  First-principles prediction of lattice thermal conductivity $\kappa_L$ of
strongly anharmonic crystals is a long-standing challenge in solid state
physics. Making use of recent advances in information science, we propose a
systematic and rigorous approach to this problem, compressive sensing lattice
dynamics (CSLD). Compressive sensing is used to select the physically important
terms in the lattice dynamics model and determine their values in one shot.
Non-intuitively, high accuracy is achieved when the model is trained on
first-principles forces in {\it quasi-random\/} atomic configurations. The
method is demonstrated for Si, NaCl, and Cu$_{12}$Sb$_4$S$_{13}$, an
earth-abundant thermoelectric with strong phonon-phonon interactions that limit
the room-temperature $\kappa_L$ to values near the amorphous limit.
