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We present a randomized iterative algorithm that exponentially converges in expectation to the minimum Euclidean norm least squares solution of a given linear system of equations.
Angenäherte Auflösung von Systemen Linearer Gleichungen
S. Kaczmarz · 1937
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Projection Methods in Calculation
C. Tompkins · 1955
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Chebyshev Semi-iterative Methods, Successive Overrelaxation Iterative Methods, and Second order Richardson Iterative Methods
G. H. Golub and R. S. Varga · 1961
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Two Algorithms related to the Method of Steepest Descent
T. Whitney and R. Meany · 1967
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Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography
R. Gordon, R. Bender, and G. T. Herman · 1970
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Projection Method for Solving a Singular System of Linear Equations and its Applications
K. Tanabe · 1971
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An Iterative Procedure for the Solution of Constrained Nonlinear Equations with Application to Optimization Problems
S. F. McCormick · 1975
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An Efficient Method for Generating Discrete Random Variables with General Distributions
A. J. Walker · 1977
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Image Reconstruction from Projections The Fundamentals of Computerized Tomography
G. T. Herman · 1980
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Row-Action Methods for Huge and Sparse Systems and Their Applications
Y. Censor · 1981
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LSQR: An algorithm for sparse linear equations and sparse least squares
C.C. Paige and M.A. Saunders · 1982
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Strong Underrelaxation in Kaczmarz’s Method for Inconsistent Systems
Y. Censor, P. Eggermont, and D. Gordon · 1983
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Connections between the Cimmino-method and the Kaczmarz-method for the Solution of Singular and Regular Systems of Equations
R. Ansorge · 1984
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Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst · 1987
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The Probability that a Numerical Analysis Problem is Difficult
J. W. Demmel · 1988
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A Collection of Test Matrices in MATLAB
N. J. Higham · 1989
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LAPACK: a portable linear algebra library for high-performance computers
E. Anderson, Z. Bai, J. Dongarra, A. Greenbaum, A. McKenney, J. Du Croz, S. Hammerling, J. Demmel, C. Bischof, and D. Sorensen · 1990
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On the Acceleration of Kaczmarz’s Method for Inconsistent Linear Systems
M. Hanke and Wilhelm N · 1990
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A Linear Algorithm for Generating Random Numbers with a given Distribution
M. D. Vose · 1991
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New Variants of the POCS Method using Affine Subspaces of Finite Codimension with Applications to Irregular Sampling
H. G. Feichtinger, C. Cenker, M. Mayer, H. Steier, and T. Strohmer · 1992
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Algebraic Reconstruction Techniques Can Be Made Computationally Efficient
G. T. Herman and L. B. Meyer · 1993
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Error Analysis of Stationary Iteration and Associated Problems
Sampling Algorithms for ℓ 2 \ell_{2} -regression and Applications
P. Drineas, M. W. Mahoney, and S. Muthukrishnan · 2006
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Improved Approximation Algorithms for Large Matrices via Random Projections
T. Sarlòs · 2006
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A Fast Randomized Algorithm for Overdetermined Linear Least-squares Regression
V. Rokhlin and M. Tygert · 2008
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Numerical Linear Algebra in the Streaming Model
K. L. Clarkson and D. P. Woodruff · 2009
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A Fast and Efficient Algorithm for Low-rank Approximation of a Matrix
N. H. Nguyen, T. T. Do, and T. D. Tran · 2009
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A Randomized Kaczmarz Algorithm with Exponential Convergence
T. Strohmer and R. Vershynin · 2009
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P. A. Knight · 1993
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Numerical Methods for Least Squares Problems
A. Björck · 1996
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Matrix Computations
G. H. Golub and C. F. Van Loan · 1996
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Accuracy and Stability of Numerical Algorithms
Nicholas J. Higham · 1996
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A Rounding Error Analysis of Row-Action Methods
P. A. Knight · 1996
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Parallel Optimization: Theory, Algorithms, and Applications
Y. Censor and S. A. Zenios · 1997
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Characterization of the Solutions Set of Inconsistent Least-squares Problems by an Extended Kaczmarz Algorithm
C. Popa · 1999
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Blendenpik: Supercharging LAPACK’s Least-squares Solver
H. Avron, P. Maymounkov, and S. Toledo · 2010
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Randomized Methods for Linear Constraints: Convergence Rates and Conditioning
D. Leventhal and A. S. Lewis · 2010
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Randomized Kaczmarz Solver for Noisy Linear Systems
D. Needell · 2010
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A Fast Randomized Algorithm for Orthogonal Projection
E. S. Coakley, V. Rokhlin, and M. Tygert · 2011
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Faster Least Squares Approximation
P. Drineas, M. W. Mahoney, S. Muthukrishnan, and T. Sarlòs · 2011
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LSRN: A Parallel Iterative Solver for Strongly Over- and Under-Determined Systems
X. Meng, M. A. Saunders, and M. W. Mahoney · 2011
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Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication
A. Magen and A. Zouzias · 2011
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Low Rank Approximation and Regression in Input Sparsity Time
K. L. Clarkson and D. P. Woodruff · 2012
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Fast Distributed Smoothing for Network Clock Synchronization
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