Sharif Digital Repository

Based on the notion of maximal correlation, we introduce a new measure of correlation between two different rankings of the same group of items. Our measure captures various types of correlation detected in previous measures of rank correlation like the Spearman correlation and the Kendall tau correlation. We show that the maximal rank correlation satisfies the data processing and tensorization properties (that make ordinary maximal correlation applicable to problems in information theory). Furthermore, MRC is shown to be intimately related to the FKG inequality. Finally, we pose the problem of the complexity of the computation of this new measure. We make partial progress by giving a simple... 

In this paper we provide the correct tight constant to a data-processing inequality claimed by Erkip and Cover. The correct constant turns out to be a particular hypercontractivity parameter of (X,Y), rather than their squared maximal correlation. We also provide alternate geometric characterizations for both maximal correlation as well as the hypercontractivity parameter that characterizes the data-processing inequality  

Non-locality is the phenomenon of observing strong correlations among the outcomes of local measurements of a multipartite physical system. No-signaling boxes are the abstract objects for studying non-locality, and wirings are local operations on the space of no-signaling boxes. This means that, no matter how non-local the nature is, the set of physical non-local correlations must be closed under wirings. Then, one approach to identify the non-locality of nature is to characterize the closed sets of non-local correlations. Although non-trivial examples of wirings of no-signaling boxes are known, there is no systematic way to study wirings. In particular, given a set of no-signaling boxes, we... 

In this paper, we consider the problem of simulating one copy of a zero-capacity two-way channel from arbitrary many copies of another two-way channel. The motivation for this problem comes from foundation of quantum physics where zero-capacity two-way channels (a.k.a. no-signaling boxes) serve as the abstract objects for studying non-locality. We provide necessary conditions for the possibility of channel simulation using two measures of correlation, namely maximal correlation and hypercontractivity ribbon. It is shown that when these measures are defined appropriately for two-way channels, they are monotonically decreasing under local operations. Our result allows us to establish a... 

A function is said to be additive if, similar to mutual information, expands by a factor of n, when evaluated on n i.i.d. repetitions of a source or channel. On the other hand, a function is said to satisfy the tensorization property if it remains unchanged when evaluated on i.i.d. repetitions. Additive rate regions are of fundamental importance in network information theory, serving as capacity regions or upper bounds thereof. Tensorizing measures of correlation have also found applications in distributed source and channel coding problems as well as the distribution simulation problem. Prior to our work only two measures of correlation, namely the hypercontractivity ribbon and maximal... 

A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in “non-degenerate” cases. Next, we turn to a distributed setting. In... 

A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to s "nondegenerate" dice that are c-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness... 

A measure of correlation is said to have the tensorization property if it does not change when computed for i.i.d. copies. More precisely, a measure of correlation between two random variables X, Y denoted by rho (X, Y), has the tensorization property if ρ(Xn, Yn)=ρ (X, Y) where (Xn, Yn) denotes n i.i.d. copies of (X, Y). Two well-known examples of such measures are the maximal correlation and the hypercontractivity ribbon (HC ribbon). We show that the maximal correlation and the HC ribbon are special cases of the new notion of Φ-ribbons, defined in this paper for a class of convex functions Φ. Φ-ribbon reduces to the HC ribbon and the maximal correlation for special choices of Φ, and is a...