Compressive sensing lattice dynamics. I. General formalism

{\it Ab initio\/} calculations have been successfully used for evaluating lattice dynamical properties of solids within the (quasi-)harmonic approximation (i.e., assuming non-interacting phonons with infinite lifetimes), but it remains difficult to treat anharmonicity in all but the simplest compounds. We detail a systematic information theory based approach to deriving {\it ab initio\/} anharmonic force constants: compressive sensing lattice dynamics (CSLD). The non-negligible terms that are necessary to reproduce the first-principles calculated interatomic forces are automatically selected by minimizing the $\ell_1$ norm (sum of absolute values) of the scaled force constants. By using efficient sampling of the configuration space using a modest number of atomic configurations with quasi-random displacements, CSLD is well suited for deriving accurate anharmonic potentials for complex multicomponent compounds with large unit cells. We demonstrate the power and generality of CSLD by calculating the phonon lifetimes and thermal transport properties of Type-I Si clathrates.