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Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences

Authors Ming Ding, Peng Zhang

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LIPIcs.ESA.2023.41.pdf
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Author Details

Ming Ding
  • ETH Zürich, Switzerland
Peng Zhang
  • Rutgers University, New Brunswick, NJ, USA

Funding

This work was supported in part by NSF Grant CCF-2238682.

Acknowledgements

We thank Rasmus Kyng for valuable discussions and the reviewers for their insightful comments.

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Ming Ding and Peng Zhang. Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.41

Abstract

We present efficient algorithms for solving systems of linear equations in 1-Laplacians of well-shaped simplicial complexes. 1-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in ℝ³ (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022], Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides quadratic time solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in ℝ³.
We generalize the specialized solvers for 1-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint" and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Computations on matrices
  • Mathematics of computing → Algebraic topology
Keywords
  • 1-Laplacian Solvers
  • Simplicial Complexes
  • Incomplete Nested Dissection

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References

  1. Owe Axelsson. A survey of preconditioned iterative methods for linear systems of algebraic equations. BIT Numerical Mathematics, 25(1):165-187, 1985. Google Scholar
  2. Mitali Bafna and Nikhil Vyas. Optimal fine-grained hardness of approximation of linear equations. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:19, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  3. Sergio Barbarossa and Stefania Sardellitti. Topological signal processing over simplicial complexes. IEEE Transactions on Signal Processing, 68:2992-3007, 2020. Google Scholar
  4. Gill Barequet and Sariel Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. Journal of Algorithms, 38(1):91-109, 2001. Google Scholar
  5. 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. SIAM, 2022. Google Scholar
  6. 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). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Google Scholar
  7. Erik G Boman, Bruce Hendrickson, and Stephen Vavasis. Solving elliptic finite element systems in near-linear time with support preconditioners. SIAM Journal on Numerical Analysis, 46(6):3264-3284, 2008. Google Scholar
  8. Michael M Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18-42, 2017. Google Scholar
  9. Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009. Google Scholar
  10. Li Chen, Rasmus Kyng, Yang P Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 612-623. IEEE, 2022. Google Scholar
  11. David RJ Chillingworth. Collapsing three-dimensional convex polyhedra. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 63, pages 353-357. Cambridge University Press, 1967. Google Scholar
  12. David RJ Chillingworth. Collapsing three-dimensional convex polyhedra: correction. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 88, pages 307-310. Cambridge University Press, 1980. Google Scholar
  13. 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 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 204-216. SIAM, 2014. Google Scholar
  14. Michael B Cohen, Jonathan Kelner, Rasmus Kyng, John Peebles, Richard Peng, Anup B Rao, and Aaron Sidford. Solving directed laplacian systems in nearly-linear time through sparse lu factorizations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 898-909. IEEE, 2018. Google Scholar
  15. Michael B Cohen, Jonathan Kelner, John Peebles, Richard Peng, Anup B Rao, Aaron Sidford, and Adrian Vladu. Almost-linear-time algorithms for markov chains and new spectral primitives for directed graphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 410-419, 2017. Google Scholar
  16. 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 m log^1/2 n time. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 343-352. ACM, 2014. Google Scholar
  17. Samuel I Daitch and Daniel A Spielman. Support-graph preconditioners for 2-dimensional trusses. arXiv preprint cs/0703119, 2007. Google Scholar
  18. Samuel I Daitch and Daniel A Spielman. Faster approximate lossy generalized flow via interior point algorithms. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 451-460, 2008. Google Scholar
  19. Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, and Peng Zhang. Hardness results for laplacians of simplicial complexes via sparse-linear equation complete gadgets. In Mikołaj Bojańczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 53:1-53:19, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  20. Ming Ding and Peng Zhang. Weighted 1-laplacian solvers for well-shaped simplicial complexes. arXiv preprint arXiv:2302.06499, 2023. Google Scholar
  21. Beno Eckmann. Harmonische Funktionen und Randwertaufgaben in einem Komplex. Commentarii Mathematici Helvetici, 17(1):240-255, December 1944. URL: https://doi.org/10.1007/BF02566245.
  22. Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Soc., 2010. Google Scholar
  23. Herbert Edelsbrunner and Salman Parsa. On the computational complexity of betti numbers: reductions from matrix rank. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms, pages 152-160. SIAM, 2014. Google Scholar
