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