The embedding dimension of Laplacian eigenfunction maps

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ 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 $\mathbb{R}^3$, 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.

• Publication year

  2014

• Venue

  arXiv.org

• Publication date

  2014-11-01

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.1016/j.acha.2014.03.002arXiv 1605.01643
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Eigenfunctions of the Laplacian on a Riemannian manifold

  2017cited by this paper
• EMBEDDING RIEMANNIAN MANIFOLDS VIA THEIR EIGENFUNCTIONS AND THEIR HEAT KERNEL

  2012cited by this paper
• Heat Kernel and Analysis on Manifolds

  2012cited by this paper
• A spectral notion of Gromov–Wasserstein distance and related methods

  2011influential reference
• Scale-Space Spectral Representation of Shape

  2010cited by this paper
• Hierarchical Shape Segmentation and Registration via Topological Features of Laplace-Beltrami Eigenfunctions

  2010cited by this paper
• Shape matching based on diffusion embedding and on mutual isometric consistency

  2010cited by this paper
• An Analysis of the Convergence of Graph Laplacians

  2010cited by this paper
• LOCAL AND GLOBAL ANALYSIS OF EIGENFUNCTIONS ON RIEMANNIAN MANIFOLDS

  2009influential reference
• Registration of contours of brain structures through a heat-kernel representation of shape

  2009cited by this paper
• Consistency of spectral clustering

  2008cited by this paper
• Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration

  2008cited by this paper
• Relating diameter and mean curvature for submanifolds of Euclidean space

  2008cited by this paper
• Measuring Graph Similarity Using Spectral Geometry

  2008cited by this paper
• C α -COMPACTNESS FOR MANIFOLDS WITH RICCI CURVATURE AND INJECTIVITY RADIUS BOUNDED BELOW

  2008cited by this paper
• Kernel functions for robust 3D surface registration with spectral embeddings

  2008cited by this paper
• Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels

  2008cited by this paper
• Non-Rigid Spectral Correspondence of Triangle Meshes

  2007cited by this paper
• Universal Local Parametrizations via Heat Kernels and Eigenfunctions of the Laplacian

  2007cited by this paper
• Laplace-Beltrami eigenfunctions for deformation invariant shape representation

  2007cited by this paper
• A spectral approach to shape-based retrieval of articulated 3D models

  2007cited by this paper
• Convergence of Laplacian Eigenmaps

  2006cited by this paper
• Diffusion maps

  2006cited by this paper
• Laplace-Beltrami Eigenfunctions Towards an Algorithm That "Understands" Geometry

  2006cited by this paper
• Convergence of Riemannian Manifolds and Laplace Operators, II

  2006cited by this paper
• SPECTRAL CONVERGENCE OF RIEMANNIAN MANIFOLDS

  2006cited by this paper
• Diffusion maps, spectral clustering and reaction coordinates of dynamical systems

  2005cited by this paper
• Sequence-independent segmentation of magnetic resonance images.

  2004cited by this paper
• Heat Kernels, Manifolds and Graph Embedding

  2004cited by this paper
• Convergence of Riemannian manifolds and Laplace operators. I@@@Convergence des variétés riemanniennes et des opérateurs laplaciens. I

  2002cited by this paper
• Neurotechnique Whole Brain Segmentation: Automated Labeling of Neuroanatomical Structures in the Human Brain

  2002cited by this paper
• Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering

  2001cited by this paper
• Spectral Correspondence for Deformed Point-Set Matching

  2000cited by this paper
• The Heat Kernel on Hyperbolic Space

  1998cited by this paper
• CONVERGENCE OF HEAT KERNELS ON A COMPACT MANIFOLD

  1997cited by this paper
• The Laplacian on a Riemannian Manifold: The Laplacian on a Riemannian Manifold

  1997cited by this paper
• SPECTRAL CONVERGENCE OF RIEMANNIAN MANIFOLDS, II

  1994cited by this paper
• Embedding Riemannian manifolds by their heat kernel

  1994influential reference
• $C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below

  1992influential reference
• Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups

  1988cited by this paper
• Collapsing of Riemannian manifolds and eigenvalues of Laplace operator

  1987cited by this paper
• Spectral Geometry: Direct and Inverse Problems

  1986cited by this paper
• On the parabolic kernel of the Schrödinger operator

  1986cited by this paper
• Volume des ensembles nodaux des fonctions propres du laplacien

  1985cited by this paper
• EIGENVALUES IN RIEMANNIAN GEOMETRY (Pure and Applied Mathematics: A Series of Monographs and Textbooks, 115)

  1985cited by this paper
• Eigenvalues in Riemannian geometry

  1984influential reference
• Some regularity theorems in riemannian geometry

  1981influential reference
• A lower bound for the heat kernel

  1981cited by this paper
• Some isoperimetric inequalities and eigenvalue estimates

  1980cited by this paper
• The connections between the order of smoothness of a surface and its metric

  1976cited by this paper
• FINITENESS THEOREMS FOR RIEMANNIAN MANIFOLDS.

  1970cited by this paper

• Manifold learning in metric spaces

  2025cites this paper
• Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces

  2025cites this paper
• Sketching the Heat Kernel: Using Gaussian Processes to Embed Data

  2024cites this paper
• How well behaved is finite dimensional Diffusion Maps?

  2024cites this paper
• Boundary Detection Algorithm Inspired by Locally Linear Embedding

  2024cites this paper
• From Varadhan’s Limit to Eigenmaps: A Guide to the Geometric Analysis behind Manifold Learning

  2022cites this paper
• Augmentation Invariant Manifold Learning

  2022cites this paper
• Continuous normalizing flows on manifolds

  2021influential citation
• Convergence of Laplacian Eigenmaps and Its Rate for Submanifolds with Singularities

  2021cites this paper
• Manifold Embeddings by Heat Kernels of Connection Laplacian

  2021cites this paper
• Manifold Learning with Arbitrary Norms

  2020cites this paper
• A survey on spectral embeddings and their application in data analysis

  2020cites this paper
• Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions

  2020cites this paper
• Neural Ordinary Differential Equations on Manifolds

  2020cites this paper
• A Geometric Approach to Biomedical Time Series Analysis

  2020cites this paper
• Wave-shape oscillatory model for nonstationary periodic time series analysis

  2020cites this paper
• Product Manifold Learning

  2020cites this paper
• Data-driven efficient solvers for Langevin dynamics on manifold in high dimensions

  2020cites this paper
• Wave-shape oscillatory model for biomedical time series with applications

  2019cites this paper
• Explore Intrinsic Geometry of Sleep Dynamics and Predict Sleep Stage by Unsupervised Learning Techniques

  2019cites this paper
• Selecting the independent coordinates of manifolds with large aspect ratios

  2019cites this paper
• Intrinsic Topological Transforms via the Distance Kernel Embedding

  2019cites this paper
• Diffuse to fuse EEG spectra - Intrinsic geometry of sleep dynamics for classification

  2018cites this paper
• Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

  2018cites this paper
• Embeddings of Riemannian manifolds with finite eigenvector fields of connection Laplacian

  2016cites this paper
• Spherical Parameterization for Genus Zero Surfaces Using Laplace-Beltrami Eigenfunctions

  2015cites this paper
• Connection graph Laplacian methods can be made robust to noise

  2014cites this paper
• Applied and Computational Harmonic Analysis

  year unknowncites this paper