The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into well-conditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined least-squares problem.
Most of the literature on block Kaczmarz methods assumes that a partition @math of the rows of the matrix is provided as part of the problem data. We are aware of some research on the prospects for partitioning a matrix in a manner that is favorable for block Kaczmarz methods. In particular, Popa @cite_17 has introduced an algorithm for partitioning a sparse matrix so that each block contains mutually orthogonal rows. Popa has pursued this idea in a sequence of papers, including @cite_46 @cite_30 . We believe that our work is the first to recognize the natural connection between the paving literature and the block Kaczmarz method.
@cite_30 @cite_46 @cite_17
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The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving of the matrix, a partition of the rows into well-conditioned blocks. The operator theory literature provides detailed information about the existence and construction of good row pavings. Together, these results yield an efficient block Kaczmarz scheme that applies to many overdetermined least-squares problem. Abstract The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations A x = b . The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges linearly 1 in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, designed to converge in expectation to the least squares solution, often faster than the standard variants. Our approach utilizes both a row and column-paving of the matrix A to guarantee linear convergence when the matrix has consistent row norms (called nearly standardized), and a single column-paving when the row norms are unbounded. The proposed methods are an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our results demonstrate convergence to the least squares solution of inconsistent systems (both methods in the nearly standardized case and the second method in other cases). By using appropriate blocks of the matrix this convergence can be significantly accelerated, as is demonstrated by numerical experiments. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V, E) be a graph whose vertex set has a ‘latent’ partition V1,. . ., Vk. Moreover, consider a ‘density matrix’ Ɛ = (Ɛvw)v, sw∈V such that, for v ∈ Vi and w ∈ Vj, the entry Ɛvw is the fraction of all possible Vi−Vj-edges that are actually present in G. We show that on input (G, k) the partition V1,. . ., Vk can (very nearly) be recovered in polynomial time via spectral methods, provided that the following holds: Ɛ approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj ≠ Vi the corresponding column vectors Ɛv, Ɛw are separated, and G is sufficiently ‘regular’ with respect to the matrix Ɛ. This result in particular applies to sparse graphs with bounded average degree as n = #V → ∞, and it has various consequences on partitioning random graphs. This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison-Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and (noncommutative) Khinchin inequalities to estimate the norms of some random matrices.