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A Novel Greedy Kaczmarz Method For Solving Consistent Linear Systems
With a quite different way to determine the working rows, we propose a n...
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Greed Works: An Improved Analysis of Sampling Kaczmarz-Motkzin
Stochastic iterative algorithms have gained recent interest in machine l...
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Sparse Sampling Kaczmarz-Motzkin Method with Linear Convergence
The randomized sparse Kaczmarz method was recently proposed to recover s...
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The CCP Selector: Scalable Algorithms for Sparse Ridge Regression from Chance-Constrained Programming
Sparse regression and variable selection for large-scale data have been ...
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Block sampling Kaczmarz-Motzkin methods for consistent linear systems
The sampling Kaczmarz-Motzkin (SKM) method is a generalization of the ra...
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Efficient Data-Driven Geologic Feature Detection from Pre-stack Seismic Measurements using Randomized Machine-Learning Algorithm
Conventional seismic techniques for detecting the subsurface geologic fe...
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Preconditioning Kaczmarz method by sketching
We propose a new method for preconditioning Kaczmarz method by sketching...
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A Kaczmarz Method with Simple Random Sampling for Solving Large Linear Systems
The Kaczmarz method is a popular iterative scheme for solving large-scale linear systems. The randomized Kaczmarz method (RK) greatly improves the convergence rate of the Kaczmarz method, by using the rows of the coefficient matrix in random order rather than in their given order. An obvious disadvantage of the randomized Kaczmarz method is its probability criterion for selecting the active or working rows in the coefficient matrix. In [Z.Z. Bai, W. Wu, On greedy randomized Kaczmarz method for solving large sparse linear systems, SIAM Journal on Scientific Computing, 2018, 40: A592–A606], the authors proposed a greedy randomized Kaczmarz method (GRK). However, this method may suffer from heavily computational cost when the size of the matrix is large, and the overhead will be prohibitively large for big data problems. The contribution of this work is as follows. First, from the probability significance point of view, we present a partially randomized Kaczmarz method, which can reduce the computational overhead needed in greedy randomized Kaczmarz method. Second, based on Chebyshev's law of large numbers and Z-test, we apply a simple sampling approach to the partially randomized Kaczmarz method. The convergence of the proposed method is established. Third, we apply the new strategy to the ridge regression problem, and propose a partially randomized Kaczmarz method with simple random sampling for ridge regression. Numerical experiments demonstrate the superiority of the new algorithms over many state-of-the-art randomized Kaczmarz methods for large linear systems problems and ridge regression problems.
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