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The M-P (Moore-Penrose) pseudoinverse has as a key application thecomputation of least-squares solutions of inconsistent systems of linearequations. Irrespective of whether a given input matrix is sparse, its M-Ppseudoinverse can be dense, potentially leading to high computational burden,especially when we are dealing with high-dimensional matrices. The M-Ppseudoinverse is uniquely characterized by four properties, but only two ofthem need to be satisfied for the computation of least-squares solutions. Fampaand Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-searchprocedures to construct sparse block-structured generalized inverses thatsatisfy the two key M-P properties, plus one more (the so-called reflexiveproperty). That additional M-P property is equivalent to imposing aminimum-rank condition on the generalized inverse. (Vector) 1-norm minimizationis used to induce sparsity and, importantly, to keep the magnitudes of entriesunder control for the generalized-inverses constructed. Here, we investigatethe trade-off between low 1-norm and low rank for generalized inverses that canbe used in the computation of least-squares solutions. We propose severalalgorithmic approaches that start from a $1$-norm minimizing generalizedinverse that satisfies the two key M-P properties, and gradually decrease itsrank, by iteratively imposing the reflexive property. The algorithms iterateuntil the generalized inverse has the least possible rank. During theiterations, we produce intermediate solutions, trading off low 1-norm (andtypically high sparsity) against low rank.

Trading off 1-norm and sparsity against rank for linear models using  mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric  generalized inverses