Related papers: Krylov Subspace Recycling for Fast Iterative Least…
In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan…
Matrices arising in scientific applications frequently admit linear low-rank approximations due to smoothness in the physical and/or temporal domain of the problem. In large-scale problems, computing an optimal low-rank approximation can be…
A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any…
The efficient solution of large-scale multiterm linear matrix equations is a challenging task in numerical linear algebra, and it is a largely open problem. We propose a new iterative scheme for symmetric and positive definite operators,…
Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear…
In this paper, we propose a new algorithm for recovery of low-rank matrices from compressed linear measurements. The underlying idea of this algorithm is to closely approximate the rank function with a smooth function of singular values,…
A matrix algorithm runs superfast (aka at sublinear cost) if it involves much fewer flops and memory cells than an input matrix has entries. Big Data are frequently represented by matrices of immense sizes that cannot be handled directly…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the…
For many applications involving a sequence of linear systems with slowly changing system matrices, subspace recycling, which exploits relationships among systems and reuses search space information, can achieve huge gains in iterations…
The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically…
With the emergence of mixed precision capabilities in hardware, iterative refinement schemes for solving linear systems $Ax=b$ have recently been revisited and reanalyzed in the context of three or more precisions. These new analyses show…
Positive semi-definite matrices commonly occur as normal matrices of least squares problems in statistics or as kernel matrices in machine learning and approximation theory. They are typically large and dense. Thus algorithms to solve…
Iterative refinement is particularly popular for numerical solution of linear systems of equations. We extend it to Low Rank Approximation of a matrix (LRA) and observe close link of the resulting algorithm to oversampling techniques,…
Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $M$, the goal is to compute a matrix $M'$ of given rank $r$ in a linear or affine…
Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known…
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 paper presents a new algorithmic framework for computing sparse solutions to large-scale linear discrete ill-posed problems. The approach is motivated by recent perspectives on iteratively reweighted norm schemes, viewed through the…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local…