Local intrinsic dimensionality II: Multivariate analysis and distributional support

Abstract

Distance-based expansion models of intrinsic dimensionality have had recent application in the analysis of complexity of similarity applications, and in the design of efficient heuristics. This theory paper extends one such model, the local intrinsic dimension (LID), to a multivariate form that can account for the contributions of different distributional components towards the intrinsic dimensionality of the entire feature set, or equivalently towards the discriminability of distance measures defined in terms of these feature combinations. Formulas are established for the effect on LID under summation, product, composition, and convolution operations on smooth functions in general, and cumulative distribution functions in particular. For some of these operations, the dimensional or discriminability characteristics of the result are also shown to depend on a form of distributional support. As an example, an analysis is provided that quantifies the impact of introduced random Gaussian noise on the intrinsic dimension of data. Finally, a theoretical relationship is established between the LID model and the classical correlation dimension.