Atomic Cluster Expansion: Completeness, Efficiency and Stability.

The {\em Atomic Cluster Expansion} (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation in a way that yields immediate guarantees that a complete basis is indeed obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.