On Sparse Reflexive Generalized Inverses
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We study sparse generalized inverses H of a rank-r real matrix A. We give a "block construction" for reflexive generalized inverses having at most r^2 nonzeros. When r=1, we demonstrate how minimizing the (vector) 1-norm of H among generalized inverses can be achieved by a particular reflexive generalized inverse following our block construction. When r=2 and A is equivalent to a nonnegative matrix by signing rows and columns, we again demonstrate how minimizing the (vector) 1-norm of H among generalized inverses can be achieved by a particular reflexive generalized inverse following our block construction. Finally, for general r, we demonstrate how to efficiently find a reflexive generalized inverse following our block construction that is within approximately a factor of r^2 of the (vector) 1-norm of the generalized inverse having minimum (vector) 1-norm.
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