But we already saw that some random variables (e.g. the number of balls in a bin) fall off exponentially with distance from the mean. So Markov and Chebyshev are very poor bounds for those kinds of random variables. The Chernoff bound applies to a class of random variables and does give exponential fall-off of probability with distance from the mean. The critical condition that’s needed for a C...

It is well known that lattice-valued rough sets are important branches of fuzzy sets. The axiomatic characterization and related topology the main research directions For L=(L,⊛), a complete co-residuated lattice (CCRL), Qiao recently defined an L-fuzzy lower approximation operator (LFLAO) on basis relation. In this article, we give further study Qiao’s LFLAO around induced L-topology. Firstly,...

Shannon entropy is the most crucial foundation of Information Theory, which has been proven to be effective in many fields such as communications. Rényi entropy and Chernoff information are other two popular measures of information with wide applications. The mutual information is effective to measure the channel information for the fact that it reflects the relation between output variables an...

Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of...

We present a performance analysis of three linear dimensionality reduction techniques: Fisher’s discriminant analysis (FDA), and two methods introduced recently based on the Chernoff distance between two distributions, the Loog and Duin (LD) method, which aims to maximize a criterion derived from the Chernoff distance in the original space, and the one introduced by Rueda and Herrera (RH), whic...

Today we will look at a matrix analog of the standard scalar Chernoff bounds. This matrix analog will be used in the next lecture when we talk about graph sparsification. While we’re more interested in the application of the theorem than its proof, it’s still useful to see the similarities and the differences of moving from the proof of the result for scalars to the same result for matrices. Sc...

In this paper we present a probabilistic approach to some geometric problems in asymptotic convex geometry. The aim of this paper is to demonstrate that the well known Chernoff bounds from probability theory can be used in a geometric context for a very broad spectrum of problems, and lead to new and improved results. We begin by briefly describing Chernoff bounds, and the way we will use them....

We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the minimal probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long-standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernof...