1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two

10/20/2020
by   Luze Xu, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/09/2018

On Sparse Reflexive Generalized Inverses

We study sparse generalized inverses H of a rank-r real matrix A. We giv...
research
08/31/2010

Penalty Decomposition Methods for L0-Norm Minimization

In this paper we consider general l0-norm minimization problems, that is...
research
05/22/2021

On Symmetric Invertible Binary Pairing Functions

We construct a symmetric invertible binary pairing function F(m,n) on th...
research
05/07/2018

A Generalized Matrix Inverse with Applications to Robotic Systems

It is well-understood that the robustness of mechanical and robotic cont...
research
10/18/2018

Concentration of the Frobenius norm of generalized matrix inverses

In many applications it is useful to replace the Moore-Penrose pseudoinv...
research
04/19/2018

A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity

There has recently been renewed recognition of the need to understand th...

Please sign up or login with your details

Forgot password? Click here to reset