Effectiveness of the Euclidean distance in high dimensional spaces

This paper presents analysis of applicability and performance of the Euclidean distance in relation to the dimensionality of the space. The effect of dimensionality on the behavior of Euclidean distance is explored; Furthermore, it is shown that the minimum distance approaches the maximum distance under a broader set of conditions without requiring the calculation of variance of random variables. It is demonstrated that the minimum distance approaches the maximum distance even for some low dimensional distributions, such as normal distribution. Many proposed measures not based directly on Euclidean distance cannot enlarger the difference between closest point and farthest point. The analysis has been performed on a wide range of artificial and publicly available datasets. As the variables of different distributions have different convergence rates, the results should not be interpreted to mean that the Euclidean distance is not applicable. In fact, it is shown in experiments that the Euclidean distance is very useful in the noncentral t-distribution even for the dimensionality higher than 10,000. Furthermore, it is observed that the behavior of Euclidean distance becomes more useful with increased number of samples. (C) 2015 Elsevier GmbH. All rights reserved.