Efficient electronic structure methods can be built around efficient tensor representations of the wavefunction. Here we first describe a general view of tensor factorization for the compact representation of electronic wavefunctions. Next, we use this language to construct a low-complexity representation of the doubles amplitudes in local second-order Møller-Plesset perturbation theory. We introduce two approximations--the direct orbital-specific virtual approximation and the full orbital-specific virtual approximation. In these approximations, each occupied orbital is associated with a small set of correlating virtual orbitals. Conceptually, the representation lies between the projected atomic orbital representation in Pulay-Saebø local correlation theories and pair natural orbital correlation theories. We have tested the orbital-specific virtual approximations on a variety of systems and properties including total energies, reaction energies, and potential energy curves. Compared to the Pulay-Saebø ansatz, we find that these approximations exhibit favorable accuracy and computational times while yielding smooth potential energy curves.