Exact Computation of the Matching Distance on 2-Parameter Persistence Modules
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.
Topological Data Analysis
Multi-Parameter Persistence
Line arrangements
Mathematics of computing~Algebraic topology
Mathematics of computing~Mathematical optimization
46:1-46:15
Regular Paper
A full version of this paper is available at https://arxiv.org/abs/1812.09085.
This work was initiated at the BIRS workshop "Multiparameter Persistent Homology" (18w55140) in Oaxaca, Mexico (Aug. 2018). We thank Jan Reininghaus and the other members of the discussion group on this topic for fruitful initial exchanges. We thank Matthew Wright for helpful discussions about line arrangements, slices, and the computational aspects of 2-parameter persistence.
Michael
Kerber
Michael Kerber
Graz University of Technology, Graz, Austria
https://orcid.org/0000-0002-8030-9299
Supported by Austrian Science Fund (FWF) grant number P 29984-N35.
Michael
Lesnick
Michael Lesnick
University at Albany, SUNY, United States
https://orcid.org/0000-0003-1924-3283
Steve
Oudot
Steve Oudot
Inria Saclay - Île-de-France, Palaiseau, France
https://orcid.org/0000-0003-2939-9417
10.4230/LIPIcs.SoCG.2019.46
J. Bentley and T. Ottmann. Algorithms for Reporting and Counting Geometric Intersections. IEEE Transactions on Computers, 28:643-647, 1979.
M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, Santa Clara, CA, USA, 3rd ed. edition, 2008.
S. Biasotti, A. Cerri, P. Frosini, and D. Giorgi. A new algorithm for computing the 2-dimensional matching distance between size functions. Pattern Recognition Letters, 32(14):1735-1746, 2011.
H. Bjerkevik, M. Botnan, and M. Kerber. Computing the interleaving distance is NP-hard. arXiv, abs/1811.09165, 2018. URL: http://arxiv.org/abs/1811.09165.
http://arxiv.org/abs/1811.09165
G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009.
A. Cerri, B. Di Fabio, M. Ferri, P. Frosini, and C. Landi. Betti numbers in multidimensional persistent homology are stable functions. Mathematical Methods in the Applied Sciences, 36(12):1543-1557, 2013.
R. Corbet and M. Kerber. The representation theorem of persistence revisited and generalized. Journal of Applied and Computational Topology, 2(1):1-31, 2018.
W. Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and its Applications, 14(05):1550066, 2015.
The RIVET Developers. RIVET: Software for visualization and analysis of 2-parameter persistent homology. http://repo.rivet.online/, 2014-2018.
http://repo.rivet.online/
T. Dey and C. Xin. Computing Bottleneck Distance for 2-D Interval Decomposable Modules. In International Symposium on Computational Geometry (SoCG), pages 32:1-32:15, 2018.
H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, Providence, RI, USA, 2010.
A. Efrat, A. Itai, and M. Katz. Geometry Helps in Bottleneck Matching and Related Problems. Algorithmica, 31(1):1-28, 2001. URL: http://dx.doi.org/10.1007/s00453-001-0016-8.
http://dx.doi.org/10.1007/s00453-001-0016-8
B. Keller, M. Lesnick, and T. L. Willke. Persistent Homology for Virtual Screening. ChemRxiv preprint, October 2018.
M. Kerber, M. Lesnick, and S. Oudot. Exact computation of the matching distance on 2-parameter persistence modules. CoRR, abs/1812.09085, 2018.
M. Kerber, D. Morozov, and A. Nigmetov. Geometry Helps to Compare Persistence Diagrams. Journal of Experimental Algorithms, 22:1.4:1-1.4:20, September 2017.
C. Landi. The Rank Invariant Stability via Interleavings. In Research in Computational Topology, pages 1-10. Springer International Publishing, 2018.
M. Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 15(3):613-650, 2015.
M. Lesnick and M. Wright. Interactive visualization of 2-D persistence modules. arXiv:1512.00180, 2015. URL: http://arxiv.org/abs/1512.00180.
http://arxiv.org/abs/1512.00180
M. Lesnick and M. Wright. Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent Homology. arXiv:1902.05708, 2019. URL: http://arxiv.org/abs/1902.05708.
http://arxiv.org/abs/1902.05708
N. Milosavljevic, D. Morozov, and P. Skraba. Zigzag persistent homology in matrix multiplication time. In ACM Symposium on Computational Geometry (SoCG), pages 216-225, 2011.
S. Oudot. Persistence theory: From Quiver Representation to Data Analysis, volume 209 of Mathematical Surveys and Monographs. American Mathematical Society, 2015.
A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete and Computational Geometry, 33(2):249-274, 2005.
Michael Kerber, Michael Lesnick, and Steve Oudot
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode