  The adaptively compressed exchange (ACE) method provides an efficient way for
solving Hartree-Fock-like equations in quantum physics, chemistry, and
materials science. The key step of the ACE method is to adaptively compress an
operator that is possibly dense and full-rank. In this paper, we present a
detailed study of the adaptive compression operation and establish rigorous
convergence properties of the adaptive compression method in the context of
solving linear eigenvalue problems. Our analysis also elucidates the potential
use of the adaptive compression method in a wide range of problems.
