It is demonstrated that the "corrected Hartree-Fock" (CHF) density matrix functional proposed by Csanyi and Arias is identical with the Hartree-Fock-Bogoliubov (HFB) functional of the generalized density matrix up to the sign of the pairing energy term. Using this analogy, variational CHF calculations can be performed much more efficiently by solving the HFB equations for the generalized density matrix than by optimizing separately the natural orbitals and their occupations numbers. A family of CHF-type functionals with a scaled pairing energy is introduced and compared to the closely related antisymmetrized geminal power method.