## Efficient Simplicial Reconstructions of Manifolds from their Samples (2002)

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Citations: | 11 - 1 self |

### BibTeX

@MISC{Freedman02efficientsimplicial,

author = {Daniel Freedman},

title = {Efficient Simplicial Reconstructions of Manifolds from their Samples},

year = {2002}

}

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### Abstract

A new algorithm for manifold learning is presented. Given only samples of a finite-dimensional differentiable manifold, and no a priori knowledge of the manifold's geometry or topology except for its dimension, the goal is to find a description of the manifold. The learned manifold must approximate the true manifold well, both geometrically and topologically, when the sampling density is sufficiently high. The proposed algorithm constructs a simplicial complex based on approximations to the tangent bundle of the manifold. An important property of the algorithm is that its complexity depends on the dimension of the manifold, rather than that of the embedding space. Successful examples are presented in the cases of learning curves in the plane, curves in space, and surfaces in space; in addition, a case when the algorithm fails is analyzed.

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Citation Context ...st neighbors leaves holes. (b) Greedy nearest neighbors (see description in the text) leads to a different kind of problem. concept; perhaps the most useful definition of sampling density is given in =-=[5]-=- and is based on the notion of local feature size. The local feature size at a given point on the manifold is defined to be the distance from that point to the medial axis of the manifold. This defini... |

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Citation Context ...ces (2-manifolds) in IR 3 . The solutions have used machinery from computational geometry, both in terms of construction and proofs. Other papers in the same vein have followed suit [13], [14], [15], =-=[16]-=-, [17], [18], [19], [20]. The crust is defined to be the set of edges from the Delaunay Triangulation of the set composed of the samples and their Voronoi vertices such that both vertices of the edge ... |

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Citation Context ... in IR 2 and surfaces (2-manifolds) in IR 3 . The solutions have used machinery from computational geometry, both in terms of construction and proofs. Other papers in the same vein have followed suit =-=[13]-=-, [14], [15], [16], [17], [18], [19], [20]. The crust is defined to be the set of edges from the Delaunay Triangulation of the set composed of the samples and their Voronoi vertices such that both ver... |

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Citation Context ...-manifolds) in IR 3 . The solutions have used machinery from computational geometry, both in terms of construction and proofs. Other papers in the same vein have followed suit [13], [14], [15], [16], =-=[17]-=-, [18], [19], [20]. The crust is defined to be the set of edges from the Delaunay Triangulation of the set composed of the samples and their Voronoi vertices such that both vertices of the edge belong... |

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Citation Context ...our trackers. A contour tracker seeks to follow an object as it moves through a video stream by following its contour, i.e., the curve which represents its silhouette; examples include [1], [2], [3], =-=[4]-=-. It is often useful, and sometimes necessary, for such trackers to have an a priori model of the dynamics of the object to be tracked. Such a model may occasionally be known or derived from first pri... |

2 |
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Citation Context ...in IR 3 . The solutions have used machinery from computational geometry, both in terms of construction and proofs. Other papers in the same vein have followed suit [13], [14], [15], [16], [17], [18], =-=[19]-=-, [20]. The crust is defined to be the set of edges from the Delaunay Triangulation of the set composed of the samples and their Voronoi vertices such that both vertices of the edge belong to the set ... |

1 |
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Citation Context ...3 . The solutions have used machinery from computational geometry, both in terms of construction and proofs. Other papers in the same vein have followed suit [13], [14], [15], [16], [17], [18], [19], =-=[20]-=-. The crust is defined to be the set of edges from the Delaunay Triangulation of the set composed of the samples and their Voronoi vertices such that both vertices of the edge belong to the set of sam... |