Invertible nonlinear cluster unwrapping

We propose that the orthogonal curvilinear coordinate systems revealed locally by the eigenvector flow derived from the Hessian of data density can be used as a mean to obtain local charts around modes. These charts can be stitched to form an atlas to define a global map of the data space, providing a base for data classification or clustering. A given point is projected to the ridges of the probability density by solving a differential equation which forces the gradient to be in the direction of the eigenvector corresponding to the largest eigenvalue of the Hessian. A curvilinear coordinate is then determined as the curve length along each ridge from the projection point to the mode. Since solving such differential equations numerically could be computationally prohibitive for large number of samples to be projected, we also present a diffeomorphic coordinate transformation model to approximate these Cartesian-to-curvilinear coordinate mappings. The model is primarily conceived as an interpolator, and the landmark training data are transformed exactly. The interpolation model is regularized in the Tikhonov sense using a user-specified differential operator. The proposed interpolation methodology is adapted from landmark-matching-based deformable image registration literature.