Nonparametric ridge estimation

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.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

Manifold learning and unwrapping using density ridges

“…A kernel density ridge can be used to completely describe a biased version, called a surrogate, of a manifold embedded in R D given sufficient samples and bounded noise [20]. The main motivation for using density ridges to locally unwrap nonlinear manifolds comes from the following.…”

“…Of great value is the recent paper by Genovese et al, [20], which showed that the Hausdorff distance between a d-dimensional manifold embedded in D dimensions and the d-dimensional ridge of the density is bounded under certain restrictions wrt. noise and the closeness of the density estimate to the true density.…”

“…The most intuitive setting, which is most commonly used [2,20,38] is to use the top eigenvector(s) corresponding to eigenvalue(s) as the orthogonal subspace. We recall that the eigenvalues and eigenvectors of the Hessian matrix of a function points to the directions of largest second order change.…”

“…[3] -can be interpreted in a differential geometry setting. We recall that Genovese et al, [20], has shown that kernel density derivatives can be used to estimate a smoothed version of an underlying manifold sampled from data with noise.…”

“…Genovese et al have provided a theoretical analysis of non-parametric density ridge estimation [20] and Chen provides asymptotic theory and formulates density ridges as so-called solution manifolds [21].…”