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The atomic cluster expansion is developed as a complete descriptor of the local atomic environment, including multicomponent materials, and its relation to a number of other descriptors and potentials is discussed. The effort for evaluating the atomic cluster expansion is shown to scale linearly with the number of neighbors, irrespective of the order of the expansion. Application to small Cu clusters demonstrates smooth convergence of the atomic cluster expansion to meV accuracy. By introducing nonlinear functions of the atomic cluster expansion an interatomic potential is obtained that is comparable in accuracy to state-of-the-art machine learning potentials. Because of the efficient convergence of the atomic cluster expansion relevant subspaces can be sampled uniformly and exhaustively. This is demonstrated by testing against a large database of density functional theory calculations for copper.