Tight Bounds on the R\'enyi Entropy via Majorization with Applications to Guessing and Compression

This paper provides tight bounds on the R\'enyi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the R\'enyi entropy of a discrete random variable with a finite support is derived as a function of the size of the support, and the ratio of the maximal to minimal probability masses. This work was inspired by the recently published paper by Cicalese et al., which is focused on the Shannon entropy, and it strengthens and generalizes the results of that paper to R\'enyi entropies of arbitrary positive orders. In view of these generalized bounds and the works by Arikan and Campbell, non-asymptotic bounds are derived for guessing moments and lossless data compression of discrete memoryless sources.Comment: The paper was published in the Entropy journal (special issue on Probabilistic Methods in Information Theory, Hypothesis Testing, and Coding), vol. 20, no. 12, paper no. 896, November 22, 2018. Online available at https://www.mdpi.com/1099-4300/20/12/89

Information-Distilling Quantizers

Let $X$ and $Y$ be dependent random variables. This paper considers the problem of designing a scalar quantizer for $Y$ to maximize the mutual information between the quantizer's output and $X$, and develops fundamental properties and bounds for this form of quantization, which is connected to the log-loss distortion criterion. The main focus is the regime of low $I(X;Y)$, where it is shown that, if $X$ is binary, a constant fraction of the mutual information can always be preserved using $\mathcal{O}(\log(1/I(X;Y)))$ quantization levels, and there exist distributions for which this many quantization levels are necessary. Furthermore, for larger finite alphabets $2 < |\mathcal{X}| < \infty$, it is established that an $\eta$-fraction of the mutual information can be preserved using roughly $(\log(| \mathcal{X} | /I(X;Y)))^{\eta\cdot(|\mathcal{X}| - 1)}$ quantization levels

Bottleneck Problems: Information and Estimation-Theoretic View

Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems (e.g., Mrs. Gerber's Lemma), strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding and dependence dilution. Leveraging these connections, we prove a new cardinality bound for the auxiliary variable in IB, making its computation more tractable for discrete random variables. In the second part, we introduce a general family of optimization problems, termed as \textit{bottleneck problems}, by replacing mutual information in IB and PF with other notions of mutual information, namely $f$-information and Arimoto's mutual information. We then argue that, unlike IB and PF, these problems lead to easily interpretable guarantee in a variety of inference tasks with statistical constraints on accuracy and privacy. Although the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to binary case, we derive closed form expressions for several bottleneck problems

On Data-Processing and Majorization Inequalities for ff-Divergences with Applications

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.Comment: This paper is published in the Entropy journal, vol. 21, no. 10, paper 1022, pages 1-80, October 21, 2019 (https://www.mdpi.com/1099-4300/21/10/1022

Bounds on the Entropy of a Function of a Random Variable and Their Applications

It is well known that the entropy H(X) of a discrete random variable X is always greater than or equal to the entropy H(f(X)) of a function f of X, with equality if and only if f is one-to-one. In this paper, we give tight bounds on H(f(X)), when the function f is not one-to-one, and we illustrate a few scenarios, where this matters. As an intermediate step toward our main result, we derive a lower bound on the entropy of a probability distribution, when only a bound on the ratio between the maximal and minimal probabilities is known. The lower bound improves on previous results in the literature, and it could find applications outside the present scenario