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Kernel-based non-linear dimensionality reduction methods, such as LocalLinear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwisedistances or similarity scores, with which one can construct and study aweighted graph associated with the dataset. When each individual data objectcarries additional structural details, however, the correspondence relationsbetween these structures provide extra information that can be leveraged forstudying the dataset using the graph. Based on this observation, we generalizeDiffusion Maps (DM) in manifold learning and introduce the framework ofHorizontal Diffusion Maps (HDM). We model a dataset with pairwise structuralcorrespondences as a fibre bundle equipped with a connection. We demonstratethe advantage of incorporating such additional information and study theasymptotic behavior of HDM on general fibre bundles. In a broader context, HDMreveals the sub-Riemannian structure of high-dimensional datasets, and providesa nonparametric learning framework for datasets with structuralcorrespondences.

The Diffusion Geometry of Fibre Bundles: Horizontal Diffusion Maps