  Nonlinear dimensionality reduction embeddings computed from datasets do not
provide a mechanism to compute the inverse map. In this paper, we address the
problem of computing a stable inverse map to such a general bi-Lipschitz map.
Our approach relies on radial basis functions (RBFs) to interpolate the inverse
map everywhere on the low-dimensional image of the forward map. We demonstrate
that the scale-free cubic RBF kernel performs better than the Gaussian kernel:
it does not suffer from ill-conditioning, and does not require the choice of a
scale. The proposed construction is shown to be similar to the Nystr\"om
extension of the eigenvectors of the symmetric normalized graph Laplacian
matrix. Based on this observation, we provide a new interpretation of the
Nystr\"om extension with suggestions for improvement.

Numerical Analysis;Numerical Analysis;Data Analysis, Statistics and Probability;Machine Learning