AUTHOR=Wang Jianzhong 

TITLE=Least Square Approach to Out-of-Sample Extensions of Diffusion Maps

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=5

YEAR=2019

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00024

DOI=10.3389/fams.2019.00024

ISSN=2297-4687

ABSTRACT=<p>Let <italic>X</italic> = <bold>X</bold> ∪ <bold>Z</bold> be a data set in ℝ<sup><italic>D</italic></sup>, where <bold>X</bold> is the training set and <bold>Z</bold> the testing one. Assume that a kernel method produces a dimensionality reduction (DR) mapping 𝔉: <bold>X</bold> → ℝ<sup><italic>d</italic></sup> (<italic>d</italic> ≪ <italic>D</italic>) that maps the high-dimensional data <bold>X</bold> to its row-dimensional representation <bold>Y</bold> = 𝔉(<bold>X</bold>). The out-of-sample extension of dimensionality reduction problem is to find the dimensionality reduction of <italic>X</italic> using the extension of 𝔉 instead of re-training the whole data set <italic>X</italic>. 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. This analysis also provides a uniform treatment of many popular out-of-sample algorithms based on kernel methods. We illustrate the validity of the developed out-of-sample DR algorithms in several examples.</p>