On the computation of sparse reflexive generalized inverses

The well-known Moore-Penrose (M-P) pseudoinverse is used in several linear-algebra applications; for example, to compute least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given matrix is sparse, its M-P pseudoinverse can be completely dense, potentially leading to high computational burden and numerical difficulties, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. We apply mathematical optimization to induce general sparsity and structured sparsity on generalized inverses of a given matrix, which satisfy only specific subsets of the M-P properties. We use norm minimization to induce (unstructured) sparsity and (structured) row-sparsity. We also consider the application of local-search procedures that produce generalized inverses with guaranteed structured sparsity, with low rank, and with entries under control.