The Annals of Statistics

Nonparametric ridge estimation

Christopher R. Genovese, Marco Perone-Pacifico, Isabella Verdinelli, and Larry Wasserman

Full-text: Open access

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.

Article information

Source
Ann. Statist. Volume 42, Number 4 (2014), 1511-1545.

Dates
First available in Project Euclid: 7 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1407420007

Digital Object Identifier
doi:10.1214/14-AOS1218

Mathematical Reviews number (MathSciNet)
MR3262459

Zentralblatt MATH identifier
1310.62045

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62H12: Estimation

Keywords
Ridges density estimation manifold learning

Citation

Genovese, Christopher R.; Perone-Pacifico, Marco; Verdinelli, Isabella; Wasserman, Larry. Nonparametric ridge estimation. Ann. Statist. 42 (2014), no. 4, 1511--1545. doi:10.1214/14-AOS1218. https://projecteuclid.org/euclid.aos/1407420007.


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