Local angles and dimension estimation from data on manifolds
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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 of vectors X_i-p and X_j-p, for data points X_i, X_j, near p, and 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.
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