Nonparametric ridge estimation

TL;DR

This paper introduces a method for estimating density ridges that extend mode finding, demonstrating consistency under mild conditions and applicability to hidden manifold structures, with practical algorithms validated through experiments.

Contribution

It presents a consistent nonparametric ridge estimation method that adapts the mean-shift algorithm for practical use in structure discovery.

Findings

Ridges of kernel density estimators consistently estimate true density ridges.

Estimated ridges are topologically similar to underlying manifolds in noisy data.

Numerical experiments confirm the accuracy of the adapted mean-shift algorithm.

Abstract

We study the problem of estimating the ridges of a density function. Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density. It can also be used to find hidden structure in point cloud data. We show that, under mild regularity conditions, the ridges of the kernel density estimator consistently estimate the ridges of the true density. When the data are noisy measurements of a manifold, we show that the ridges are close and topologically similar to the hidden manifold. To find the estimated ridges in practice, we adapt the modified mean-shift algorithm proposed by Ozertem and Erdogmus [J. Mach. Learn. Res. 12 (2011) 1249-1286]. Some numerical experiments verify that the algorithm is accurate.