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Connecting dots: from
local covariance to empirical intrinsic geometry and locally
linear embedding
John Malik, Chao Shen, Hau-Tieng Wu and Nan Wu |
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Vol. 1 (2019), No. 4, 515–542
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DOI: 10.2140/paa.2019.1.515
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Abstract |
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Local covariance structure under the manifold setup has been widely applied in the machine-learning community. Based on the established theoretical results, we provide an extensive study of two relevant manifold learning algorithms, empirical intrinsic geometry (EIG) and locally linear embedding (LLE) under the manifold setup. Particularly, we show that without an accurate dimension estimation, the geodesic distance estimation by EIG might be corrupted. Furthermore, we show that by taking the local covariance matrix into account, we can more accurately estimate the local geodesic distance. When understanding LLE based on the local covariance structure, its intimate relationship with the curvature suggests a variation of LLE depending on the “truncation scheme”. We provide a theoretical analysis of the variation. |
Keywords
local covariance matrix, empirical intrinsic geometry,
locally linear embedding, geodesic distance, latent space
model, Mahalanobis distance
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Mathematical Subject Classification 2010
Primary: 62-04, 62-07, 68P01
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Milestones
Received: 8 April 2018
Revised: 21 November 2018
Accepted: 2 January 2019
Published: 12 October 2019
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Authors |
| John Malik | |
| Department of Mathematics Duke University Durham, NC United States |
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| Chao Shen | |
| Department of Mathematics Duke University Durham, NC United States |
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| Hau-Tieng Wu | |
| Department of Mathematics and
Department of Statistical Science Duke University Durham, NC United States |
Mathematics Division National Center for Theoretical Sciences Taipei Taiwan |
| Nan Wu | |
| Department of Mathematics Duke University Durham, NC United States |
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