Related papers: Approximating matrix eigenvalues by subspace itera…

In this paper, we describe a new algorithm to build a few sparse principal components from a given data matrix. Our approach does not explicitly create the covariance matrix of the data and can be viewed as an extension of the Kogbetliantz…

The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…

This survey highlights the recent advances in algorithms for numerical linear algebra that have come from the technique of linear sketching, whereby given a matrix, one first compresses it to a much smaller matrix by multiplying it by a…

In this paper, we study first-order methods on a large variety of low-rank matrix optimization problems, whose solutions only live in a low dimensional eigenspace. Traditional first-order methods depend on the eigenvalue decomposition at…

With appropriately chosen sampling probabilities, sampling-based random projection can be used to implement large-scale statistical methods, substantially reducing computational cost while maintaining low statistical error. However,…

Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a…

Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize…

We propose a quantum algorithm for finding eigenvalues of non-unitary matrices. We show how to construct, through interactions in a quantum system and projective measurements, a non-Hermitian or non-unitary matrix and obtain its eigenvalues…

Randomized algorithms are overwhelming methods for low-rank approximation that can alleviate the computational expenditure with great reliability compared to deterministic algorithms. A crucial thought is generating a standard Gaussian…

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…

We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix…

We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we…

Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…

We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation. Our results indicate that spectral sparsification…

In light of recent data science trends, new interest has fallen in alternative matrix factorizations. By this, we mean various ways of factorizing particular data matrices so that the factors have special properties and reveal insights into…

In distributed systems, communication is a major concern due to issues such as its vulnerability or efficiency. In this paper, we are interested in estimating sparse inverse covariance matrices when samples are distributed into different…

We develop an algorithm for sampling from the unitary invariant random matrix ensembles. The algorithm is based on the representation of their eigenvalues as a determinantal point process whose kernel is given in terms of orthogonal…

This paper discusses new simulation algorithms for stochastic chemical kinetics that exploit the linearity of the chemical master equation and its matrix exponential exact solution. These algorithms make use of various approximations of the…

Iterative phase estimation has long been used in quantum computing to estimate Hamiltonian eigenvalues. This is done by applying many repetitions of the same fundamental simulation circuit to an initial state, and using statistical…

We present a quantum algorithm for estimating the matrix determinant based on quantum spectral sampling. The algorithm estimates the logarithm of the determinant of an $n \times n$ positive sparse matrix to an accuracy $\epsilon$ in time…