Wigner kernels: body-ordered equivariant machine learning without a basis

TL;DR

This paper introduces Wigner kernels, a novel, fully equivariant, body-ordered density-based machine learning method that efficiently models local atomic environments with linear computational complexity, achieving state-of-the-art results.

Contribution

The paper presents Wigner kernels, a new iterative approach for body-ordered equivariant kernels that avoids exponential complexity growth, unlike traditional feature-space models.

Findings

Achieves state-of-the-art accuracy on QM9 dataset

Computational cost grows linearly with body-order

Applicable to scalar and tensorial targets

Abstract

Machine-learning models based on a point-cloud representation of a physical object are ubiquitous in scientific applications and particularly well-suited to the atomic-scale description of molecules and materials. Among the many different approaches that have been pursued, the description of local atomic environments in terms of their neighbor densities has been used widely and very succesfully. We propose a novel density-based method which involves computing ``Wigner kernels''. These are fully equivariant and body-ordered kernels that can be computed iteratively with a cost that is independent of the radial-chemical basis and grows only linearly with the maximum body-order considered. This is in marked contrast to feature-space models, which comprise an exponentially-growing number of terms with increasing order of correlations. We present several examples of the accuracy of models based on Wigner kernels in chemical applications, for both scalar and tensorial targets, reaching state-of-the-art accuracy on the popular QM9 benchmark dataset, and we discuss the broader relevance of these ideas to equivariant geometric machine-learning.