Related papers: The matrix-vector complexity of $Ax=b$
Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of…
We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$…
This paper presents an efficient algorithm to solve total variation (TV) regularizations of images contaminated by a both blur and noise. The unconstrained structure of the problem suggests that one can solve a constrained optimization…
We study the optimal lower and upper complexity bounds for finding approximate solutions to the composite problem $\min_x\ f(x)+h(Ax-b)$, where $f$ is smooth and $h$ is convex. Given access to the proximal operator of $h$, for strongly…
This paper introduces new solvers for the computation of low-rank approximate solutions to large-scale linear problems, with a particular focus on the regularization of linear inverse problems. Although Krylov methods incorporating explicit…
The computation of approximating e^tA B, where A is a large sparse matrix and B is a rectangular matrix, serves as a crucial element in numerous scientific and engineering calculations. A powerful way to consider this problem is to use…
The eigenproblem of low-rank updated matrices are of crucial importance in many applications. Recently, an upper bound on the number of distinct eigenvalues of a perturbed matrix was established. The result can be applied to estimate the…
We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are…
We consider the low-rank alternating directions implicit (ADI) iteration for approximately solving large-scale algebraic Sylvester equations. Inside every iteration step of this iterative process a pair of linear systems of equations has to…
In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program,…
The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the well-known T-product for two tensors to define tensor global Arnoldi and tensor global…
We consider optimization problems involving the multiplication of variable matrices to be selected from a given family, which might be a discrete set, a continuous set or a combination of both. Such nonlinear, and possibly discrete,…
We introduce and compare new compression approaches to obtain regularized solutions of large linear systems which are commonly encountered in large scale inverse problems. We first describe how to approximate matrix vector operations with a…
We present a novel Krylov subspace method for approximating $L_f(A, E) \vc{b}$, the matrix-vector product of the Fr\'echet derivative $L_f(A, E)$ of a large-scale matrix function $f(A)$ in direction $E$, a task that arises naturally in the…
Matrix sensing has many real-world applications in science and engineering, such as system control, distance embedding, and computer vision. The goal of matrix sensing is to recover a matrix $A_\star \in \mathbb{R}^{n \times n}$, based on a…
We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this…
Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times, especially…
This paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted, potentially by a malicious adversary. It is well-known that standard algorithms for matrix completion can return…