Sketching as a Tool for Numerical Linear Algebra

Abstract

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 (usually) random matrix with certain properties. Much of the expensive computation can then be performed on the smaller matrix, thereby accelerating the solution for the original problem. In this survey we consider least squares as well as robust regression problems, low rank approximation, and graph sparsification. We also discuss a number of variants of these problems. Finally, we discuss the limitations of sketching methods.

fixed minor errors/typos in section 4.3, e.g., Fact 6 and its propagation, clarified when Lemma 4.2 can be applied, typos in section 4.2.3 (G should be applied on the left), other typos throughout

References in corpus (5)

• Row Sampling for Matrix Algorithms via a Non-Commutative Bernstein Bound
• Moments of the Gaussian Chaos
• On the Sensitivity of Shape Fitting Problems
• Dimensionality Reduction for k-Means Clustering and Low Rank Approximation
• Using a Non-Commutative Bernstein Bound to Approximate Some Matrix Algorithms in the Spectral Norm

Cited by in corpus (7)

• Sub-sampled Newton Methods with Non-uniform Sampling
• A Random Matrix Theoretical Approach to Early Event Detection Using Experimental Data
• Fourier-sparse interpolation without a frequency gap
• Sketching Meets Random Projection in the Dual: A Provable Recovery Algorithm for Big and High-dimensional Data
• Effective sketching methods for value function approximation
• Adjusting Leverage Scores by Row Weighting: A Practical Approach to Coherent Matrix Completion
• Massive MIMO as a Big Data System: Random Matrix Models and Testbed