Intrinsic dimension identification via graph-theoretic methods

Three graph theoretical statistics are considered for the problem of estimating the intrinsic dimension of a data set. The first is the ‘‘reach’’ statistic, r j,k, proposed in Brito et al. (2002) [4] for the problem of identification of Euclidean dimension. The second,Mn, is the sample average of squared degrees in the minimum spanning tree of the data, while the third statistic, Uk n , is based on counting the number of common neighbors among the knearest, for each pair of sample points {Xi, Xj}, i < j ≤ n. For the first and third of these statistics, central limit theorems are proved under general assumptions, for data living in an m-dimensional C1 submanifold of Rd, and in this setting, we establish the consistency of intrinsic dimension identification procedures based on r j,k and Uk n . For Mn, asymptotic results are provided whenever data live in an affine subspace of Euclidean space. The graph theoretical methods proposed are compared, via simulations, with a host of recently proposed nearest neighbor alternatives. © 2013 Elsevier Inc. All rights reserved.