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。主要结果证明了脊之间的包含关系：$\cR(f\circ p)\subseteq \cR(p)$，条件是在函数$p$的值域上变换$f$严格递增且凹。此外，给定一个底层真实流形$\cM$，我们证明$\cR(f\circ p)$与其在$\cM$上投影之间的Hausdorff距离小于$\cR(p)$与相应投影之间的Hausdorff距离。这一结果启发我们在进行脊估计之前应用递增且凹的变换。具体地，我们证明幂变换$f^{q}(y)=y^q/q,-\infty<q\leq 1$在$\RR_+$上递增且凹，因此当$p$严格大于零时可以利用此类幂变换。数值实验展示了所提方法的优越性。