期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2003
卷号：100
期号：10
页码：5591-5596
DOI：10.1073/pnas.1031596100
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：We describe a method for recovering the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based locally linear embedding, derives from a conceptual framework of local isometry in which the manifold M, viewed as a Riemannian submanifold of the ambient Euclidean space [R]n, is locally isometric to an open, connected subset {Theta} of Euclidean space [R]d. Because {Theta} does not have to be convex, this framework is able to handle a significantly wider class of situations than the original ISOMAP algorithm. The theoretical framework revolves around a quadratic form [H](f)={int}M ||Hf(m)||[IMG]f1.gif" ALT="Formula" BORDER="0">dm defined on functions f : M [↦] [R]. Here Hf denotes the Hessian of f, and [H](f) averages the Frobenius norm of the Hessian over M. To define the Hessian, we use orthogonal coordinates on the tangent planes of M. The key observation is that, if M truly is locally isometric to an open, connected subset of [R]d, then [H](f) has a (d + 1)-dimensional null space consisting of the constant functions and a d-dimensional space of functions spanned by the original isometric coordinates. Hence, the isometric coordinates can be recovered up to a linear isometry. Our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the Laplacian eigenmaps framework, where we substitute a quadratic form based on the Hessian in place of one based on the Laplacian.
