An $\ell_0\ell_2$-norm regularized regression model for construction of robust cluster expansions in multicomponent systems

We introduce the $\ell_0\ell_2$-norm regularization and hierarchy constraints into linear regression for the construction of cluster expansion to describe configurational disorder in materials. The approach is implemented through mixed integer quadratic programming (MIQP). The $\ell_2$-norm regularization is used to suppress intrinsic data noise, while $\ell_0$-norm is used to penalize the number of non-zero elements in the solution. The hierarchy relation between clusters imposes relevant physics and is naturally included by the MIQP paradigm. As such, sparseness and cluster hierarchy can be well optimized to obtain a robust, converged, and effective cluster interactions with improved physical meaning. We demonstrate the effectiveness of $\ell_0\ell_2$-norm regularization in two high-component disordered rocksalt cathode material systems, where we compare the cross-validation and convergence speed, reproduction of phase diagrams, voltage profiles, and Li-occupancy energies with those of the conventional $\ell_1$-norm regularized cluster expansion model.