  The M-P (Moore-Penrose) pseudoinverse has as a key application the
computation of least-squares solutions of inconsistent systems of linear
equations. Irrespective of whether a given input matrix is sparse, its M-P
pseudoinverse can be dense, potentially leading to high computational burden,
especially when we are dealing with high-dimensional matrices. The M-P
pseudoinverse is uniquely characterized by four properties, but only two of
them need to be satisfied for the computation of least-squares solutions. Fampa
and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search
procedures to construct sparse block-structured generalized inverses that
satisfy the two key M-P properties, plus one more (the so-called reflexive
property). That additional M-P property is equivalent to imposing a
minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization
is used to induce sparsity and, importantly, to keep the magnitudes of entries
under control for the generalized-inverses constructed. Here, we investigate
the trade-off between low 1-norm and low rank for generalized inverses that can
be used in the computation of least-squares solutions. We propose several
algorithmic approaches that start from a $1$-norm minimizing generalized
inverse that satisfies the two key M-P properties, and gradually decrease its
rank, by iteratively imposing the reflexive property. The algorithms iterate
until the generalized inverse has the least possible rank. During the
iterations, we produce intermediate solutions, trading off low 1-norm (and
typically high sparsity) against low rank.
