Computation and applications of sparse generalized inverses

Given $A\in\mathbb{R}^{m\times n}$, its Moore-Penrose (M-P) pseudoinverse $H\in\mathbb{R}^{n\times m}$ satisfies the properties: (P1) $AHA=A$, (P2) $HAH=H$, (P3) $AH=(AH)^\top$ and (P4) $HA=(HA)^\top$.  We say that $H$ is a generalized inverse of $A$ if it satisfies (P1). We propose different approaches for computing sparse generalized inverses that can be used to efficiently compute least-squares solutions of overdetermined systems of linear equations and minimum 2-norm solutions of underdetermined systems, with comprehensive applications. We investigate mathematical models for computing these generalized inverses based on a theoretical study of the representation of proper subsets of the M-P properties as reduced systems of linear equations. We propose an ADMM (Alternating Direction Method of Multipliers) algorithm for computing generalized inverses that can be used to compute least-squares and minimum 2-norm solutions. We investigate the generalized Tikhonov problem and use its analysis to propose an alternative approach for computing least-squares solutions using generalized inverses.