The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=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 exponentially 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, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, and suggests that paving yields a significant improvement in performance in certain regimes. The proposed method is 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 methods converge to the least-squares solution of inconsistent systems, and by using appropriate blocks of the matrix this convergence can be significantly accelerated. