Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Scientific Computing
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A Levinson--Galerkin Algorithm for Regularized Trigonometric Approximation
SIAM Journal on Scientific Computing
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
Database-friendly random projections: Johnson-Lindenstrauss with binary coins
Journal of Computer and System Sciences - Special issu on PODS 2001
Sampling algorithms for l2 regression and applications
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Relative-Error $CUR$ Matrix Decompositions
SIAM Journal on Matrix Analysis and Applications
The Fast Johnson-Lindenstrauss Transform and Approximate Nearest Neighbors
SIAM Journal on Computing
Johnson-Lindenstrauss lemma for circulant matrices
Random Structures & Algorithms
A randomized solver for linear systems with exponential convergence
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
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The Kaczmarz method is an algorithm for finding the solution to an overdetermined consistent system of linear equations Ax驴=驴b by iteratively projecting onto the solution spaces. The randomized version put forth by Strohmer and Vershynin yields provably exponential convergence in expectation, which for highly overdetermined systems even outperforms the conjugate gradient method. In this article we present a modified version of the randomized Kaczmarz method which at each iteration selects the optimal projection from a randomly chosen set, which in most cases significantly improves the convergence rate. We utilize a Johnson---Lindenstrauss dimension reduction technique to keep the runtime on the same order as the original randomized version, adding only extra preprocessing time. We present a series of empirical studies which demonstrate the remarkable acceleration in convergence to the solution using this modified approach.