  24. Joel Friedman. Computing betti numbers via combinatorial laplacians. In Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing, pages 386-391, 1996. Google Scholar
  25. Alan George. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis, 10(2):345-363, 1973. Google Scholar
  26. Robert Ghrist. Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society, 45(1):61-75, 2008. Google Scholar
  27. Arun Jambulapati and Aaron Sidford. Ultrasparse ultrasparsifiers and faster laplacian system solvers. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 540-559. SIAM, 2021. Google Scholar
  28. Xiaoye Jiang, Lek-Heng Lim, Yuan Yao, and Yinyu Ye. Statistical ranking and combinatorial hodge theory. Mathematical Programming, 127(1):203-244, 2011. Google Scholar
  29. 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, pages 911-920, 2013. Google Scholar
  30. Ioannis Koutis, Gary L. Miller, and Richard Peng. Approaching optimality for solving SDD linear systems. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS '10, pages 235-244, Washington, DC, USA, 2010. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2010.29.
  31. Ioannis Koutis, Gary L. Miller, and Richard 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, FOCS '11, pages 590-598, Washington, DC, USA, 2011. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2011.85.
  32. Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, and Daniel A Spielman. Sparsified cholesky and multigrid solvers for connection laplacians. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 842-850, 2016. Google Scholar
  33. Rasmus Kyng, Richard Peng, Robert Schwieterman, and Peng Zhang. Incomplete nested dissection. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 404-417, 2018. Google Scholar
  34. Rasmus Kyng and Sushant Sachdeva. Approximate gaussian elimination for laplacians-fast, sparse, and simple. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 573-582. IEEE, 2016. Google Scholar
  35. Hyekyoung Lee, Moo K Chung, Hongyoon Choi, Hyejin Kang, Seunggyun Ha, Yu Kyeong Kim, and Dong Soo Lee. Harmonic holes as the submodules of brain network and network dissimilarity. In Computational Topology in Image Context: 7th International Workshop, CTIC 2019, Málaga, Spain, January 24-25, 2019, Proceedings 7, pages 110-122. Springer, 2019. Google Scholar
  36. Yin Tat Lee and Aaron Sidford. Efficient accelerated coordinate descent methods and faster algorithms for solving linear systems. In 2013 ieee 54th annual symposium on foundations of computer science, pages 147-156. IEEE, 2013. Google Scholar
  37. Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in o (vrank) iterations and faster algorithms for maximum flow. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 424-433. IEEE, 2014. Google Scholar
  38. Lek-Heng Lim. Hodge laplacians on graphs. Siam Review, 62(3):685-715, 2020. Google Scholar
  39. Richard J Lipton, Donald J Rose, and Robert Endre Tarjan. Generalized nested dissection. SIAM journal on numerical analysis, 16(2):346-358, 1979. Google Scholar
  40. Wenye Ma, Jean-Michel Morel, Stanley Osher, and Aichi Chien. An l 1-based variational model for retinex theory and its application to medical images. In CVPR 2011, pages 153-160. IEEE, 2011. Google Scholar
  41. Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 253-262. IEEE, 2013. Google Scholar
  42. Aleksander Madry. Computing maximum flow with augmenting electrical flows. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 593-602. IEEE, 2016. Google Scholar
  43. Gary L Miller and William Thurston. Separators in two and three dimensions. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 300-309, 1990. Google Scholar
  44. Zipei Nie. Matrix anti-concentration inequalities with applications. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 568-581, 2022. Google Scholar
  45. Braxton Osting, Jérôme Darbon, and Stanley Osher. Statistical ranking using the l1-norm on graphs. AIMS Journal on Inverse Problems and Imaging, 7(3):907-926, 2013. Google Scholar
  46. Richard Peng and Daniel A Spielman. An efficient parallel solver for sdd linear systems. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 333-342. ACM, 2014. Google Scholar
  47. Richard Peng and Santosh Vempala. Solving sparse linear systems faster than matrix multiplication. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 504-521. SIAM, 2021. Google Scholar
  48. Gil Shklarski and Sivan Toledo. Rigidity in finite-element matrices: Sufficient conditions for the rigidity of structures and substructures. SIAM Journal on Matrix Analysis and Applications, 30(1):7-40, 2008. Google Scholar
  49. Daniel A Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 563-568, 2008. Google Scholar
  50. Daniel A Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835-885, 2014. Google Scholar
  51. Martin Tancer. Recognition of collapsible complexes is np-complete. Discrete & Computational Geometry, 55(1):21-38, 2016. Google Scholar
  52. Shang-Hua Teng. The laplacian paradigm: Emerging algorithms for massive graphs. In Theory and Applications of Models of Computation: 7th Annual Conference, TAMC 2010, Prague, Czech Republic, June 7-11, 2010. Proceedings 7, pages 2-14. Springer, 2010. Google Scholar
  53. Yiying Tong, Santiago Lombeyda, Anil N Hirani, and Mathieu Desbrun. Discrete multiscale vector field decomposition. ACM transactions on graphics (TOG), 22(3):445-452, 2003. Google Scholar
  54. Ke Ye and Lek-Heng Lim. Cohomology of cryo-electron microscopy. SIAM Journal on Applied Algebra and Geometry, 1(1):507-535, 2017. Google Scholar
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