Revisiting the mathematical foundations of cluster expansion

The cluster expansion (CE) has been a mainstay of alloy modeling for nearly four decades. Early work includes Connolly and Williams who proposed the idea of fitting first-principles data to a generalized Ising model. The mathematical foundation for the cluster expansion was laid by SDG84 (Sanchez, Ducastelle, Gratias, Physica A, v. 128 p. 334-350, 1984). Since that time, CE has been refined into a practical tool. But important questions remain: 1) What is the optimal basis? 2) How can rank deficiency be avoided in the correlation matrix? 3) Can the linearly independent clusters be efficiently enumerated? 4) What fitting data is needed for a robust model? We propose a mathematically simple approach to CE modeling. This new approach is numerically stable, algorithmically simple, and mitigates the need for regularization (genetic algorithms, compressive sensing, etc.)