Global InformationSubdivision bifiltration information
In topological data analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that captures shape and density information about the underlying data set. The subdivision bifiltration relies on a natural filtration of the barycentric subdivision of a simplicial complex by flags of minimum dimension, which encodes density information about the metric space upon which the complex is built. The subdivision bifiltration was first introduced by Donald Sheehy in 2011 as part of his doctoral thesis[1] (later subsumed by a conference paper in 2012[2]) as a discrete model of the multicover bifiltration, a continuous construction whose underlying framework dates back to the 1970s.[3] In particular, Sheehy applied the construction to both the Vietoris-Rips and Čech filtrations, two common objects in the field of topological data analysis.[4][5][6] Whereas single parameter filtrations are not robust with respect to outliers in the data,[7] the subdivision-Rips and -Cech bifiltrations satisfy several desirable stability properties.[8]
- ^ Sheehy, D. R. (2011). Mesh generation and geometric persistent homology (Doctoral dissertation, Carnegie Mellon University).
- ^ Sheehy, Donald R. 2012. “A Multicover Nerve for Geometric Inference.” in CCCG: Canadian conference in computational geometry.
- ^ Fejes Tóth, G. (March 1976). "Multiple packing and covering of the plane with circles". Acta Mathematica Academiae Scientiarum Hungaricae. 27 (1–2): 135–140. doi:10.1007/BF01896768. ISSN 0001-5954. S2CID 189778121.
- ^ Chazal, Frédéric; Oudot, Steve Yann (2008). "Towards persistence-based reconstruction in euclidean spaces". Proceedings of the twenty-fourth annual symposium on Computational geometry. pp. 232–241. arXiv:0712.2638. doi:10.1145/1377676.1377719. ISBN 9781605580715. S2CID 1020710.
- ^ Ghrist, Robert (2007). "Barcodes: The persistent topology of data". Bulletin of the American Mathematical Society. 45: 61–76. doi:10.1090/s0273-0979-07-01191-3.
- ^ Chazal, Frédéric; De Silva, Vin; Oudot, Steve (2014). "Persistence stability for geometric complexes". Geometriae Dedicata. 173: 193–214. arXiv:1207.3885. doi:10.1007/s10711-013-9937-z. S2CID 254508455.
- ^ Chazal, Frédéric; Cohen-Steiner, David; Mérigot, Quentin (December 2011). "Geometric Inference for Probability Measures". Foundations of Computational Mathematics. 11 (6): 733–751. doi:10.1007/s10208-011-9098-0. ISSN 1615-3375. S2CID 15371638.
- ^ Blumberg, Andrew J.; Lesnick, Michael (2022-10-17). "Stability of 2-Parameter Persistent Homology". Foundations of Computational Mathematics. arXiv:2010.09628. doi:10.1007/s10208-022-09576-6. ISSN 1615-3375. S2CID 224705357.