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2020

Graph Laplacians, Riemannian Manifolds and their Machine-Learning

He, Yang-Hui, Yau, Shing-Tung

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Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics.

We apply some of the latest techniques in data science such as supervised and unsupervised machine-learning and topological data analysis to the Wolfram database of some 8000 finite graphs in light of studying these correspondences.
Encouragingly, we find that neural classifiers, regressors and networks can perform, with high efficiently and accuracy, a multitude of tasks ranging from recognizing graph Ricci-flatness, to predicting the spectral gap, to detecting the presence of Hamiltonian cycles, etc.

Graph Laplacians, Riemannian Manifolds and their Machine-Learning · Around

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V. Jejjala, A. Kar and O. Parrikar, “Deep Learning the Hyperbolic Volume of a Knot,” arXiv:1902.05547 [hep-th]
Original
1902
Earlier work this paper cites.
K. Bull, Y. H. He, V. Jejjala and C. Mishra, “Machine Learning CICY Threefolds,” Phys. Lett. B 785
Original
1903
Earlier work this paper cites.
J. Halverson, B. Nelson and F. Ruehle, “Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning,” JHEP 06
Original
1903
Earlier work this paper cites.
Y. H. He and S. J. Lee, “Distinguishing Elliptic Fibrations with AI,” arXiv:1904.08530 [hep-th]
Original
1904
Earlier work this paper cites.
Y. H. He and M. Kim, “Learning Algebraic Structures: Preliminary Investigations,” arXiv:1905.02263 [cs.LG]
Original
1905
Earlier work this paper cites.
C. R. Brodie, A. Constantin, R. Deen and A. Lukas, “Machine Learning Line Bundle Cohomology,” arXiv:1906.08730 [hep-th]
Original
1906
Earlier work this paper cites.
A. Cole, A. Schachner and G. Shiu, “Searching the Landscape of Flux Vacua with Genetic Algorithms,” JHEP 1911
Original
1907
Earlier work this paper cites.
Ryunosuke Ozawa, Yohei Sakurai, Taiki Yamada “Geometric and spectral properties of directed graphs under a lower Ricci curvature bound”, arXiv:1909.07715 [math.DG]
Original
1909
Earlier work this paper cites.
A. Ashmore, Y. H. He and B. Ovrut, “Machine learning Calabi-Yau metrics,” arXiv:1910.08605 [hep-th]
Original
1910
Earlier work this paper cites.
T. W. Grimm, F. Ruehle and D. van de Heisteeg, “Classifying Calabi-Yau threefolds using infinite distance limits,” arXiv:1910.02963 [hep-th]
Original
1910
Earlier work this paper cites.
L. Alessandretti, A. Baronchelli, Y. H. He, “ML meets Number Theory: The Data Science of Birch-Swinnerton-Dyer,” arXiv:1911.02008 [math.NT]
Original
1911
Earlier work this paper cites.
Calabi, Eugenio, “The space of Kähler metrics”, Proc. Internat. Congress Math. Amsterdam, 2, pp. 206 - 207 (1954) – “On Kähler manifolds with vanishing canonical class”, in Fox, Spencer, Tucker, Algebraic geometry and topology. A symposium in honor of S. Lefschetz
1957
Earlier work this paper cites.
S.-T. Yau, “Calabi’s conjecture and some new results in algebraic geometry,” Proc. Nat. Acad., USA, 74 (5), pp 1798-9, (1977) –, “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I”, Comm. Pure and Applied Maths, 31 (3), pp 339-411, (1978)
1978
Earlier work this paper cites.
P. Li, S.-T. Yau, “Estimates of eigenvalues of a compact Riemannian manifold,” AMS Symp. on the Geometry of the Laplace Operator, U. Hawaii at Manoa, 1979, 205-239
1979
Earlier work this paper cites.
Appel, Kenneth; Haken, Wolfgang; J. Joch, “Every Planar Map is Four-Colorable”, Contemporary Maths, 98, AMS (1989)
1989
Earlier work this paper cites.
B. Mohar, “The Laplacian spectrum of graphs”, in Graph Theory, Combinatorics, and Applications
1991
Earlier work this paper cites.
Robert Grone, Russell Merris, “The Laplacian spectrum of a graph II, SIAM J. Disc.”, Math. 7 (1994), 221 - 229
1994
Earlier work this paper cites.
Fan Chung and S.-T. Yau, “Logarithmic Harnack inequalities”, Math. Res. Lett. (1996), 793 - 812
1996
Earlier work this paper cites.
Chung, Fan R. K., Spectral graph theory
1997
Earlier work this paper cites.
J. Bao, Y. H. He, E. Hirst and S. Pietromonaco, “Lectures on the Calabi-Yau Landscape,” arXiv:2001.01212 [hep-th]
Original
2001
Earlier work this paper cites.
P. Betzler and S. Krippendorf, “Connecting Dualities and Machine Learning,” Fortsch. Phys. 68
Original
2002
Earlier work this paper cites.

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2016
Later among the works it cites.
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Cited alongside, same era.

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Original

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