Topology representing networks
Neural Networks
A simple algorithm for homeomorphic surface reconstruction
Proceedings of the sixteenth annual symposium on Computational geometry
Detecting undersampling in surface reconstruction
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Neural Networks for Pattern Recognition
Neural Networks for Pattern Recognition
Shape dimension and approximation from samples
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Smooth-surface reconstruction in near-linear time
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Nonlinear manifold learning for visual speech recognition
ICCV '95 Proceedings of the Fifth International Conference on Computer Vision
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Provable dimension detection using principal component analysis
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Texture design using a simplicial complex of morphable textures
ACM SIGGRAPH 2005 Papers
Meshless geometric subdivision
Graphical Models
Cycle bases of graphs and sampled manifolds
Computer Aided Geometric Design
Constructing Laplace operator from point clouds in Rd
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Integral estimation from point cloud in d-dimensional space: a geometric view
Proceedings of the twenty-fifth annual symposium on Computational geometry
Manifold reconstruction using tangential Delaunay complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Bounds on the k-neighborhood for locally uniformly sampled surfaces
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Distance preserving embeddings for general n-dimensional manifolds
The Journal of Machine Learning Research
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We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some (potentially very high-dimensional) Euclidean space Rd. We assume that M is known to us only through a finite sample P? M, as it is often the case in applications. The adaptive neighborhood graph is a geometric graph on P. Its complexity is at most min[2O(k)n, n2], where n=|P| and k=dim M, as opposed to the n[d/2] complexity of the Delaunay triangulation, which is often used to model manifolds. We show that we can provably correctly infer the connectivity of M and the dimension of M from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is d2O(k7log k) for each connected component of M. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on theintrinsic dimension k, not on the ambient dimension d. This is of particular interest if the co-dimension is high, i.e., if k is much smaller than d, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in P.