Randomized Numerical Linear Algebra: Foundations & Algorithms

Abstract

This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper treats both the theoretical foundations of the subject and the practical computational issues. Topics covered include norm estimation; matrix approximation by sampling; structured and unstructured random embeddings; linear regression problems; low-rank approximation; subspace iteration and Krylov methods; error estimation and adaptivity; interpolatory and CUR factorizations; Nyström approximation of positive-semidefinite matrices; single view ("streaming") algorithms; full rank-revealing factorizations; solvers for linear systems; and approximation of kernel matrices that arise in machine learning and in scientific computing.

References in corpus (5)

• Coherence Motivated Sampling and Convergence Analysis of Least-Squares Polynomial Chaos Regression
• The Gaussian min-max theorem in the Presence of Convexity
• The PowerURV algorithm for computing rank-revealing full factorizations
• Simple and Almost Assumption-Free Out-of-Sample Bound for Random Feature Mapping
• Estimating the Algorithmic Variance of Randomized Ensembles via the Bootstrap

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• Efficient algorithms for computing a rank-revealing UTV factorization on parallel computing architectures
• Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning
• Effective and Sparse Count-Sketch via k-means clustering