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 [IMG]img001.gif" ALT="">Hf(m)[IMG]img001.gif" ALT="">[IMG]img002.gif" ALT="<UP><SUB><IT>F</IT></SUB><SUP>2</SUP></UP>">dm defined on functions f : M [maps to] [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. 10.1073/pnas.1031596100
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