Related papers: Solution of Least Squares Problems with Randomized…
We study optimal diagonal preconditioning using the classical worst-case $\kappa$-condition number and the averaging-based $\omega$-condition number. For the $\kappa$-optimal preconditioning problem, we derive an affine-based pseudoconvex…
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and…
We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations,…
This paper derives a new variational equation for the linear least-squares backward error by expressing the backward error in terms of a generalized eigenvalue problem and using results from indefinite linear algebra. For problems with…
We a present and analyze rpCholesky-QR, a randomized preconditioned Cholesky-QR algorithm for computing the thin QR factorization of real mxn matrices with rank n. rpCholesky-QR has a low orthogonalization error, a residual on the order of…
We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$…
Contrary to an assumption made by Bojanczyk, Higham, and Patel [SIAM J. Matrix Anal. Appl., 24(4):914-931, 2003], a perturbation bound for indefinite least square problems is capable of arbitrarily large overestimates for all perturbations…
The Total Least Squares solution of an overdetermined, approximate linear equation $Ax \approx b$ minimizes a nonlinear function which characterizes the backward error. We show that a globally convergent variant of the Gauss--Newton…
Sparse linear regression is a fundamental problem in high-dimensional statistics, but strikingly little is known about how to efficiently solve it without restrictive conditions on the design matrix. We consider the (correlated) random…
We use the practical framework for abstract perturbed saddle point problems recently introduced by Hong et al. to analyze the mixed formulation of the Hodge Laplace problem. We compose two parameter-dependent norms in which the uniform…
Many real-world applications are addressed through a linear least-squares problem formulation, whose solution is calculated by means of an iterative approach. A huge amount of studies has been carried out in the optimization field to…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…
We consider a setting in which it is desired to find an optimal complex vector $\mathbf{x}\in\mathbb{C}^N$ that satisfies $\mathcal{A}(\mathbf{x}) \approx \mathbf{b}$ in a least-squares sense, where $\mathbf{b} \in \mathbb{C}^M$ is a data…
The paper introduces a novel, hierarchical preconditioner based on nested dissection and hierarchical matrix compression. The preconditioner is intended for continuous and discontinuous Galerkin formulations of elliptic problems. We exploit…
We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously we want this enclosure to be as tight as…
The least-squares support vector machine is a frequently used kernel method for non-linear regression and classification tasks. Here we discuss several approximation algorithms for the least-squares support vector machine classifier. The…
This work provides simple algorithms for multi-class (and multi-label) prediction in settings where both the number of examples n and the data dimension d are relatively large. These robust and parameter free algorithms are essentially…
We present and analyze an efficient implementation of an iteratively reweighted least squares algorithm for recovering a matrix from a small number of linear measurements. The algorithm is designed for the simultaneous promotion of both a…
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it,…
Ill-posed linear inverse problems appear frequently in various signal processing applications. It can be very useful to have theoretical characterizations that quantify the level of ill-posedness for a given inverse problem and the degree…