Geometric proximity graphs for improving nearest neighbor methods in instance-based learning and data mining

In the typical nonparametric approach to classification in instance-based learning and data mining, 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 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? How can the nearest neighbors of a new point be computed efficiently? What is the smallest neural network that can implement nearest neighbor decision rules? Geometric proximity graphs such as Voronoi diagrams and their many relatives provide elegant solutions to these problems, as well as other related data mining problems such as outlier detection. After a non-exhaustive review of some of the classical canonical approaches to these problems, the methods that use proximity graphs are discussed, some new observations are made, and open problems are listed.