Summary: | Let X = X ∪ Z be a data set in ℝD, where X is the training set and Z the testing one. Assume that a kernel method produces a dimensionality reduction (DR) mapping 𝔉: X → ℝd (d ≪ D) that maps the high-dimensional data X to its row-dimensional representation Y = 𝔉(X). The out-of-sample extension of dimensionality reduction problem is to find the dimensionality reduction of X using the extension of 𝔉 instead of re-training the whole data set X. 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.
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