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

2013/05/01 by Monnig, Nathan D., Fornberg, Bengt, Meyer, Francois G.
#Data Analysis #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Statistics and Probability (physics.data-an)

paper · doi:10.48550/arxiv.1305.0258

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 Nyström extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nyström extension with suggestions for improvement.

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