Graph Laplacians, Riemannian Manifolds and their Machine-Learning

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 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.

From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook

“…In [48], MLPs, decision trees and graph NNs could distinguish table/non-table ideals to 100% accuracy, whereby suggesting the existence of a yet-unknown formula. Graphs & Combinatorics: Various properties of finite graphs, such as cyclicity, genus, existence of Euler or Hamilton cycles, etc., were explored by "looking" at the the adjacency matrix with MLPs and SVMs [49]. The algorithms determining some of these quantities are quite involved indeed.…”

Machine Learning Symmetry

“…The works [22,23] are in a sense part of a more general program of applying methods from data science and machine learning to study problems in mathematical physics, see [24,25] for a pedagogical treatment. By now, machine learning and data science have been used to learn various aspects of number theory [26,27,28], quiver gauge theories and cluster algebras [29], knot theory [30,31,32] as well as graph theory [33] and commutative algebra [34]. Importantly, and in hindsight somewhat as a precursor of [23], support vector machines and neural networks were used to learn the algebraic structures of discrete groups and rings [35].…”

Hilbert Series, Machine Learning, and Applications to Physics

“…In these, the effectiveness of ML regressor and classifier techniques in various branches of mathematics and mathematical physics has been investigated. Applications of ML include: finding bundle cohomology on varieties [22,24,25]; distinguishing elliptic fibrations [26] and invariants of Calabi-Yau threefolds [27]; the Donaldson algorithm for numerical Calabi-Yau metrics [28]; the algebraic structures of groups and rings [29]; arithmetic geometry and number theory [30][31][32]; quiver gauge theories and cluster algebras [33]; patterns in particle masses [34]; statistical predictions and model-building in string theory [35][36][37]; and classifying combinatorial properties of finite graphs [38].…”