Continuous machine learning on Euclidean graphs with unordered vertices

Yury Elkin, Vitaliy Kurlin

TL;DR: The paper develops complete invariants with Lipschitz continuous metrics for embedded graphs under geometric isomorphism (or rigid motion) in any Euclidean space, illustrated on the world's largest databases of molecules with 3D coordinates.

Abstract: Molecular graphs can change their chemical properties under non-rigid deformations in Euclidean space. Hence it is vitally important to distinguish rigid classes of molecular graphs under compositions of translations and rotations. Also, robust outputs of machine learning on molecular graphs embedded in Euclidean space should continuously change under perturbations, motivated by atomic vibrations and experimental noise. We developed a complete invariant that can be inverted back to an embedded graph, uniquely under rigid motion, and has a Lipschitz continuous distance satisfying all metric axioms. For a fixed dimension, the invariant and metric can be computed in polynomial time of the number $m$ of unordered vertices and hence avoiding exponentially many permutations. The new invariants distinguish all chemically different graphs in the world's largest databases of 3D molecules in a few hours on a modest desktop.

Primary Area: Applications->Chemistry, Physics, and Earth Sciences

Keywords: application-driven machine learning, molecular graph, geometric isomorphism, complete SE(n)-invariant, continuous metric

Application-Driven Machine Learning: This submission is on Application-Driven Machine Learning.

Submission Number: 13537