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 (18)

• Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method
• Coherence Motivated Sampling and Convergence Analysis of Least-Squares Polynomial Chaos Regression
• Randomized QR with Column Pivoting
• Scalable Kernel K-Means Clustering with Nystrom Approximation: Relative-Error Bounds
• Compact Random Feature Maps
• The achievable performance of convex demixing
• Fast Parallel Randomized QR with Column Pivoting Algorithms for Reliable Low-rank Matrix Approximations
• Generalized Spectral Kernels
• Active Regression via Linear-Sample Sparsification
• The Gaussian min-max theorem in the Presence of Convexity
• A Rank-1 Sketch for Matrix Multiplicative Weights
• On spectral and numerical properties of random butterfly matrices
• Simple and Almost Assumption-Free Out-of-Sample Bound for Random Feature Mapping
• The PowerURV algorithm for computing rank-revealing full factorizations
• A Generalized Randomized Rank-Revealing Factorization
• Gaussian Elimination with Randomized Complete Pivoting
• Superlinear Convergence of Randomized Block Lanczos Algorithm
• Estimating the Algorithmic Variance of Randomized Ensembles via the Bootstrap

Cited by in corpus (12)

• Efficient estimation of Pauli observables by derandomization
• A literature survey of matrix methods for data science
• Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs
• Spectral estimation from simulations via sketching
• Structured random sketching for PDE inverse problems
• Solving non-linear integral equations for laser pulse retrieval with Newton's method
• Matrix Sketching for Secure Collaborative Machine Learning
• Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations
• Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning
• Efficient algorithms for computing a rank-revealing UTV factorization on parallel computing architectures
• Effective and Sparse Count-Sketch via k-means clustering
• A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs