Papers
Topics
Authors
Recent
2000 character limit reached

Learning 3-Manifold Triangulations (2405.09610v2)

Published 15 May 2024 in math.GT, hep-th, and stat.ML

Abstract: Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $<8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (77)
  1. W. P. Thurston, “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bulletin of the American Mathematical Society 6 no. 3, (1982) 357–381.
  2. G. Perelman, “The Entropy formula for the Ricci flow and its geometric applications,” arXiv:math/0211159.
  3. G. Perelman, “Ricci flow with surgery on three-manifolds,” arXiv:math/0303109.
  4. E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys. 121 (1989) 351–399.
  5. World Scientific, March, 2019.
  6. A. J. Zomorodian, Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2005.
  7. C. R. Brodie, A. Constantin, R. Deen, and A. Lukas, “Machine Learning Line Bundle Cohomology,” Fortsch. Phys. 68 no. 1, (2020) 1900087, arXiv:1906.08730 [hep-th].
  8. M. R. Douglas, S. Lakshminarasimhan, and Y. Qi, “Numerical Calabi-Yau metrics from holomorphic networks,” arXiv:2012.04797 [hep-th].
  9. S. Krippendorf and M. Spannowsky, “A duality connecting neural network and cosmological dynamics,” Mach. Learn. Sci. Tech. 3 no. 3, (2022) 035011, arXiv:2202.11104 [gr-qc].
  10. B. Aslan, D. Platt, and D. Sheard, “Group invariant machine learning by fundamental domain projections,” in NeurIPS Workshop on Symmetry and Geometry in Neural Representations, pp. 181–218, PMLR. 2023.
  11. D. S. Berman, M. S. Klinger, and A. G. Stapleton, “Bayesian renormalization,” Mach. Learn. Sci. Tech. 4 no. 4, (2023) 045011, arXiv:2305.10491 [hep-th].
  12. J. Halverson and F. Ruehle, “Metric Flows with Neural Networks,” arXiv:2310.19870 [hep-th].
  13. R. Alawadhi, D. Angella, A. Leonardo, and T. S. Gherardini, “Constructing and Machine Learning Calabi-Yau Five-Folds,” Fortsch. Phys. 72 no. 2, (2024) 2300262, arXiv:2310.15966 [hep-th].
  14. G. Butbaia, D. Mayorga Peña, J. Tan, P. Berglund, T. Hübsch, V. Jejjala, and C. Mishra, “Physical Yukawa Couplings in Heterotic String Compactifications,” arXiv:2401.15078 [hep-th].
  15. Y.-H. He, E. Hirst, and T. Peterken, “Machine-learning dessins d’enfants: explorations via modular and Seiberg–Witten curves,” J. Phys. A 54 no. 7, (2021) 075401, arXiv:2004.05218 [hep-th].
  16. J. Bao, S. Franco, Y.-H. He, E. Hirst, G. Musiker, and Y. Xiao, “Quiver Mutations, Seiberg Duality and Machine Learning,” Phys. Rev. D 102 no. 8, (2020) 086013, arXiv:2006.10783 [hep-th].
  17. J. Bao, Y.-H. He, E. Hirst, J. Hofscheier, A. Kasprzyk, and S. Majumder, “Hilbert series, machine learning, and applications to physics,” Phys. Lett. B 827 (2022) 136966, arXiv:2103.13436 [hep-th].
  18. J. Bao, Y.-H. He, and E. Hirst, “Neurons on Amoebae,” J. Symb. Comput. 116 (2022) 1–38, arXiv:2106.03695 [math.AG].
  19. J. Bao, Y.-H. He, E. Hirst, J. Hofscheier, A. Kasprzyk, and S. Majumder, “Polytopes and Machine Learning,” Math. Sci. 01 (2023) 181–211, arXiv:2109.09602 [math.CO].
  20. D. S. Berman, Y.-H. He, and E. Hirst, “Machine learning Calabi-Yau hypersurfaces,” Phys. Rev. D 105 no. 6, (2022) 066002, arXiv:2112.06350 [hep-th].
  21. G. Arias-Tamargo, Y.-H. He, E. Heyes, E. Hirst, and D. Rodriguez-Gomez, “Brain webs for brane webs,” Phys. Lett. B 833 (2022) 137376, arXiv:2202.05845 [hep-th].
  22. P.-P. Dechant, Y.-H. He, E. Heyes, and E. Hirst, “Cluster Algebras: Network Science and Machine Learning,” J. Comput. Algebra 8 (2023) , arXiv:2203.13847 [math.CO].
  23. D. Aggarwal, Y.-H. He, E. Heyes, E. Hirst, H. N. S. Earp, and T. S. R. Silva, “Machine learning Sasakian and G2 topology on contact Calabi-Yau 7-manifolds,” Phys. Lett. B 850 (2024) 138517, arXiv:2310.03064 [math.DG].
  24. Y.-H. He, V. Jejjala, C. Mishra, and M. Sharnoff, “Learning to be Simple,” arXiv:2312.05299 [cs.LG].
  25. E. Hirst and T. S. Gherardini, “Calabi-Yau four-, five-, sixfolds as ℙwnsuperscriptsubscriptℙ𝑤𝑛\mathbb{P}_{w}^{n}blackboard_P start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT hypersurfaces: Machine learning, approximation, and generation,” Phys. Rev. D 109 no. 10, (2024) 106006, arXiv:2311.17146 [hep-th].
  26. R.-K. Seong, “Unsupervised machine learning techniques for exploring tropical coamoeba, brane tilings and Seiberg duality,” Phys. Rev. D 108 no. 10, (2023) 106009, arXiv:2309.05702 [hep-th].
  27. E. Choi and R.-K. Seong, “Machine learning regularization for the minimum volume formula of toric Calabi-Yau 3-folds,” Phys. Rev. D 109 no. 4, (2024) 046015, arXiv:2310.19276 [hep-th].
  28. S. Chen, P.-P. Dechant, Y.-H. He, E. Heyes, E. Hirst, and D. Riabchenko, “Machine Learning Clifford Invariants of ADE Coxeter Elements,” Adv. Appl. Clifford Algebras 34 no. 3, (2024) 20, arXiv:2310.00041 [cs.LG].
  29. J. Halverson, B. Nelson, and F. Ruehle, “Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning,” JHEP 06 (2019) 003, arXiv:1903.11616 [hep-th].
  30. S. Abel, A. Constantin, T. R. Harvey, and A. Lukas, “Evolving Heterotic Gauge Backgrounds: Genetic Algorithms versus Reinforcement Learning,” Fortsch. Phys. 70 no. 5, (2022) 2200034, arXiv:2110.14029 [hep-th].
  31. A. Cole, S. Krippendorf, A. Schachner, and G. Shiu, “Probing the Structure of String Theory Vacua with Genetic Algorithms and Reinforcement Learning,” in 35th Conference on Neural Information Processing Systems. 11, 2021. arXiv:2111.11466 [hep-th].
  32. V. Niarchos, C. Papageorgakis, P. Richmond, A. G. Stapleton, and M. Woolley, “Bootstrability in line-defect CFTs with improved truncation methods,” Phys. Rev. D 108 no. 10, (2023) 105027, arXiv:2306.15730 [hep-th].
  33. P. Berglund, Y.-H. He, E. Heyes, E. Hirst, V. Jejjala, and A. Lukas, “New Calabi–Yau manifolds from genetic algorithms,” Phys. Lett. B 850 (2024) 138504, arXiv:2306.06159 [hep-th].
  34. S. Gukov, J. Halverson, and F. Ruehle, “Rigor with Machine Learning from Field Theory to the Poincaré Conjecture,” arXiv:2402.13321 [hep-th].
  35. M. C. Hughes, “A neural network approach to predicting and computing knot invariants,” 2016.
  36. V. Jejjala, A. Kar, and O. Parrikar, “Deep Learning the Hyperbolic Volume of a Knot,” Phys. Lett. B 799 (2019) 135033, arXiv:1902.05547 [hep-th].
  37. S. Gukov, J. Halverson, F. Ruehle, and P. Sułkowski, “Learning to Unknot,” Mach. Learn. Sci. Tech. 2 no. 2, (2021) 025035, arXiv:2010.16263 [math.GT].
  38. J. Craven, V. Jejjala, and A. Kar, “Disentangling a deep learned volume formula,” JHEP 06 (2021) 040, arXiv:2012.03955 [hep-th].
  39. J. Craven, M. Hughes, V. Jejjala, and A. Kar, “Learning knot invariants across dimensions,” SciPost Phys. 14 no. 2, (2023) 021, arXiv:2112.00016 [hep-th].
  40. A. Davies, P. Veličković, L. Buesing, S. Blackwell, D. Zheng, N. Tomašev, R. Tanburn, P. Battaglia, C. Blundell, A. Juhász, M. Lackenby, G. Williamson, D. Hassabis, and P. Kohli, “Advancing mathematics by guiding human intuition with ai,” Nature 600 no. 7887, (12, 2021) 70–74.
  41. S. Gukov, J. Halverson, C. Manolescu, and F. Ruehle, “Searching for ribbons with machine learning,” 2023.
  42. P. Putrov and S. J. Ri, “Graph neural networks and 3-dimensional topology,” 2023.
  43. Casler, “An embedding theorem for connected 3 –manifolds with boundary,” Proc. Amer. Math. Soc. no. 16, (1965) 559–566.
  44. B. A. Burton, “Structures of small closed non-orientable 3-manifold triangulations,” Journal of Knot Theory and Its Ramifications 16 no. 05, (May, 2007) 545–574.
  45. W. Lickorish, “Simplicial moves on complexes and manifolds,” Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest (1999) 299–320.
  46. M. Culler, N. M. Dunfield, M. Goerner, and J. R. Weeks, “SnapPy, a computer program for studying the geometry and topology of 3333-manifolds.” Available at http://snappy.computop.org.
  47. B. A. Burton, R. Budney, W. Pettersson, et al., “Regina: Software for low-dimensional topology.” HTTP://REGINA-NORMAL.GITHUB.IO/, 1999–2021.
  48. B. A. Burton, “The Pachner graph and the simplification of 3-sphere triangulations,” in Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11. ACM Press, 2011.
  49. A. A. Hagberg, D. A. Schult, and P. Swart, “Exploring network structure, dynamics, and function using networkx,” 2008.
