Local angles and dimension estimation from data on manifolds

TL;DR

This paper introduces a new local angle-based statistic for estimating the intrinsic dimension of data on manifolds, demonstrating its consistency and asymptotic properties, and compares its performance with existing methods.

Contribution

The paper proposes a novel angle-based statistic for local dimension estimation on manifolds, with proven consistency and asymptotic distribution, validated through simulations.

Findings

The statistic consistently estimates local dimension.

Asymptotic distribution derived under minimal assumptions.

Performs favorably compared to existing methods on simulated data.

Abstract

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.