Calculating Wannier functions via basis pursuit using a machine learned dictionary

One of the earliest methods for calculating Wanner functions enforced localization by maximizing their projection onto a carefully chosen set of localized functions. Eventually use of this method faded with the emergence of more systematic approaches such as maximal localization. In this work we provide a modern method for calculating Wannier functions via projection by incorporating basis pursuit into the quantum variational method to automatically generate a set of localized functions needed to enforce localization while obtaining the ground state. We determine the dictionary to use for basis pursuit by performing sparse coding on a set of known Wannier functions. The algorithmic simplicity of our method suggests that it may be suited for automated calculation of Wannier functions. Other applications in physics that involve the variational principle could benefit from the inclusion of basis pursuit and sparse coding.