Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation

Nathan D. Monnig; Bengt Fornberg; Francois G. Meyer

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arXiv · 1305.0258

Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation

Nathan D. Monnig·Bengt Fornberg·Francois G. Meyer

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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.

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BibTeX RIS

Nathan D. Monnig, Bengt Fornberg, Francois G. Meyer. 2013-05-01. Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation. https://arxiv.org/abs/1305.0258

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