Local angles and dimension estimation from data on manifolds

For data living in a manifold $M\subseteq \mathbb{R}^m$ and a point $p\in M$ we consider a statistic $U{k,n}$ which estimates the variance of the angle between pairs of vectors $Xi-p$ and $Xj-p$, for data points $Xi$, $Xj$, 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.