Inverting Non-Linear Dimensionality Reduction with Scale-Free Radial Basis Interpolation

A numerical method is proposed to approximate the inverse of a general bi-Lipschitz nonlinear dimensionality reduction mapping, where the forward and consequently the inverse mappings are only explicitly defined on a discrete dataset. A radial basis function (RBF) interpolant is used to independently interpolate each component of the high-dimensional representation of the data as a function of its low-dimensional representation. The scale-free cubic RBF kernel is shown to perform better than the Gaussian kernel, as it does not require the difficult-to-choose scale parameter as an input, and does not suffer from illconditioning. The proposed numerical inverse is shown to be mathematically similar to the eigenvector interpolation known as the Nyström method, a commonly used numerical method for rapid approximation of eigenvectors of a dense weight matrix. Based on this observation, a critique of the Nyström method is provided, with suggestions for improvement.