Experimental analysis of local searches for sparse reflexive generalized inverses

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 (2018) and Xu, Fampa, and Lee (2019) 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.

[1]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[2]  Rémi Gribonval,et al.  Beyond Moore-Penrose Part II: The Sparse Pseudoinverse , 2017, ArXiv.

[3]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[4]  J. Gondzio.  Stable algorithm for updating dense LU factorization after row or column exchange and row and column addition or deletion , 1992 .

[5]  Jon Lee,et al.  On Sparse Reflexive Generalized Inverses , 2018, Oper. Res. Lett..

[6]  Jon Lee,et al.  On Sparse Reflexive Generalized Inverses , 2018, ArXiv.

[7]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[8]  Jon Lee,et al.  Sparse pseudoinverses via LP and SDP relaxations of Moore-Penrose , 2016, 1606.06969.

[9]  Rémi Gribonval,et al.  Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix Norms , 2017, ArXiv.

[10]  C. A. Rohde.  Contributions to the theory, computation and application of generalized inverses. , 1964 .

[11]  Jon Lee,et al.  Approximate 1-Norm Minimization and Minimum-Rank Structured Sparsity for Various Generalized Inverses via Local Search , 2019, SIAM J. Optim..

[12]  Nikolaos V. Sahinidis,et al.  A review of the LU update in the simplex algorithm , 2012, Int. J. Math. Oper. Res..

[13]  Mihailo Kolundzija,et al.  Beyond Moore-Penrose: Sparse pseudoinverse , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[14]  J. Meyer.  Generalized Inversion of Modified Matrices , 1973 .

[15]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[16]  Jon Lee,et al.  Diving for Sparse Partially-Reflexive Generalized Inverses , 2019, WCGO.