{"id":147436,"date":"2003-01-01T00:00:00","date_gmt":"2003-01-01T00:00:00","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/msr-research-item\/spectral-clustering-and-kernel-pca-are-learning-eigenfunctions\/"},"modified":"2018-10-16T21:14:12","modified_gmt":"2018-10-17T04:14:12","slug":"spectral-clustering-and-kernel-pca-are-learning-eigenfunctions","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/spectral-clustering-and-kernel-pca-are-learning-eigenfunctions\/","title":{"rendered":"Spectral Clustering and Kernel PCA are Learning Eigenfunctions"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In this paper, we show a direct equivalence between spectral clustering and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of a kernel, when the functions are from a function space whose scalar product is defined with respect to a density model. This defines a natural mapping for new data points, for methods that only provided an embedding, such as spectral clustering and Laplacian eigenmaps. The analysis hinges on a notion of generalization for embedding algorithms based on the estimation of underlying eigenfunctions, and suggests ways to improve this generalization by smoothing the data empirical distribution.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper, we show a direct equivalence between spectral clustering and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of a kernel, when the functions are from a function space whose scalar product is defined with respect to a density model. 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