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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Vydané v:	Applied and computational harmonic analysis Ročník 34; číslo 1; s. 1 - 14
Hlavní autori:	Wolf, Guy, Averbuch, Amir
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier Inc 01.01.2013
Predmet:	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
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Abstract	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.
AbstractList	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. 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.
Author	Averbuch, Amir Wolf, Guy
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Cites_doi	10.1126/science.290.5500.2323 10.1037/h0071325 10.1162/089976698300017467 10.1073/pnas.1031596100 10.1137/040616024 10.1016/j.acha.2010.10.001 10.1126/science.290.5500.2319 10.1109/CSSE.2008.1332 10.1162/089976603321780317 10.1016/j.acha.2006.04.006 10.1109/SSP.2009.5278634 10.1007/BF02289565 10.1016/j.acha.2005.07.005 10.1023/A:1023705401078 10.1016/j.acha.2011.11.003 10.1007/s11263-007-0056-x
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Keywords	Vector processing Stochastic processing Manifold learning Diffusion maps Kernel method
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SubjectTerms	Density Diffusion Diffusion maps Generators Kernel method Manifold learning Manifolds Mathematical analysis Operators Projection Stochastic processing Tangents Vector processing
Title	Linear-projection diffusion on smooth Euclidean submanifolds
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