The problem of nearest-neighbor  condensation deals with finding a subset R from a set of labeled points P such that for every point p \in P the nearest-neighbor of p in R has the same label as p. This is motivated by applications in classification, where the nearest-neighbor rule assigns to an unlabeled query point the label of its nearest-neighbor in the point set. In this context, condensation aims to reduce the size of the setneeded to classify new points . However, finding such subsets of minimum cardinality is NP-hard, and most research has focused on practical heuristics without performance guarantees. Additionally, the use of exact nearest-neighbors is always assumed, ignoring the effect of condensation in the classification accuracy when nearest-neighbors are computed approximately. In this paper, we address these shortcomings by proposing new approximation-sensitive criteria for the nearest-neighbor condensationproblem, along with practical algorithms with provable performance guarantees. We characterize sufficient conditions to guarantee correct classification of unlabeled points using approximate nearest-neighbor queries on these subsets, which introduces the notion of coresets for classification with the nearest-neighbor rule. Moreover , we prove that it is NP-hard to compute subsets with these characteristics, whose cardinality approximates that of the minimum cardinality subset. Additionally, we propose new algorithms for computing such subsets, with tight approximation factors in general metrics, and improved factors for doubling metrics and \ell_p metrics with p%DIFDELCMD < \geq2%%% . Finally , we show an alternative implementation scheme that reduces the worst-case time complexity of one of these algorithms , becoming the first truly subquadratic approximation algorithm for the nearest-neighbor condensationproblem .
Given a training set P of labeled points, the nearest-neighbor rule predicts the class of an unlabeled query point as the label of its closest point in the set. To improve the time and space complexity of classification, a natural question is how to reduce the training set without significantly affecting the accuracy of the nearest-neighbor rule. Nearest-neighbor condensation deals with finding a subset R \subseteq P such that for every point p \in P , p's nearest-neighbor in R has the same label as p. This relates to the concept of coresets, which can be broadly defined as subsets of the set, such that an exact result on the coreset corresponds to an approximate result on the original set . However, the guarantees of a coreset hold for any query point, and not only for the points of the training set. This paper introduces the concept of coresets for nearest-neighbor classification. We extend existing criteria used for condensation, and prove sufficient conditions to correctly classify any query point when using these subsets. Additionally , we prove that %DIFDELCMD < \geq2%%% finding such subsets of minimum cardinality is NP-hard, and propose quadratic-time approximation algorithms with provable upper-bounds on the size of their selected subsets. Moreover , we show how to improve one of these algorithms to have subquadratic runtime, being the first of this kind for condensation .
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finding such subsets of minimum cardinality is NP-hard, and most research has focused on practical heuristics without performance guarantees. Additionally, the use of exact nearest-neighbors is always assumed, ignoring the effect of condensation in the classification accuracy when nearest-neighbors are computed approximately. In this paper, we address these shortcomings by proposing new approximation-sensitive criteria for
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training set. This paper introduces the concept of coresets for nearest-neighbor classification. We extend existing criteria used for condensation, and prove
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