Dimensionality, Discriminability, Density and Distance Distributions

For many large-scale applications in data mining, machine learning, and multimedia, fundamental operations such as similarity search, retrieval, classification, clustering, and anomaly detection generally suffer from an effect known as the `curse of dimensionality´. As the dimensionality of the data increases, distance values tend to become less discriminative, due to their increasing relative concentration about the mean of their distribution. For this reason, researchers have considered the analysis of structures and methods in terms of measures of the intrinsic dimensionality of the data sets. This paper is concerned with a generalization of a discrete measure of intrinsic dimensionality, the expansion dimension, to the case of continuous distance distributions. This notion of the intrinsic dimensionality of a distribution is shown to precisely coincide with a natural notion of the indiscriminability of distances and features. Furthermore, for any distance distribution with differentiable cumulative density function, a fundamental relationship is shown to exist between probability density, the cumulative density (cumulative probability divided by distance), intrinsic dimensionality, and discriminability.