×

Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. (English) Zbl 1130.62337

Summary: We describe a method for recovering the underlying parametrization of scattered data \((m_i)\) 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 \(\mathbb R^n\), is locally isometric to an open, connected subset \(\Theta\) of Euclidean space \(\mathbb R^d\). 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 \({\mathcal H}(f)= \int_M \|H_f(m)\|_F^2\,dm\) defined on functions \(f:M\to \mathbb R\). Here \(Hf\) denotes the Hessian of \(f\), and \({\mathcal H}(f)\) 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 \(\mathbb R^d\), then \({\mathcal H}(f)\) 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.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62-07 Data analysis (statistics) (MSC2010)
53B20 Local Riemannian geometry
62H99 Multivariate analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Tenenbaum, Science 290 (5500) pp 2319– (2000)
[2] PROCEEDINGS OF THE EUROPEAN SYMPOSIUM ON ARTIFICIAL NEURAL NETWORKS 10 pp 199– (2002)
[3] Roweis, Science 290 (5500) pp 2323– (2000)
[4] 14 pp 585– (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.