  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.
