Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation
Abstract
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.
Cite
@article{arxiv.1305.0258,
title = {Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation},
author = {Nathan D. Monnig and Bengt Fornberg and Francois G. Meyer},
journal= {arXiv preprint arXiv:1305.0258},
year = {2013}
}
Comments
Accepted for publication in Applied and Computational Harmonic Analysis