Local angles and dimension estimation from data on manifolds

For data living in a manifold M⊆R ^m and a point p∈M, we consider a statistic U _k,n which estimates the variance of the angle between pairs (X _i −p,X _j −p) of vectors, for data points X _i , X _j , near p, and we evaluate this statistic as a tool for estimation of the intrinsic dimension of M at p. Consistency of the local dimension estimator is established and the asymptotic distribution of U _k,n is found under minimal regularity assumptions. Performance of the proposed methodology is compared against state-of-the-art methods on simulated data and real datasets.