  We study sparse generalized inverses $H$ of a rank-$r$ real matrix $A$. We
give a construction for reflexive generalized inverses having at most $r^2$
nonzeros. For $r=1$ and for $r=2$ with $A$ nonnegative, we demonstrate how to
minimize the (vector) 1-norm over reflexive generalized inverses. For general
$r$, we efficiently find reflexive generalized inverses with 1-norm within
approximately a factor of $r^2$ of the minimum 1-norm generalized inverse.
