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- W2891372043 abstract "Kernel-based non-linear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph associated with the dataset. When each individual data object carries additional structural details, however, the correspondence relations between these structures provide extra information that can be leveraged for studying the dataset using the graph. Based on this observation, we generalize Diffusion Maps (DM) in manifold learning and introduce the framework of Horizontal Diffusion Maps (HDM). We model a dataset with pairwise structural correspondences as a fibre bundle equipped with a connection. We demonstrate the advantage of incorporating such additional information and study the asymptotic behavior of HDM on general fibre bundles. In a broader context, HDM reveals the sub-Riemannian structure of high-dimensional datasets, and provides a nonparametric learning framework for datasets with structural correspondences." @default.
- W2891372043 created "2018-09-27" @default.
- W2891372043 creator A5057450352 @default.
- W2891372043 date "2016-02-06" @default.
- W2891372043 modified "2023-09-26" @default.
- W2891372043 title "The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps" @default.
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- W2891372043 doi "https://doi.org/10.48550/arxiv.1602.02330" @default.
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