Theoretical analysis of nonparametric filament estimation

Abstract

This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density f . Points on the filament are considered as local extrema of the density when traversing the support of f along the integral curve driven by the vector field of second eigenvectors of the Hessian of f . We 'parametrize' points on the filaments by such integral curves, and thus both the estimation of integral curves and of filaments will be considered via a plug-in method using kernel density estimation. We establish rates of convergence and asymptotic distribution results for the estimation of both the integral curves and the filaments. The main theoretical result establishes the asymptotic distribution of the uniform deviation of the estimated filament from its theoretical counterpart. This result utilizes the extreme value behavior of non-stationary Gaussian processes indexed by manifolds M h , h ∈ (0, 1] as h → 0.

Extrema of rescaled locally stationary Gaussian fields on manifolds

“…It is shown in Qiao and Polonik (2015a) that the assumptions formulated in that paper insure that the processes U h (x) satisfy the assumptions of our main theorem in the special case of r = 1, n = 2, α = 2, and Q(δ) = 0 for δ > δ 0 . This in particular means that the function v(δ) = log(δ) −β for β > 0 works in this case.…”

“…This implies that the covariances of U h (x 1 ) and U h (x 2 ) become zero once the distance x 1 − x 2 exceeds a certain threshold. The derivation of the corresponding matrices D s can also be found in Qiao and Polonik (2015a).…”

A tutorial on kernel density estimation and recent advances

“…Ridges. Another interesting geometric structures are ridges (Genovese et al, 2014;Chen et al, 2015b;Qiao et al, 2016) of the density functions. Formally, ridges are defined as follows.…”

“…The convergence rate and topological characteristics were discussed in Genovese et al (2014). Chen et al (2015b) and Qiao et al (2016) both studied the asymptotic theory, and Chen et al (2015b) further proposed methods of constructing confidence sets of R. Ozertem and Erdogmus (2011) introduced the subspace-constrained mean shift (SCMS) algorithm to compute R. The red curves in the right panel of Figure 7 are estimated ridges from the SCMS algorithm.…”

Asymptotic theory for density ridges

“…In a recent paper, Qiao and Polonik (2014) give another asymptotic analysis for density ridges (called filaments in that paper). Their approach is quite different; they prove an extreme value distribution as the limiting result for estimating gradient ascent.…”