Open AccessPosted Content
Blocked rank-revealing QR factorizations: How randomized sampling can be used to avoid single-vector pivoting
Reads0
Chats0
TLDR
The manuscript describes a algorithm for computing a QR factorization where $P$ is a permutation matrix, $Q$ is orthonormal, and $R$ is upper triangular, and the algorithm is blocked, to allow it to be implemented efficiently.Abstract:
Given a matrix $A$ of size $m\times n$, the manuscript describes a algorithm for computing a QR factorization $AP=QR$ where $P$ is a permutation matrix, $Q$ is orthonormal, and $R$ is upper triangular. The algorithm is blocked, to allow it to be implemented efficiently. The need for single vector pivoting in classical algorithms for computing QR factorizations is avoided by the use of randomized sampling to find blocks of pivot vectors at once. The advantage of blocking becomes particularly pronounced when $A$ is very large, and possibly stored out-of-core, or on a distributed memory machine. The manuscript also describes a generalization of the QR factorization that allows $P$ to be a general orthonormal matrix. In this setting, one can at moderate cost compute a \textit{rank-revealing} factorization where the mass of $R$ is concentrated to the diagonal entries. Moreover, the diagonal entries of $R$ closely approximate the singular values of $A$. The algorithms described have asymptotic flop count $O(m\,n\,\min(m,n))$, just like classical deterministic methods. The scaling constant is slightly higher than those of classical techniques, but this is more than made up for by reduced communication and the ability to block the computation.read more
Citations
More filters
Journal ArticleDOI
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns
TL;DR: This article explores how to design optimal sensor locations for signal reconstruction in a framework that scales to arbitrarily large problems, leveraging modern techniques in machine learning and sparse sampling.
Journal ArticleDOI
Randomized numerical linear algebra: Foundations and algorithms
TL;DR: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.
Journal ArticleDOI
Data-Driven Sparse Sensor Placement for Reconstruction
TL;DR: In this paper, the singular value decomposition and QR pivoting are used to find sparse point sensors for signal reconstruction in high-dimensional high-bandwidth systems, and a tailored library of features extracted from training data is used.
Journal ArticleDOI
Randomized QR with Column Pivoting
Jed A. Duersch,Ming Gu +1 more
TL;DR: This work proposes a truncated QR factorization with column pivoting that avoids trailing matrix updates which are used in current implementations of level-3 BLAS QR and QRCP and demonstrates strong parallel scalability on shared-memory multiple core systems using an implementation in Fortran with OpenMP.
Posted Content
Randomized Numerical Linear Algebra: Foundations & Algorithms.
TL;DR: This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problem instances and treats both the theoretical foundations and the practical computational issues.
References
More filters
Journal ArticleDOI
The approximation of one matrix by another of lower rank
Carl Eckart,Gale Young +1 more
TL;DR: In this paper, the problem of approximating one matrix by another of lower rank is formulated as a least-squares problem, and the normal equations cannot be immediately written down, since the elements of the approximate matrix are not independent of one another.
Journal ArticleDOI
Efficient algorithms for computing a strong rank-revealing QR factorization
Ming Gu,Stanley C. Eisenstat +1 more
TL;DR: 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.
Journal ArticleDOI
Rank revealing QR factorizations
TL;DR: An algorithm is presented for computing a column permutation Pi and a QR-factorization of an m by n (m or = n) matrix A such that a possible rank deficiency of A will be revealed in the triangular factor R having a small lower right block.
The WY representation for products of householder matrices
TL;DR: A new way to represent products of Householder matrices is given that makes a typical Householder matrix algorithm rich in matrix-matrix multiplication.
Journal ArticleDOI
The WY representation for products of householder matrices
TL;DR: In this article, a new way to represent products of Householder matrices is given that makes a typical Householder matrix algorithm rich in matrix-matrix multiplication, which is very desirable in that matrixmatrix...