## Towards a theoretical foundation for Laplacian-based manifold methods (2005)

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Citations: | 111 - 10 self |

### BibTeX

@INPROCEEDINGS{Belkin05towardsa,

author = {Mikhail Belkin and Partha Niyogi},

title = {Towards a theoretical foundation for Laplacian-based manifold methods},

booktitle = {},

year = {2005},

pages = {486--500},

publisher = {Springer}

}

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### Abstract

Abstract. In recent years manifold methods have attracted a considerable amount of attention in machine learning. However most algorithms in that class may be termed “manifold-motivated ” as they lack any explicit theoretical guarantees. In this paper we take a step towards closing the gap between theory and practice for a class of Laplacian-based manifold methods. We show that under certain conditions the graph Laplacian of a point cloud converges to the Laplace-Beltrami operator on the underlying manifold. Theorem 1 contains the first result showing convergence of a random graph Laplacian to manifold Laplacian in the machine learning context. 1

### Citations

2775 | Normalized cuts and image segmentation
- SHI, MALIK
- 1997
(Show Context)
Citation Context ...raph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and =-=[25, 28, 24, 18, 14]-=- among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The problem of estimating geometric and topological invariants from ... |

1792 |
A global geometric framework for non- linear dimensionality reduction
- TENENBAUM, SILVA, et al.
- 2000
(Show Context)
Citation Context ...ing empirical convergence of the graph Laplacian to the manifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including =-=[22, 27, 3, 12]-=- for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spe... |

1725 | Nonlinear dimensionality reduction by locally linear embedding
- ROWEIS, SAUL
- 2000
(Show Context)
Citation Context ...ing empirical convergence of the graph Laplacian to the manifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including =-=[22, 27, 3, 12]-=- for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spe... |

1186 | On spectral clustering: Analysis and an algorithm
- Ng, Jordan, et al.
- 2002
(Show Context)
Citation Context ...raph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and =-=[25, 28, 24, 18, 14]-=- among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The problem of estimating geometric and topological invariants from ... |

785 | Laplacian eigenmaps for dimensionality reduction and data representation
- BELKIN, NIYOGI
- 1998
(Show Context)
Citation Context ...ing empirical convergence of the graph Laplacian to the manifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including =-=[22, 27, 3, 12]-=- for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spe... |

533 | Semi-supervised learning using Gaussian field and harmonic functions
- Zhu, Ghahramani, et al.
- 2003
(Show Context)
Citation Context ...ifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, =-=[30, 29, 9, 23, 4, 2, 26]-=- for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The... |

266 | On clusterings — good, bad and spectral
- Kannan, Vempala, et al.
- 2000
(Show Context)
Citation Context ...raph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for partially supervised classification and =-=[25, 28, 24, 18, 14]-=- among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The problem of estimating geometric and topological invariants from ... |

194 | Diffusion kernels on graphs and other discrete input spaces
- Kondor, Lafferty
- 2002
(Show Context)
Citation Context ...1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed for various problems and applications, including [26,32,3,14,12] for visualization and data representation, =-=[38,34,37,10,27,5,2,31]-=- for partially supervised classification and [30,33,28,21,17], among others, for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [6]. The p... |

159 | Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps
- Coifman, Lafon, et al.
(Show Context)
Citation Context ... converges to it as the amount of data goes to infinity. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed for various problems and applications, including =-=[26,32,3,14,12]-=- for visualization and data representation, [38,34,37,10,27,5,2,31] for partially supervised classification and [30,33,28,21,17], among others, for spectral clustering. A discussion of various spectra... |

153 | Spectral partitioning works: Planar graphs and finite element meshes
- Spielman, Teng
- 1996
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Citation Context |

127 |
The Laplacian on a Riemannian Manifold
- Rosenberg
- 1997
(Show Context)
Citation Context ...ferential operator. The family of diffusion operators Ht M satisfies the following properties: ∆MH t M(f) = ∂ ∂t Ht M(f) Heat Equation lim t→0 HtM(f) = f δ-family property It can be shown (see, e.g., =-=[21]-=-) that Ht M (f) is an integral operator, a convolution with the heat kernel. Our proof hinges on the fact that in geodesic coordinates the heat kernel can be approximated by a Gaussian for small value... |

