Mathematics, Computation and Geometry of Data

Volume 2 (2022)

Number 1

Graph Laplacians, Riemannian manifolds, and their machine-learning

Pages: 1 – 48

DOI: https://dx.doi.org/10.4310/MCGD.2022.v2.n1.a1

Authors

Yang-Hui He (London Institute for Mathematical Sciences, London, United Kingdom; Merton College, University of Oxford, United Kingdom; Dept. of Mathematics, City University of London, U.K.; and School of Physics, NanKai University, Tianjin, China)

Shing-Tung Yau (Yau Math. Sci. Center, Tsinghua Univ., Beijing, China; Depts. of Math. and Physics & Center of Math. Sci. and App. (CMSA), Harvard University; and Beijing Institute of Math. Sci. & Appl., Huairou Science City, Beijing, China)

Abstract

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 efficiency 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.

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Y.-H. H. is indebted to the Science and Technology Facilities Council, UK, for grant ST/J00037X/1, as well as the Chang-Jiang chair professorship from Nankai University where this work began.

The work of S.-T. Y. is supported in part by a grant from the Simons Foundation in Homological Mirror Symmetry

Received 9 August 2020

Published 21 October 2022