Least Square Approach to Out-of-Sample Extensions of Diffusion Maps - Wang, Jianzhong (Frontiers in Applied Mathematics and Statistics, 2019)
Let Z be a union of a training set X and a testing set Y. Assume that a kernel method produces a dimensionality reduction (DR) mapping P that maps the high-dimensional data X to its row-dimensional representation P(X). The out-of-sample extension of dimensionality reduction problem is to find the dimensionality reduction of Y using the extension of P instead of re-training the whole set Z. In this paper, utilizing the framework of reproducing kernel Hilbert space theory, we introduce a least-square approach to extensions of the popular DR mappings called Diffusion maps (Dmaps). We establish a theoretic analysis for the out-of-sample DR Dmaps, which also provides a uniform treatment of many popular out-of-sample algorithms based on kernel methods. We also illustrate the validity of the developed out-of-sample DR algorithms in several examples.