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.

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