Document Open Access Logo

Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes

Authors Mitchell Black, Amir Nayyeri

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.23.pdf
  • Filesize: 0.72 MB
  • 17 pages

Document Identifiers

Author Details

Mitchell Black
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
Amir Nayyeri
  • School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA

Funding

The authors were supported in part by NSF grants CCF-1941086 and CCF-1816442.

Acknowledgements

The authors would like to thank the reviewers for their helpful comments, especially for an observation that improved the dependence on β in the runtime.

Cite AsGet BibTex

Mitchell Black and Amir Nayyeri. Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.23

Abstract

We describe a nearly-linear time algorithm to solve the linear system L₁x = b parameterized by the first Betti number of the complex, where L₁ is the 1-Laplacian of a simplicial complex K that is a subcomplex of a collapsible complex X linearly embedded in ℝ³. Our algorithm generalizes the work of Black et al. [SODA2022] that solved the same problem but required that K have trivial first homology. Our algorithm works for complexes K with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of K in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex K embedded in ℝ³, as K can always be expanded to a collapsible embedded complex of quadratic complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Numerical analysis
Keywords
  • Computational Topology
  • Laplacian solvers
  • Combinatorial Laplacian
  • Hodge decomposition
  • Parameterized Complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Douglas Arnold, Richard Falk, and Ragnar Winther. Finite element exterior calculus: from hodge theory to numerical stability. Bulletin of the American Mathematical Society, 47(2):281-354, January 2010. URL: https://doi.org/10.1090/s0273-0979-10-01278-4.
  2. Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. SIAM J. Matrix Anal. Appl., 27(4):930-951, December 2005. URL: https://doi.org/10.1137/S0895479801384019.
  3. Mitchell Black, William Maxwell, Amir Nayyeri, and Eli Winkelman. Computational topology in a collapsing universe: Laplacians, homology, cohomology. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 226-251, 2022. URL: https://doi.org/10.1137/1.9781611977073.12.
  4. Erik G. Boman and Bruce Hendrickson. Support theory for preconditioning. SIAM J. Matrix Anal. Appl., 25(3):694-717, March 2003. URL: https://doi.org/10.1137/S0895479801390637.
  5. Erik G. Boman, Bruce Hendrickson, and Stephen Vavasis. Solving elliptic finite element systems in near-linear time with support preconditioners. SIAM J. Numer. Anal., 46(6):3264-3284, October 2008. URL: https://doi.org/10.1137/040611781.
  6. Oleksiy Busaryev, Sergio Cabello, Chao Chen, Tamal K. Dey, and Yusu Wang. Annotating simplices with a homology basis and its applications. In Fedor V. Fomin and Petteri Kaski, editors, Algorithm Theory - SWAT 2012, pages 189-200, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. Google Scholar
  7. S. L. (Stephen La Vern) Campbell. Generalized inverses of linear transformations. Surveys and reference works in mathematics. Pitman, London, 1979. Google Scholar
  8. Ozan Candogan, Ishai Menache, Asuman E. Ozdaglar, and Pablo A. Parrilo. Flows and decompositions of games: harmonic and potential games. Math. Oper. Res., 36(3):474-503, 2011. URL: https://doi.org/10.1287/moor.1110.0500.
  9. Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 273-282, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993674.
  10. Michael B. Cohen, Brittany Terese Fasy, Gary L. Miller, Amir Nayyeri, Richard Peng, and Noel Walkington. Solving 1-laplacians in nearly linear time: collapsing and expanding a topological ball. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 204-216, USA, 2014. Society for Industrial and Applied Mathematics. Google Scholar
  11. Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, and Shen Chen Xu. Solving sdd linear systems in nearly mlog1/2n time. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 343-352, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591833.
  12. Keenan Crane, Mathieu Desbrun Fernando de Goes, and Peter Schröder. Digital geometry processing with discrete exterior calculus. In ACM SIGGRAPH 2013 courses, SIGGRAPH '13, New York, NY, USA, 2013. ACM. Google Scholar
