Vol. 1, No. 4, 2019

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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

Vol. 1 (2019), No. 4, 515–542
DOI: 10.2140/paa.2019.1.515
Abstract

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
Mathematical Subject Classification 2010
Primary: 62-04, 62-07, 68P01
Milestones
Received: 8 April 2018
Revised: 21 November 2018
Accepted: 2 January 2019
Published: 12 October 2019
Authors
John Malik
Department of Mathematics
Duke University
Durham, NC
United States
Chao Shen
Department of Mathematics
Duke University
Durham, NC
United States
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