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.

Nonparametric Ridge Estimation / Genovese Christopher, R.; Perone Pacifico, Marco; Verdinelli, Isabella; Wasserman, Larry. - In: ANNALS OF STATISTICS. - ISSN 0090-5364. - 42:4(2014), pp. 1511-1545. [10.1214/14-AOS1218]

Nonparametric Ridge Estimation

Perone-Pacifico Marco;
2014

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.
2014
Ridges; density estimation; manifold learning
Nonparametric Ridge Estimation / Genovese Christopher, R.; Perone Pacifico, Marco; Verdinelli, Isabella; Wasserman, Larry. - In: ANNALS OF STATISTICS. - ISSN 0090-5364. - 42:4(2014), pp. 1511-1545. [10.1214/14-AOS1218]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11385/182612
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