119 | Finding the homology of submanifolds with high confidence from random samples
- Niyogi, Smale, et al.
- 2008
(Show Context)
Citation Context ...ng geometric and topological invariants from point cloud data has recently attracted some attention. Some of the recent work includes estimating geometric invariants of the manifold, such as homology =-=[31, 20]-=-, geodesic distances [6], and comparing point clouds using Gromov-Hausdorff distance [15]. In particular, we note the closely related Ph.D. thesis of Lafon, [16], which generalized the convergence res... |

109 | Computing persistent homology
- Zomorodian, Carlsson
- 2005
(Show Context)
Citation Context ...ng geometric and topological invariants from point cloud data has recently attracted some attention. Some of the recent work includes estimating geometric invariants of the manifold, such as homology =-=[31, 20]-=-, geodesic distances [6], and comparing point clouds using Gromov-Hausdorff distance [15]. In particular, we note the closely related Ph.D. thesis of Lafon, [16], which generalized the convergence res... |

103 | Hessian Eigenmaps: New locally linear embedding techniques for highdimensional data
- Donoho, Grimes
- 2003
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Citation Context |

102 | Graph approximations to geodesics on embedded manifolds
- Bernstein, Silva, et al.
- 2000
(Show Context)
Citation Context ...invariants from point cloud data has recently attracted some attention. Some of the recent work includes estimating geometric invariants of the manifold, such as homology [31, 20], geodesic distances =-=[6]-=-, and comparing point clouds using Gromov-Hausdorff distance [15]. In particular, we note the closely related Ph.D. thesis of Lafon, [16], which generalized the convergence results from [1] to the imp... |

87 |
Out-of-sample extensions for
- Bengio, Paiement, et al.
- 2004
(Show Context)
Citation Context ...�2 � 4t f(y) dµy = ∆Mf(p) Proof. Consider the constant function g(y) = f(p). By applying the Eq. 6 to this function we obtain � � �� ∂ k �p−y�2 ���0 − (4πt) 2 − e 4t f(p) dµy = ∂t 1 ks(p)f(p) + Cf(p) =-=(8)-=- 3 B k+2 − To simplify the formulas put A(t) = (4πt) 2 δ-family property of the heat kernel, we see that � k − A(0) = lim (4πtn) 2 t→0 B � M e �p−y�2 4t f(y) dµy. Using the �p−y�2 − e 4t f(p) dµy = f(... |

79 |
Using manifold structure for partially labeled classification
- Belkin, Niyogi
- 2003
(Show Context)
Citation Context ...ifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, =-=[30, 29, 9, 23, 4, 2, 26]-=- for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The... |

62 |
Problems of Learning on Manifolds
- Belkin
- 2003
(Show Context)
Citation Context ...ons the graph Laplacian is directly related to the manifold Laplace-Beltrami operator and converges to it as data goes to infinity. This paper presents and extends the unpublished results obtained in =-=[1]-=-. A version of Theorem 1 showing empirical convergence of the graph Laplacian to the manifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been ... |

41 |
From graph to manifold Laplacian: The convergence rate
- Singer
(Show Context)
Citation Context ...itrary probability distribution in the 2004 Ph.D. thesis of Stefan Lafon [19] (see also [12]). Further generalization and an empirical convergence result was presented in [16] and an improved rate in =-=[29]-=-. A more subtle analysis of uniform convergence has recently been made in [15]. We also note the closely related work [35], where similar functional objects were studied in a different context. In a n... |

28 | Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results
- Giné, Koltchinskii
(Show Context)
Citation Context ...(see also [12]). Further generalization and an empirical convergence result was presented in [16] and an improved rate in [29]. A more subtle analysis of uniform convergence has recently been made in =-=[15]-=-. We also note the closely related work [35], where similar functional objects were studied in a different context. In a non-geometric setting convergence of the graph Laplacian was observed in [8]. W... |

