Linear-projection diffusion on smooth Euclidean submanifolds

To process massive high-dimensional datasets, we utilize the underlying assumption that data on manifold is approximately linear in sufficiently small patches (or neighborhoods of points) that are sampled with sufficient density from the manifold. Under this assumption, each patch can be represented...

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Bibliographic Details
Published in:Applied and computational harmonic analysis Vol. 34; no. 1; pp. 1 - 14
Main Authors: Wolf, Guy, Averbuch, Amir
Format: Journal Article
Language:English
Published: Elsevier Inc 01.01.2013
Subjects:
Density
Diffusion
Diffusion maps
Generators
Kernel method
Manifold learning
Manifolds
Mathematical analysis
Operators
Projection
Stochastic processing
Tangents
Vector processing
Vector processing
Stochastic processing
Manifold learning
Diffusion maps
Kernel method
ISSN:1063-5203, 1096-603X
Online Access:Get full text
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Summary:To process massive high-dimensional datasets, we utilize the underlying assumption that data on manifold is approximately linear in sufficiently small patches (or neighborhoods of points) that are sampled with sufficient density from the manifold. Under this assumption, each patch can be represented (up to a small approximation error) by a tangent space of the manifold in its area and the tangential point of this tangent space. We extend previously obtained results (Salhov et al., 2012 [18]) for the finite construction of a linear-projection diffusion (LPD) super-kernel by exploring its properties when it becomes continuous. Specifically, its infinitesimal generator and the stochastic process defined by it are explored. We show that the resulting infinitesimal generator of this super-kernel converges to a natural extension of the original diffusion operator from scalar functions to vector fields. This operator is shown to be locally equivalent to a composition of linear projections between tangent spaces and the vector-Laplacians on them. We define a LPD process by using the LPD super-kernel as a transition operator while extending the process to be continuous. The obtained LPD process is demonstrated on a synthetic manifold.
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ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2012.03.003