15th. International Conference on Pattern Recognition Weighting Prototypes. A New Editing Approach. R. Paredes and E. Vidal Instituto Tecnol´ ogico de Inform´ atica, Universidad Polit´ ecnica de Valencia, 46071 Valencia, Spain. rparedes@iti.upv.es, evidal@iti.upv.es Abstract It is well known that editing techniques can be applied to (large) sets of prototypes in order to bring the error rate of the Nearest Neighbour classifier close to the optimal Bayes risk. However, in practice, the behaviour of these tech- niques uses to be much worse than expected from the asymp- totic predictions. A novel editing technique is introduced here which explicitly aims at obtaining a good editing rule for each given prototype set. This is achieved by first learn- ing an adequate assignment of a weight to each prototype and then pruning out those prototypes having large weights. Experiments are presented which clearly show the superior- ity of this new method, specially for small data sets and/or large dimensions. Keywords: Editing, Condensing, Nearest Neighbour, Classification, Weighted Prototypes, Gradient Descent. 1 Introduction The Nearest neighbor (NN) rule has been used for many pattern recognition applications. The behaviour of the NN rule when the number of prototypes, , is going to infinity is well known [1]. Under this asymptotic assumption the error risk is bounded above by , where is the Bayes risk. The NN rule can be easily extended to the k-NN rule, which asymptotically approaches the Bayes behaviour as both and tend to infinity, while is kept small. However, for finite data sets it is difficult to fulfill at once these three requirements for and in practice. It is also well known that the performance of the plain NN rule can be boosted by using simple Editing Techniques [10, 9, 8, 2, 6, 3] which attempt “cleaning” inter-class over- lap regions, thereby leading to smooth NN-based decision boundaries between classes. Under the unbounded data-set- size (and computing time) assumption, the Multi Edit al- gorithm [2] has been shown to yield sets of prototypes for which the plain 1-NN rule matches Bayes performance. Unfortunately, all the above good asymptotic results usu- ally degrade dramatically for finite data sets and, in practice, most of these techniques often become useless since the number of available prototypes is much smaller than what would be required for the theoretical asymptotic predictions to make sense [4]. It is interesting to note that most of the efforts for devel- oping the above mentioned techniques have focused on im- proving asymptotic performance. In particular, no attempts seem to have been made so far to develop adequate edit- ing techniques that explicitly try to improve the NN perfor- mance for given (and small) sets of training data. With this purpose, a new editing approach is proposed here which, for each set of prototypes, aims at obtaining an editing rule which is adequate for this set. The main idea is to assign a weight to each prototype and, then, to edit out those prototypes having large weights. Obviously, for actually achieving our purposes, these weights need to be estimated by optimizing a suitable criterion index. 2 Approach Assume a classical pattern recognition problem with classes in a representation space and let be an appropriate metric in . Let x x x be a training set, where x and . If y is a test sample, the NN rule can be used to estimate the class of y as a class such that yx is minimum. Now suppose a weight is assigned to each prototype x by defining a weighted dissimilarity measure as: yx yx (1) where is the weight associated to the prototype x. We will refer to this function as “Weighted Prototype” (WP) dissimilarity. Weights can help defining editing criteria. Classical edit- ing techniques would amount to eliminate those prototypes x for which , while for non eliminated prototypes, x , . But now, different editing results can be