Local angles and dimension estimation from data on manifolds
This work addresses the challenge of dimension estimation for manifold data, which is crucial for machine learning and data analysis, but it appears to be an incremental improvement over existing methods.
The authors tackled the problem of estimating the intrinsic dimension of a manifold from local data by proposing a statistic based on the variance of angles between nearby vectors, establishing its consistency and asymptotic distribution. They demonstrated competitive performance against state-of-the-art methods on simulated data, though specific numerical results were not provided.
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 $X_i-p$ 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.