Power Transformed Density Ridge Estimation

Abstract

This paper proposes to control ridge estimation with nonlinear transformations. We establish an inclusion relationship between ridges with/without transformations: ${\mathcal R}(f\circ p)\subseteq {\mathcal R}(p)$, where ${\mathcal R}(p)$ is the $d$-dimensional ridge of $p:\mathbb {R}^{D}\to \mathbb {R}$ and $f$ is a strictly increasing and concave map defined on range$(p)$. This finding reveals the benefit of applying an increasing and concave transformation before ridge estimation. As $f^{q}(y)=y^{q}/q, q\leq 1$ are increasing and concave on $\mathbb {R}_+$, we use these power transformations to positive density functions, and then perform ridge estimation. The algorithm, named power-transformed subspace-constrained mean-shift (PSCMS), outperforms its competitors in numerical experiments.