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.

翻译：基于物理对象点云表征的机器学习模型在科学应用中无处不在，尤其适用于分子与材料的原子尺度描述。在众多已被探索的方法中，通过邻域密度描述局部原子环境的方法得到了广泛且成功的应用。我们提出了一种新颖的基于密度的方法，该方法涉及计算"Wigner核"。这些是完全等变且体序依赖的核函数，可通过迭代方式计算，其计算成本独立于径向化学基组，仅随所考虑的最大体序线性增长。这与特征空间模型形成鲜明对比——后者随关联阶数增加包含指数级增长的项数。我们展示了基于Wigner核的模型在化学应用中的多个精度实例，涵盖标量和张量目标，在流行的QM9基准数据集上达到了最先进的精度，并讨论了这些思想对等变几何机器学习的更广泛意义。