Dimension induced clustering

Abstract

It is commonly assumed that high-dimensional datasets contain points most of which are located in low-dimensional manifolds. Detection of low-dimensional clusters is an extremely useful task for performing operations such as clustering and classification, however, it is a challenging computational problem. In this paper we study the problem of finding subsets of points with low intrinsic dimensionality. Our main contribution is to extend the definition of fractal correlation dimension, which measures average volume growth rate, in order to estimate the intrinsic dimensionality of the data in local neighborhoods. We provide a careful analysis of several key examples in order to demonstrate the properties of our measure. Based on our proposed measure, we introduce a novel approach to discover clusters with low dimensionality. The resulting algorithms extend previous density based measures, which have been successfully used for clustering. We demonstrate the effectiveness of our algorithms for discovering low-dimensional m-flats embedded in high dimensional spaces, and for detecting low-rank sub-matrices.