Ridge Estimation with Nonlinear Transformations

Ridge estimation is an important manifold learning technique. The goal of this paper is to examine the effects of nonlinear transformations on the ridge sets. The main result proves the inclusion relationship between ridges: $\cR(f\circ p)\subseteq \cR(p)$, provided that the transformation $f$ is strictly increasing and concave on the range of the function $p$. Additionally, given an underlying true manifold $\cM$, we show that the Hausdorff distance between $\cR(f\circ p)$ and its projection onto $\cM$ is smaller than the Hausdorff distance between $\cR(p)$ and the corresponding projection. This motivates us to apply an increasing and concave transformation before the ridge estimation. In specific, we show that the power transformations $f^{q}(y)=y^q/q,-\infty<q\leq 1$ are increasing and concave on $\RR_+$, and thus we can use such power transformations when $p$ is strictly positive. Numerical experiments demonstrate the advantages of the proposed methods.