Efficient algorithms for computing a strong rank-revealing QR factorization
Ming Gu,Stanley C. Eisenstat +1 more
Reads0
Chats0
TLDR
Two algorithms are presented for computing rank-revealing QR factorizations that are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.Abstract:
Given anm n matrixM withm > n, it is shown that there exists a permutation FI and an integer k such that the QR factorization MYI= Q(Ak ckBk) reveals the numerical rank of M: the k k upper-triangular matrix Ak is well conditioned, IlCkll2 is small, and Bk is linearly dependent on Ak with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.read more
Citations
More filters
Journal ArticleDOI
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
TL;DR: This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation, and presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions.
Posted Content
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions
TL;DR: In this article, a modular framework for constructing randomized algorithms that compute partial matrix decompositions is presented, which uses random sampling to identify a subspace that captures most of the action of a matrix and then the input matrix is compressed to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization.
Posted Content
Randomized algorithms for matrices and data
TL;DR: This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis.
Journal ArticleDOI
Randomized algorithms for the low-rank approximation of matrices
TL;DR: Two recently proposed randomized algorithms for the construction of low-rank approximations to matrices are described and shown to be considerably more efficient and reliable than the classical (deterministic) ones; they also parallelize naturally.
Journal ArticleDOI
Fast Direct Methods for Gaussian Processes
TL;DR: In this paper, the authors show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\mathcal {O} (n\,\log^2, n)$ algorithm for inversion.
References
More filters
Book
Topics in Matrix Analysis
TL;DR: The field of values as discussed by the authors is a generalization of the field of value of matrices and functions, and it includes singular value inequalities, matrix equations and Kronecker products, and Hadamard products.
Book
Solving least squares problems
TL;DR: Since the lm function provides a lot of features it is rather complicated so it is going to instead use the function lsfit as a model, which computes only the coefficient estimates and the residuals.
Journal ArticleDOI
The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses
TL;DR: In this article, the use of Partial Least Squares (PLS) for handling collinearities among the independent variables X in multiple regression is discussed, and successive estimates are obtained using the residuals from previous rank as a new dependent variable y.
Book
Introduction to matrix computations
TL;DR: Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination.
Journal ArticleDOI
Numerical methods for solving linear least squares problems
TL;DR: This paper considers stable numerical methods for handling linear least squares problems that frequently involve large quantities of data, and they are ill-conditioned by their very nature.