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.

翻译：脊估计是一种重要的流形学习技术。本文旨在研究非线性变换对脊集的影响。主要结果证明了脊集之间的包含关系：$\cR(f\circ p)\subseteq \cR(p)$，前提是变换$f$在函数$p$的值域上严格递增且凹。此外，给定一个底层真实流形$\cM$，我们证明了$\cR(f\circ p)$与其在$\cM$上的投影之间的豪斯多夫距离，小于$\cR(p)$与相应投影之间的豪斯多夫距离。这促使我们在脊估计之前应用递增且凹的变换。具体而言，我们证明了幂变换$f^{q}(y)=y^q/q,-\infty<q\leq 1$在$\RR_+$上是递增且凹的，因此当$p$严格为正时，我们可以使用此类幂变换。数值实验展示了所提方法的优势。