Abstract-A new class of information-theoretic divergence measures based on the Shannon entropy is introduced. Unlike the well-known Kullback divergences, the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions in-volved. More importantly, their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds. These bounds are crucial in many applications of divergence measures. The new measures are also well characterized by the properties of nonnegativity, finiteness, semiboundedness, and boundedness. Index Terms-Divergence, dissimilarity measure, discrimination in-formation, entropy, probability of error bounds. I.

In the typical nonparametric approach to pattern classification, random data (the training set of patterns) are collected and used to design a decision rule (classifier). One of the most well known such rules is the k-nearest-neighbor decision rule (also known as instance-based learning, and lazy learning) in which an unknown pattern is classified into the majority class among its k nearest neighbors in the training set. Several questions related to this rule have received considerable attention over the years. Such questions include the following. How can the storage of the training set be reduced without degrading the performance of the decision rule? How should the reduced training set be selected to represent the different classes? How large should k be? How should the value of k be chosen? Should all k neighbors be equally weighted when used to decide the class of an unknown pattern? If not, how should the weights be chosen? Should all the features (attributes) we weighted equally and if not how should the feature weights be chosen? What distance metric should be used? How can the rule be made robust to overlapping classes or noise present in the training data? How can the rule be made invariant to scaling of the measurements? Geometric proximity graphs such as Voronoi diagrams and their many relatives provide elegant solutions to most of these problems. After a brief and non-exhaustive review of some of the classical canonical approaches to solving these problems, the methods that use proximity graphs are discussed, some new observations are made, and avenues for further research are proposed.