Ridges play a vital role in accurately approximating the underlying structure of manifolds. In this paper, we explore the ridge's variation by applying a concave nonlinear transformation to the density function. Through the derivation of the Hessian matrix, we observe that nonlinear transformations yield a rank-one modification of the Hessian matrix. Leveraging the variational properties of eigenvalue problems, we establish a partial order inclusion relationship among the corresponding ridges. We intuitively discover that the transformation can lead to improved estimation of the tangent space via rank-one modification of the Hessian matrix. To validate our theories, we conduct extensive numerical experiments on synthetic and real-world datasets that demonstrate the superiority of the ridges obtained from our transformed approach in approximating the underlying truth manifold compared to other manifold fitting algorithms.

翻译：岭在精确逼近流形底层结构中起着关键作用。本文通过将凹非线性变换应用于密度函数，探讨了岭的变化。通过Hessian矩阵的推导，我们观察到非线性变换对Hessian矩阵产生了秩一修正。利用特征值问题的变分性质，我们建立了对应岭之间的偏序包含关系。我们直观地发现，该变换通过Hessian矩阵的秩一修正能够改进切空间估计。为验证理论，我们在合成和真实数据集上进行了大量数值实验，结果表明，与其他流形拟合算法相比，经变换方法得到的岭在逼近底层真实流形方面具有优越性。