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Using prototype methods to reduce the size of training datasets candrastically reduce the computational cost of classification with instance-basedlearning algorithms like the k-Nearest Neighbour classifier. The number anddistribution of prototypes required for the classifier to match its originalperformance is intimately related to the geometry of the training data. As aresult, it is often difficult to find the optimal prototypes for a givendataset, and heuristic algorithms are used instead. However, we consider aparticularly challenging setting where commonly used heuristic algorithms failto find suitable prototypes and show that the optimal prototypes can instead befound analytically. We also propose an algorithm for finding nearly-optimalprototypes in this setting, and use it to empirically validate the theoreticalresults.

Optimal 1-NN Prototypes for Pathological Geometries