  50. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research 12 (2011) 2825–2830.
  51. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: Large-scale machine learning on heterogeneous systems,” 2015. https://www.tensorflow.org/. Software available from tensorflow.org.
  52. S. Matveev, “Transformations of special spines and the Zeeman conjecture,” Math. USSR Izvestia 31 (1988) 423–434.
  53. R. Piergallini, “Standard moves for standard polyhedra and spines,” Rend. Circ. Mat. Palermo 37 no. 19, (1988) 391–414.
  54. M. Lackenby, “Algorithms in 3333-manifold theory,” (2020) 44, arXiv:2002.02179.
  55. S. Matveev, Algorithmic topology and classification of 3-manifolds. No. 9 in Algorithms and computation in mathematics. Springer, 2003.
  56. G. Kuperberg, “Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization,” Pacific Journal of Mathematics 301 no. 1, (Sep, 2019) 189–241.
  57. P.Scott and H. Short, “The homeomorphism problem for closed 3333-manifolds,” Algebr.Geom. Topol. 14 no. 4, (2014) 2431–2444.
  58. H. Rubinstein, “An algorithm to recognize the three sphere,” in Proceedings of the International Congress of Mathematicians (Zurich 1994), pp. 601–611. Birkhäuser, 1995.
  59. A. Mijatović, “Simplifying triangulations of S3superscript𝑆3S^{3}italic_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT,” Pacific J. Math. 208 no. 2, (2003) 291–324.
  60. D. Chicco and G. Jurman, “The advantages of the matthews correlation coefficient (mcc) over f1 score and accuracy in binary classification evaluation,” BMC Genomics 21 no. 1, (Jan, 2020) 6.
  61. https://doi.org/10.5281/zenodo.438045.
  62. P. W. Holland and S. Leinhardt, “Transitivity in structural models of small groups,” Comparative Group Studies 2 no. 2, (1971) 107–124.
  63. D. J. Watts and S. Strogatz, “Collective dynamics of ’small-world’ networks,” Nature 393 no. 6684, (June, 1998) 440–442.
  64. H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society 69 no. 1, (1947) 17–20.
  65. O. Perron, “Zur theorie der matrices,” Mathematische Annalen 64 no. 2, (1907) 248–263.
  66. G. Frobenius, “Ueber matrizen aus nicht negativen elementen,” Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (May, 1912) 456–477.
  67. P. Bonacich, “Power and centrality: A family of measures,” American Journal of Sociology 92 no. 5, (1987) 1170–1182. http://www.jstor.org/stable/2780000.
  68. C. Liebchen and R. Rizzi, “Classes of cycle bases,” Discrete Applied Mathematics 155 no. 3, (2007) 337–355.
  69. A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” 2023.
  70. OpenAI, “Gpt-4 technical report,” 2023.
  71. M. R. Douglas, “Large Language Models,” arXiv:2307.05782 [cs.CL].
  72. J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” 2019.
  73. T. Wolf, L. Debut, V. Sanh, J. Chaumond, C. Delangue, A. Moi, P. Cistac, T. Rault, R. Louf, M. Funtowicz, J. Davison, S. Shleifer, P. von Platen, C. Ma, Y. Jernite, J. Plu, C. Xu, T. L. Scao, S. Gugger, M. Drame, Q. Lhoest, and A. M. Rush, “Transformers: State-of-the-art natural language processing,” in Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pp. 38–45. Association for Computational Linguistics, Online, Oct., 2020. https://www.aclweb.org/anthology/2020.emnlp-demos.6.
  74. C. M. Gordon and J. Luecke, “Knots are determined by their complements,” Bulletin (New Series) of the American Mathematical Society 20 no. 1, (1989) 83 – 87.
  75. M. Dehn, “Die Gruppe der Abbildungsklassen: Das arithmetische Feld auf Flächen,” Acta Mathematica 69 no. none, (1938) 135 – 206. https://doi.org/10.1007/BF02547712.
  76. B. A. Burton, “Simplification paths in the pachner graphs of closed orientable 3-manifold triangulations,” 2011.
  77. ANSI, “ISO-IR-6: ASCII Graphic character set,” Dec, 1975. PDF. ITSCJ/IPSJ.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now

Whiteboard

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Tweets

Sign up for free to view the 7 tweets with 3 likes about this paper.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up for Free