The embedding dimension of Laplacian eigenfunction maps

Jonathan Bates 1 Department of Mathematics, Florida State University, T allahassee, FL 32306, USAAbstract Any closed, connected Riemannian manifold M can be smoothly embedded by its Laplacian eigenfunction maps into R m for some m. We call the smallest such m the maximal embedding dimension of M. We show that the maximal embedding dimension of M is bounded from above by a constant depending only on the dimension of M, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in R 3, showing that the maximal embedding dimension only depends on bounds for the Gaussian curvature, mean curvature, and surface area. Furthermore, we consider the relevance of these results for shape registration. Keywords: spectral embedding, eigenfunction embedding, eigenmap, di ffusion map, global point signature, heat kernel embedding, shape registration, nonlinear dimensionality reduction, manifold learning 1. Introduction Let M ( M, g) be a closed (compact, without boundary), connected Riemannian manifold; we assume both M and g are smooth. The Laplacian of M is a di fferential operator given by: div grad, where div and grad are the Riemannian divergence and gradient, respectively. Since M is compact and connected, has a discrete spectrum { λ j } j N, 0 λ 0 λ 1 λ 2 ··· . We may choose an orthonormal basis for L 2 ( M) of eigenfunctions { ϕ j } j N of, where ϕ j λ j ϕ j, ϕ j C ( M),ϕ 0 V( M) 1 / 2 . We consider maps of the form Φ m: M R m x 7 { ϕ j( x)} 1 j m .