A multivariate approach to dimension estimation on manifolds with triangle area U-statistics (2025)
For data on a manifold M ⊆ ℝᵐ and a point p ∈ M, this thesis introduces a multivariate estimator that leverages angle- and triangle area-based U-statistics (U₁, V₁, V₂) to assess the intrinsic dimension of M at p. By considering the variance of angles formed by pairs of nearby data points, and the mean and variance of triangle areas formed by triplets, the proposed methodology captures the manifold's local geometry. A multivariate approach using the Mahalanobis distance ensures robust dimension estimation by incorporating covariance structure. The estimator is evaluated through testing on both simulated manifolds and real-world datasets. Robust statistical tools, such as the Minimum Volume Ellipsoid (MVE), are employed to enhance reliability.