of the quality of approximating a probability mass function, induced by the leaves of a Tunstall tree, by an equiprobable distribution. The compression rates of finite-length Tunstall codes are further analyzed for asserting their closeness to the Shannon entropy of a memoryless and stationary discrete source. Almost all the analysis is relegated to the appendices, which form a major part of this manuscript.
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. Pinskers 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 Renyi divergence in terms of the relative information spectrum are derived, leading to bounds on the Renyi divergence in terms of either the variational distance or relative entropy.
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive noninteger real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, $n$, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over $n$ dimensions, as needed in the direct calculation of the desired moment.
We consider a sub-class of the $f$-divergences satisfying a stronger convexity property, which we refer to as strongly convex, or $kappa$-convex divergences. We derive new and old relationships, based on convexity arguments, between popular $f$-divergences.
New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verd{u} for general probability measures. A second bound improves the tightness of an inequality by Csisz{a}r and Talata for arbitrary probability measures that are defined on a common finite set. The latter result is further extended, for probability measures on a finite set, leading to an upper bound on the R{e}nyi divergence of an arbitrary non-negative order (including $infty$) as a function of the total variation distance. Another lower bound by Verd{u} on the total variation distance, expressed in terms of the distribution of the relative information, is tightened and it is attained under some conditions. The effect of these improvements is exemplified.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
