Finding Singular Features

TL;DR

This paper introduces a noise-robust method for detecting low-dimensional, high-density structures called singular features in noisy point clouds by identifying density ridges and analyzing Hessian eigenvalues.

Contribution

The paper proposes a novel approach for finding singular features in noisy data using density ridges and Hessian eigenvalues, improving over spectral clustering methods.

Findings

Effective detection of singular features in noisy point clouds.

Method outperforms spectral clustering in noisy environments.

Identifies structures with zero Lebesgue measure in high-dimensional spaces.

Abstract

We present a method for finding high density, low-dimensional structures in noisy point clouds. These structures are sets with zero Lebesgue measure with respect to the $D$-dimensional ambient space and belong to a $d<D$ dimensional space. We call them "singular features." Hunting for singular features corresponds to finding unexpected or unknown structures hidden in point clouds belonging to $\R^D$. Our method outputs well defined sets of dimensions $d<D$. Unlike spectral clustering, the method works well in the presence of noise. We show how to find singular features by first finding ridges in the estimated density, followed by a filtering step based on the eigenvalues of the Hessian of the density.