Weak witnesses for Delaunay triangulations of submanifolds
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Random projection trees and low dimensional manifolds
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Discrete laplace operator on meshed surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Towards a theoretical foundation for Laplacian-based manifold methods
Journal of Computer and System Sciences
Constructing Laplace operator from point clouds in Rd
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Normal cone approximation and offset shape isotopy
Computational Geometry: Theory and Applications
Provably correct reconstruction of surfaces from sparse noisy samples
Pattern Recognition
Cut locus and topology from surface point data
Proceedings of the twenty-fifth annual symposium on Computational geometry
Random projection trees for vector quantization
IEEE Transactions on Information Theory
Recovering structure from r-sampled objects
SGP '09 Proceedings of the Symposium on Geometry Processing
Isotopic reconstruction of surfaces with boundaries
SGP '09 Proceedings of the Symposium on Geometry Processing
Which spatial partition trees are adaptive to intrinsic dimension?
UAI '09 Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
Manifold reconstruction using tangential Delaunay complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Chern numbers of smooth varieties via homotopy continuation and intersection theory
Journal of Symbolic Computation
Hardness results for homology localization
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Algorithms and theory of computation handbook
IEEE Transactions on Signal Processing
Curvature analysis of frequency modulated manifolds in dimensionality reduction
Calcolo: a quarterly on numerical analysis and theory of computation
SMI 2011: Full Paper: Geometric models with weigthed topology
Computers and Graphics
Proceedings of the twenty-seventh annual symposium on Computational geometry
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on Computational geometry
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Baby morse theory in data analysis
Proceedings of the 2011 workshop on Knowledge discovery, modeling and simulation
Reconstructing 3D compact sets
Computational Geometry: Theory and Applications
A Topological View of Unsupervised Learning from Noisy Data
SIAM Journal on Computing
Local homology transfer and stratification learning
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Preserving geometric properties in reconstructing regions from internal and nearby points
Computational Geometry: Theory and Applications
A tree-based regressor that adapts to intrinsic dimension
Journal of Computer and System Sciences
Stability of Delaunay-type structures for manifolds: [extended abstract]
Proceedings of the twenty-eighth annual symposium on Computational geometry
The Journal of Machine Learning Research
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
Homological reconstruction and simplification in R3
Proceedings of the twenty-ninth annual symposium on Computational geometry
On the convergence of maximum variance unfolding
The Journal of Machine Learning Research
Distance preserving embeddings for general n-dimensional manifolds
The Journal of Machine Learning Research
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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.