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Experimental analysis of local searches for sparse reflexive generalized inverses

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Abstract

The well-known M-P (Moore-Penrose) pseudoinverse is used in several linear-algebra applications; for example, to compute least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given matrix is sparse, its M-P pseudoinverse can be completely dense, potentially leading to high computational burden and numerical difficulties, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. In this context, Fampa and Lee (Oper. Res. Lett., 46:605–610, 2018) and Xu et al. (SIAM J. Optim., to appear) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy only some of the M-P properties. (Vector) 1-norm minimization is used to induce sparsity and to keep the magnitude of the entries under control, and theoretical results limit the distance between the 1-norm of the solution of the local searches and the minimum 1-norm of generalized inverses with corresponding properties. We have implemented several local-search procedures based on results presented in these two papers and make here an experimental analysis of them, considering their application to randomly generated matrices of varied dimensions, ranks, and densities. Further, we carried out a case study on a real-world data set.

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Notes

  1. percentage of variation of the goal variable that is linearly explained by the regression

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Acknowledgements

M. Fampa was supported in part by CNPq grant 303898/2016-0. J. Lee was supported in part by ONR grant N00014-17-1-2296. G. Ponte was supported in part by CNPq PIBIC scholarship 149149/2020-4. M. Fampa, J. Lee and L. Xu were supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program.

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  1. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Marcia Fampa & Gabriel Ponte

  2. University of Michigan, Ann Arbor, USA

    Jon Lee & Luze Xu

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  1. Marcia Fampa
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  2. Jon Lee
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  3. Gabriel Ponte
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  4. Luze Xu
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Correspondence to Jon Lee.

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Fampa, M., Lee, J., Ponte, G. et al. Experimental analysis of local searches for sparse reflexive generalized inverses. J Glob Optim 81, 1057–1093 (2021). https://doi.org/10.1007/s10898-021-01087-y

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