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Stochastic iterative algorithms have gained recent interest in machine learning and signal processing for solving large-scale systems of equations, $Ax=b$. One such example is the Randomized Kaczmarz (RK) algorithm, which acts only on single rows of the matrix $A$ at a time. While RK randomly selects a row of $A$ to work with, Motzkins Method (MM) employs a greedy row selection. Connections between the two algorithms resulted in the Sampling Kaczmarz-Motzkin (SKM) algorithm which samples a random subset of $beta$ rows of $A$ and then greedily selects the best row of the subset. Despite their variable computational costs, all three algorithms have been proven to have the same theoretical upper bound on the convergence rate. In this work, an improved analysis of the range of random (RK) to greedy (MM) methods is presented. This analysis improves upon previous known convergence bounds for SKM, capturing the benefit of partially greedy selection schemes. This work also further generalizes previous known results, removing the theoretical assumptions that $beta$ must be fixed at every iteration and that $A$ must have normalized rows.
The randomized sparse Kaczmarz method was recently proposed to recover sparse solutions of linear systems. In this work, we introduce a greedy variant of the randomized sparse Kaczmarz method by employing the sampling Kaczmarz-Motzkin method, and pro
The sampling Kaczmarz-Motzkin (SKM) method is a generalization of the randomized Kaczmarz and Motzkin methods. It first samples some rows of coefficient matrix randomly to build a set and then makes use of the maximum violation criterion within this
The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy rando
In this paper, we analyze the convergence behavior of the randomized extended Kaczmarz (REK) method for all types of linear systems (consistent or inconsistent, overdetermined or underdetermined, full-rank or rank-deficient). The analysis shows that
We propose a deterministic Kaczmarz method for solving linear systems $Ax=b$ with $A$ nonsingular. Instead of using orthogonal projections, we use reflections in the original Kaczmarz iterative method. This generates a series of points on an $n$-sphe