## 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
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### References:

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