Deformable spanners and applications
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Discrete & Computational Geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Building triangulations using ε-nets
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Searching dynamic point sets in spaces with bounded doubling dimension
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Geometric Spanner Networks
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On the Local Behavior of Spaces of Natural Images
International Journal of Computer Vision
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
An Optimal Dynamic Spanner for Doubling Metric Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
Persistence Diagrams of Cortical Surface Data
IPMI '09 Proceedings of the 21st International Conference on Information Processing in Medical Imaging
Gromov-Hausdorff stable signatures for shapes using persistence
SGP '09 Proceedings of the Symposium on Geometry Processing
The tidy set: a minimal simplicial set for computing homology of clique complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
Foundations of Computational Mathematics
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
Efficient data structure for representing and simplifying simplicial complexes in high dimensions
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric Inference for Probability Measures
Foundations of Computational Mathematics
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Zigzag zoology: rips zigzags for homology inference
Proceedings of the twenty-ninth annual symposium on Computational geometry
Graph induced complex on point data
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees.