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.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1218 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We find lower and upper bounds for the risk of estimating a manifold in
Hausdorff distance under several models. We also show that there are close
connections between manifold estimation and the problem of deconvolving a
singular measure.Comment: Published in at http://dx.doi.org/10.1214/12-AOS994 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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