1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two

Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix $H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also satisfies the M-P property $HAH=H$, then it is called \emph{reflexive}. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various \emph{sparse} reflexive generalized inverses of $A$. As is common, we use (vector) 1-norm minimization for both inducing sparsity and for keeping the magnitude of entries under control. When $A$ is symmetric, a symmetric $H$ is highly desirable, but generally such a restriction on $H$ will not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. Letting the rank of $A$ be $r$, we establish that the 1-norm minimizing generalized inverse of this type is a 1-norm minimizing symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ is nonnegative. Another aspect of symmetry that we consider relates to another M-P property: $H$ is \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property is sufficient for a generalized inverse to be used to solve the least-squares problem $\min\{\|Ax-b\|_2:~x\in\mathbb{R}^n\}$ using $H$, via $x:=Hb$. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We establish that the 1-norm minimizing ah-symmetric generalized inverse of this type is a 1-norm minimizing ah-symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ satisfies a technical condition.