1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two
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Generalized inverses are important in statistics and other areas of applied matrix algebra. A 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 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 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 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∈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.
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