Related papers: Deterministic sketching for Krylov subspace method…
We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on…
Mixed-effects models are widely used to model data with hierarchical grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, current standard computations relying on…
It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even…
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but…
In this work, we propose a reduced basis method for efficient solution of parametric linear systems. The coefficient matrix is assumed to be a linear matrix-valued function that is symmetric and positive definite for admissible values of…
We develop a randomized Newton method capable of solving learning problems with huge dimensional feature spaces, which is a common setting in applications such as medical imaging, genomics and seismology. Our method leverages randomized…
The randomized SVD is a method to compute an inexpensive, yet accurate, low-rank approximation of a matrix. The algorithm assumes access to the matrix through matrix-vector products (matvecs). Therefore, when we would like to apply the…
We present a novel distributed computing framework that is robust to slow compute nodes, and is capable of both approximate and exact computation of linear operations. The proposed mechanism integrates the concepts of randomized sketching…
Solving symmetric positive definite linear problems is a fundamental computational task in machine learning. The exact solution, famously, is cubicly expensive in the size of the matrix. To alleviate this problem, several linear-time…
In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations…
Matrix sketching is a recently developed data compression technique. An input matrix A is efficiently approximated with a smaller matrix B, so that B preserves most of the properties of A up to some guaranteed approximation ratio. In so…
We propose a new random sketching approach for embedding high-dimensional Hilbert-Schmidt operators, using random input-output pairs. Such operator can then be approximated in a low-dimensional subspace of operators by solving a small…
In recent years, randomized algorithms have established themselves as fundamental tools in computational linear algebra, with applications in scientific computing, machine learning, and quantum information science. Many randomized matrix…
Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate…
Randomized orthogonal projection methods (ROPMs) can be used to speed up the computation of Krylov subspace methods in various contexts. Through a theoretical and numerical investigation, we establish that these methods produce…
This work is concerned with the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a…
This paper develops a new class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods, such as GMRES and Rayleigh--Ritz. This approach…
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…
Tensor train decomposition is a powerful tool for dealing with high-dimensional, large-scale tensor data, which is not suffering from the curse of dimensionality. To accelerate the calculation of the auxiliary unfolding matrix, some…
We investigate iterative methods with randomized preconditioners for solving overdetermined least-squares problems, where the preconditioners are based on a random embedding of the data matrix. We consider two distinct approaches: the…