Persistent Intersection Homology
- 430 Downloads
The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection homology gives useful information about the relationship between an embedded stratified space and its singularities. We give an algorithm for the computation of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcomplexes, and we prove its correctness. We also derive, from Poincaré Duality, some structural results about persistent intersection homology.
KeywordsPersistence Intersection homology Algorithms Stratified spaces
Mathematics Subject Classification (2000)55N33 68
Unable to display preview. Download preview PDF.
- 3.M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation, in Neural Computation (2003), pp. 347–356. Google Scholar
- 4.P. Bendich, Analyzing stratified spaces using persistent versions of intersection and local homology. Dissertation, Duke University, Durham, North Carolina (2008). URL: http://hdl.handle.net/10161/680.
- 5.P. Bendich, D. Cohen-Steiner, H. Edelsbrunner, J. Harer, D. Morozov, Inferring local homology from sampled stratified spaces, in Proc. 48th Ann. Sympos. Found. Comp. Sci. (2007), pp. 536–546. Google Scholar
- 6.P. Bendich, J. Harer, Extreme elevation on a stratified surface. Manuscript, Duke University Mathematics Department (2010). Google Scholar
- 7.P. Bendich, S. Mukherjee, B. Wang, Towards stratification learning via homology inference. Manuscript, Duke University Computer Science Department (2010). Google Scholar
- 9.G. Carlsson, A. Collins, L. Guibas, A. Zomorodian, Persistence barcodes for shapes. Int. J. Shape Model. (2005). Google Scholar
- 15.D. Cohen-Steiner, H. Edelsbrunner, D. Morozov, Vines and vineyards by updating persistence in linear time, in Proc. 22nd Ann. Sympos. Comput. Geom. (2006), pp. 119–126. Google Scholar
- 16.H. Edelsbrunner, J. Harer, Persistent homology-a survey, in Twenty Years After, ed. by J.E. Goodman, J. Pach, R. Pollack (AMS, Providence, 2007). Google Scholar
- 17.H. Edelsbrunner, J. Harer, Computational Topology. An Introduction (AMS, Providence, 2009). Google Scholar
- 21.P. Frosini, C. Landi, Size theory as a topological tool for computer vision, Pattern Recognit. Image Anal. 9, 596–603 (1999). Google Scholar
- 24.M. Goresky, R. Macpherson, Stratified Morse Theory (Springer, Berlin, 1987). Google Scholar
- 32.Y. Wang, P.K. Agarwal, P. Brown, H. Edelsbrunner, J. Rudolph, Coarse and reliable geometric alignment for protein docking, in Proc. Pacific Sympos. Biocomput. (2005), pp. 65–75. Google Scholar