21 |
Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions
- Smolyanov, Weizscker, et al.
- 2000
(Show Context)
Citation Context ...an empirical convergence result was presented in [16] and an improved rate in [29]. A more subtle analysis of uniform convergence has recently been made in [15]. We also note the closely related work =-=[35]-=-, where similar functional objects were studied in a different context. In a non-geometric setting convergence of the graph Laplacian was observed in [8]. We also note [20], where convergence of spect... |

12 | Chernoff’s Theorem and discrete time approximations of Brownian motion on manifolds
- Smolyanov, Weizsacker, et al.
(Show Context)
Citation Context ...1 1 vol(M) t 1 (4πt) k 2 ∫ ˜B e − ‖ expp (x)−p‖2 4t ( ˜ f(0) − ˜ √ f(x)) det(gij)dx where det(gij) is the determinant of the metric tensor in exponential coordinates. Now we make use of the fact (see =-=[35,36]-=- and references therein) that the metric tensor has an asymptotic expansion in exponential coordinates given by det(gij) = 1 − 1 6 xT Rx + O(‖x‖ 3 ) where R is the Ricci curvature tensor. On a smooth,... |

10 | M.: Out-of-sample extensions for LLE - Bengio, Paiement, et al. - 2004 |

7 |
Tommi Jaakkola, Partially labeled classification with Markov random walks, NIPS
- Szummer
- 2001
(Show Context)
Citation Context ...ifold Laplacian was stated in [19]. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, =-=[30, 29, 9, 23, 4, 2, 26]-=- for partially supervised classification and [25, 28, 24, 18, 14] among others for spectral clustering. A discussion of various spectral methods and their out-of-sample extensions is given in [5]. The... |

6 |
Diffusion Maps and Geodesic
- Lafon
- 2004
(Show Context)
Citation Context ... of the manifold, such as homology [31, 20], geodesic distances [6], and comparing point clouds using Gromov-Hausdorff distance [15]. In particular, we note the closely related Ph.D. thesis of Lafon, =-=[16]-=-, which generalized the convergence results from [1] to the important case of an arbitrary probability distribution on a manifold. Those results are further generalized and presented with an empirical... |

5 |
von Luxburg, From Graphs to
- Hein, Audibert, et al.
(Show Context)
Citation Context ... [1] to the important case of an arbitrary probability distribution on a manifold. Those results are further generalized and presented with an empirical convergence theorem in the parallel COLT paper =-=[13]-=-. We also note [17], where convergence of a class of graph Laplacians and the associated spectral objects, such as eigenfunctions and eigenvalues, is shown, which in particular, implies consistency of... |

3 |
Estimating Functional Maps on Riemannian Submanifolds from Sampled Data
- Niyogi
- 2004
(Show Context)
Citation Context ... infinity. This paper presents and extends the unpublished results obtained in [1]. A version of Theorem 1 showing empirical convergence of the graph Laplacian to the manifold Laplacian was stated in =-=[19]-=-. 1.1 Prior Work Many manifold and graph-motivated learning methods have been recently proposed, including [22, 27, 3, 12] for visualization and data representation, [30, 29, 9, 23, 4, 2, 26] for part... |

2 |
Learning segmentation by random walks, NIPS
- Meila, Shi
- 2000
(Show Context)
Citation Context |

2 |
Heat Kernels on Weighted
- Grigoryan
(Show Context)
Citation Context ...ent following a convention that makes the Laplacian a positive operator. 4weighted Laplacian seems natural. The weighted Laplacian can then be defined as ∆M,νf(x) = ∆P f = 1 P (x) div(P (x)∇Mf). See =-=[23]-=- for details. The question addressed in this work is how to reconstruct the Laplace-Beltrami operator on M given a sample of points from the manifold. Before proceeding, we will need to fix notation a... |