High dimensional nearest neighbor classification based on mean absolute differences of inter-point distances

Abstract

Traditional nearest neighbor classifiers based on usual distance functions (e.g., Euclidean distance) often suffer in high dimension low sample size (HDLSS) situations, where phenomena like concentration of pairwise distances, violation of cluster assumptions and presence of hubs often have adverse effects on their performance. In order to cope with such problems, instead of usual distance functions, in this article we use a dissimilarity measure based on average of absolute differences between inter-point distances. Our proposed nearest neighbor classifier uses concentration of pairwise distances to its advantage, and it usually yield better performance in high dimension when such concentration occurs. Under appropriate regularity conditions, we proved the optimality of the misclassification probability of the proposed classifier in HDLSS asymptotic regime, where the training sample size remains fixed, and the dimension grows to infinity. Usefulness of the proposed method has also been demonstrated using several simulated and benchmark data sets.