  13. Vin de Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discret. Comput. Geom., 45(4):737-759, 2011. URL: https://doi.org/10.1007/s00454-011-9344-x.
  14. Tamal K. Dey. Computing height persistence and homology generators in R^3 efficiently. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2649-2662. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.164.
  15. Ming Ding, Maximillian Probst Gutenberg, Rasmus Kyng, and Peng Zhang. Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  16. Joel Friedman. Computing betti numbers via combinatorial laplacians. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 386-391, New York, NY, USA, 1996. ACM. URL: https://doi.org/10.1145/237814.237985.
  17. Allen Hatcher. Algebraic topology. Cambridge Univ. Press, Cambridge, 2000. URL: https://cds.cern.ch/record/478079.
  18. Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, USA, 2nd edition, 2012. Google Scholar
  19. Arun Jambulapati and Aaron Sidford. Ultrasparse ultrasparsifiers and faster laplacian system solvers. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '21, pages 540-559, USA, 2021. Google Scholar
  20. Xiaoye Jiang, Lek-Heng Lim, Yuan Yao, and Yinyu Ye. Statistical ranking and combinatorial hodge theory. Math. Program., 127(1):203-244, March 2011. URL: https://doi.org/10.1007/s10107-010-0419-x.
  21. Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, and Zeyuan Allen Zhu. A simple, combinatorial algorithm for solving sdd systems in nearly-linear time. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 911-920, New York, NY, USA, 2013. ACM. URL: https://doi.org/10.1145/2488608.2488724.
  22. Ioannis Koutis and Richard Miller, Gary L. and Peng. Approaching optimality for solving SDD linear systems. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pages 235-244, Washington, DC, USA, 2010. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2010.29.
  23. Ioannis Koutis and Richard Miller, Gary L. and Peng. A nearly-m log n time solver for SDD linear systems. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 590-598, Washington, DC, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.85.
  24. Rasmus Kyng and Peng Zhang. Hardness results for structured linear systems. SIAM J. Comput., 49(4), 2020. URL: https://doi.org/10.1137/17M1161774.
  25. Andrew Ng, Michael Jordan, and Yair Weiss. On spectral clustering: analysis and an algorithm. In T. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14. MIT Press, 2002. URL: https://proceedings.neurips.cc/paper/2001/file/801272ee79cfde7fa5960571fee36b9b-Paper.pdf.
  26. Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley & Sons, Inc., USA, 1998. Google Scholar
  27. Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888-905, 2000. URL: https://doi.org/10.1109/34.868688.
  28. Daniel Spielman. Spectral and algebraic graph theory. Available at http://cs-www.cs.yale.edu/homes/spielman/sagt/sagt.pdf (2021/12/01).
  29. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 563-568, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374456.
  30. Shang-Hua Spielman, Daniel A. and Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 81-90. ACM, 2004. Google Scholar
  31. John Stillwell. Classical Topology and Combinatorial Group Theory, volume 72 of Graduate Texts in Mathematics. Springer, second edition, 1993. URL: http://www.springer.com/mathematics/algebra/book/978-0-387-97970-0.
  32. Alireza Tahbaz-Salehi and Ali Jadbabaie. Distributed coverage verification in sensor networks without location information. IEEE Transactions on Automatic Control, 55(8):1837-1849, 2010. URL: https://doi.org/10.1109/TAC.2010.2047541.
  33. Yiying Tong, Santiago Lombeyda, Anil N. Hirani, and Mathieu Desbrun. Discrete multiscale vector field decomposition. ACM Trans. Graph., 22(3):445-452, July 2003. URL: https://doi.org/10.1145/882262.882290.
  34. Pravin M. Vaidya. Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners. Workshop Talk at the IMA Workshop on Graph Theory and Sparse Matrix Computation, October 1991. Minneapolis, MN. Google Scholar
  35. Qianqian Xu, Qingming Huang, Tingting Jiang, Bowei Yan, Weisi Lin, and Yuan Yao. Hodgerank on random graphs for subjective video quality assessment. IEEE Transactions on Multimedia, 14(3):844-857, 2012. URL: https://doi.org/10.1109/TMM.2012.2190924.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact