Edit Distance and Persistence Diagrams over Lattices
Abstract
We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice, and the output is a persistence diagram defined as the Möbius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are 1-Lipschitz, making our pipeline stable. Our constructions generalize the classical persistence diagram, and in this setting, the bottleneck distance is strongly equivalent to the edit distance.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
H. Adams, T. Emerson, M. Kirby, R. Neville, C. Peterson, P. Shipman, S. Chepushtanova, E. Hanson, F. Motta, and L. Ziegelmeier, Persistence images: A stable vector representation of persistent homology, J. Mach. Learn. Res., 18 (2017), pp. 1--35, http://jmlr.org/papers/v18/16-337.html.
2.
M. Barnabei, A. Brini, and G.-C. Rota, The theory of Möbius functions, Russian Math. Surveys, 41 (1986), pp. 135--188.
3.
U. Bauer, C. Landi, and F. Mémoli, The Reeb graph edit distance is universal, Found. Comput. Math., (2020), https://doi.org/10.1007/s10208-020-09488-3.
4.
L. Betthauser, P. Bubenik, and P. Edwards, Graded persistence diagrams and persistence landscapes, Discrete Comput. Geom., (2021).
5.
P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), pp. 77--102, http://jmlr.org/papers/v16/bubenik15a.html.
6.
P. Bubenik, The persistence landscape and some of its properties, in Topological Data Analysis, N. A. Baas, G. E. Carlsson, G. Quick, M. Szymik, and M. Thaule, eds., Springer-Verlag, Berlin, 2020, pp. 97--117.
7.
D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.
8.
G. Carlsson, F. Mémoli, A. Ribeiro, and S. Segarra, Axiomatic construction of hierarchical clustering in asymmetric networks, in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, BC, 2013, pp. 5219--5223, https://doi.org/10.1109/ICASSP.2013.6638658.
9.
F. Chazal, D. Cohen-Steiner, and Q. Mérigot, Geometric inference for probability measures, Found. Comput. Math., 11 (2011), pp. 733--751, https://doi.org/10.1007/s10208-011-9098-0.
10.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), pp. 103--120, https://doi.org/10.1007/s00454-006-1276-5.
11.
D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov, Vines and vineyards by updating persistence in linear time, in Proceedings of the 22nd Annual Symposium on Computational Geometry, SCG '06, Association for Computing Machinery, New York, 2006, pp. 119--126, https://doi.org/10.1145/1137856.1137877.
12.
J. Curry and A. Patel, Classification of constructible cosheaves, Theory Appl. Categ., 35 (2020), pp. 1012--1047.
13.
B. Di Fabio and C. Landi, The edit distance for Reeb graphs of surfaces, Discrete Comput. Geom., 55 (2016), pp. 423--461, https://doi.org/10.1007/s00454-016-9758-6.
14.
P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recogn. Image Anal., 9 (2001), pp. 596--603.
15.
G. Henselman and R. Ghrist, Matroid Filtrations and Computational Persistent Homology, preprint, arXiv:1606.00199, 2016.
16.
G. Henselman-Petrusek, Matroids and Canonical Forms: Theory and Applications, preprint, arXiv:1710.06084, 2017.
17.
C. Hofer, R. Kwitt, M. Niethammer, and A. Uhl, Deep learning with topological signatures, in Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS'17, Curran Associates Inc., Red Hook, NY, 2017, pp. 1633--1643.
18.
W. Kim and F. Mémoli, Generalized persistence diagrams for persistence modules over posets, J. Appl. Comput. Topol., 5 (2021), pp. 533--581.
19.
C. Landi and P. Frosini, New pseudodistances for the size function space, in Vision Geometry VI, Vol. 3168, R. A. Melter, A. Y. Wu, and L. J. Latecki, eds., International Society for Optics and Photonics, SPIE, Bellingham, WA, 1997, pp. 52--60, https://doi.org/10.1117/12.279674.
20.
A. McCleary and A. Patel, Bottleneck stability for generalized persistence diagrams, Proc. Amer. Math. Soc., 148 (2020), pp. 3149--3161.
21.
A. Patel, Generalized persistence diagrams, J. Appl. Comput. Topol., 1 (2018), pp. 397--419, https://doi.org/10.1007/s41468-018-0012-6.
22.
J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, A stable multi-scale kernel for topological machine learning, in 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Boston, 2015, pp. 4741--4748.
23.
G. C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 2 (1964), pp. 340--368, https://doi.org/10.1007/BF00531932.
Information & Authors
Information
Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 134 - 155
DOI: 10.1137/20M1373700
ISSN (online): 2470-6566
Copyright
© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 15 October 2020
Accepted: 8 December 2021
Published online: 11 April 2022
Keywords
MSC codes
Authors
Funding Information
National Science Foundation https://doi.org/10.13039/100000001 : 1717159
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Signed Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact ResolutionsFoundations of Computational Mathematics, Vol. 227 | 4 September 2024
- Bigraded Betti numbers and generalized persistence diagramsJournal of Applied and Computational Topology, Vol. 8, No. 3 | 28 June 2024
- Interleaving by Parts: Join Decompositions of Interleavings and Join-Assemblage of GeodesicsOrder, Vol. 41, No. 2 | 8 September 2023
- Extracting Persistent Clusters in Dynamic Data via Möbius InversionDiscrete & Computational Geometry, Vol. 71, No. 4 | 11 October 2023
- Poincaré duality for generalized persistence diagrams of (co)filtrationsJournal of Applied and Computational Topology, Vol. 8, No. 2 | 19 February 2024
- On the Stability of Multigraded Betti Numbers and Hilbert FunctionsSIAM Journal on Applied Algebra and Geometry, Vol. 8, No. 1 | 23 January 2024
- Homological approximations in persistence theoryCanadian Journal of Mathematics, Vol. 76, No. 1 | 12 December 2022
- Exploring Graph and Digraph PersistenceAlgorithms, Vol. 16, No. 10 | 2 October 2023
- On approximation of 2D persistence modules by interval-decomposablesJournal of Computational Algebra, Vol. 6-7 | 1 Sep 2023
- Steady and ranging sets in graph persistenceJournal of Applied and Computational Topology, Vol. 7, No. 1 | 23 September 2022
- Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spacesJournal of Applied and Computational Topology, Vol. 6, No. 4 | 21 April 2022
View Options
Get Access
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].