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Vector Diffusion Maps and the Connection Laplacian

Author(s): Singer, Amit; Wu, Hau-tieng

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Abstract: We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian eigenmaps. While existing methods are either directly or indirectly related to the heat kernel for functions over the data, VDM is based on the heat kernel for vector fields. VDM provides tools for organizing complex data sets, embedding them in a low dimensional space, and interpolating and regressing vector fields over the data. In particular, it equips the data with a metric, which we refer to as the {\em vector diffusion distance}. In the manifold learning setup, where the data set is distributed on (or near) a low dimensional manifold $\MM^d$ embedded in $\RR^{p}$, we prove the relation between VDM and the connection-Laplacian operator for vector fields over the manifold.
Publication Date: Aug-2012
Electronic Publication Date: 30-Mar-2012
Citation: Singer, Amit, Wu, Hau-tieng. (Vector Diffusion Maps and the Connection Laplacian) Comm. Pure Appl. Math., 65 (8) (2012), pp. 1067-1144
DOI: 10.1002/cpa.21395
Pages: 1067-1144
Type of Material: Journal Article
Journal/Proceeding Title: Bookmark Communications on pure and applied mathematics
Version: Author's manuscript

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