Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses

The M-P (Moore-Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a $1$-norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank

On computing sparse 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. It is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. For computational reasons, it is convenient then, to construct sparse block-structured matrices satisfying relevant properties of the M-P pseudoinverse for specific applications. (Vector) 1-norm minimization has been used to induce sparsity in this context. Aiming at row-sparse generalized inverses motivated by the least-squares application, we consider 2,1-norm minimization (and generalizations). In particular, we show that a 2,1-norm minimizing generalized inverse satisfies two additional M-P pseudoinverse properties, including the one needed for computing least-squares solutions. We present mathematical-optimization formulations related to finding row-sparse generalized inverses that can be solved very efficiently, and compare their solutions numerically to generalized inverses constructed by other methodologies, also aiming at sparsity and row-sparse structure

A survey of generalized inverses and their use in stochastic modelling

In many stochastic models, in particular Markov chains in discrete or continuous time and Markov renewal processes, a Markov chain is present either directly or indirectly through some form of embedding. The analysis of many problems of interest associated with these models, eg. stationary distributions, moments of first passage time distributions and moments of occupation time random variables, often concerns the solution of a system of linear equations involving I – P, where P is the transition matrix of a finite, irreducible, discrete time Markov chain. Generalized inverses play an important role in the solution of such singular sets of equations. In this paper we survey the application of generalized inverses to the aforementioned problems. The presentation will include results concerning the analysis of perturbed systems and the characterization of types of generalized inverses associated with Markovian kernels

Some regular sums

In this paper, we examine the question of regularity of sums of special elements that appear in the study of orthogonality and invertibility.FEDER Funds through "Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - "Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/201

Two by two units

In this paper, we will use outer inverses and the Brown-McCoy shift to characterize the existence of the inverse and group inverse of a $2\times 2$ block matrix.FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - ``Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/2011

Reliability in Constrained Gauss-Markov Models: An Analytical and Differential Approach with Applications in Photogrammetry

This report was prepared by Jackson Cothren, a graduate research associate in the Department of Civil and Environmental Engineering and Geodetic Science at the Ohio State University, under the supervision of Professor Burkhard Schaffrin.This report was also submitted to the Graduate School of the Ohio State University as a dissertation in partial fulfillment of the requirements for the Ph.D. degree.Reliability analysis explains the contribution of each observation in an estimation model to the overall redundancy of the model, taking into account the geometry of the network as well as the precision of the observations themselves. It is principally used to design networks resistant to outliers in the observations by making the outliers more detectible using standard statistical tests.It has been studied extensively, and principally, in Gauss- Markov models. We show how the same analysis may be extended to various constrained Gauss-Markov models and present preliminary work for its use in unconstrained Gauss-Helmert models. In particular, we analyze the prominent reliability matrix of the constrained model to separate the contribution of the constraints to the redundancy of the observations from the observations themselves. In addition, we make extensive use of matrix differential calculus to find the Jacobian of the reliability matrix with respect to the parameters that define the network through both the original design and constraint matrices. The resulting Jacobian matrix reveals the sensitivity of reliability matrix elements highlighting weak areas in the network where changes in observations may result in unreliable observations. We apply the analytical framework to photogrammetric networks in which exterior orientation parameters are directly observed by GPS/INS systems. Tie-point observations provide some redundancy and even a few collinear tie-point and tie-point distance constraints improve the reliability of these direct observations by as much as 33%. Using the same theory we compare networks in which tie-points are observed on multiple images (n-fold points) and tie-points are observed in photo pairs only (two-fold points). Apparently, the use of two-fold tiepoints does not significantly degrade the reliability of the direct exterior observation observations. Coplanarity constraints added to the common two-fold points do not add significantly to the reliability of the direct exterior orientation observations. The differential calculus results may also be used to provide a new measure of redundancy number stability in networks. We show that a typical photogrammetric network with n-fold tie-points was less stable with respect to at least some tie-point movement than an equivalent network with n-fold tie-points decomposed into many two-fold tie-points