We show that over a field, the homology H of an A-infinity algebra A admits a canonical A-infinity algebra structure. Given a cycle selection map f : H \to A, there is a construction involving the multiplihedra by which the A-infinity algebra structure on A pulls back along f to a canonical A-infinity algebra structure on H. Whereas our construction does not require that f have a homotopy inverse, this generalizes the Basic Perturbation Lemma. We demonstrate this with an example involving a non-homotopy equivalence. Finally, we mention that our construction transfers directly to the category of A-infinity bialgebras. Consequently, the homology of a loop space over a field is canonically an A-infinity bialgebra.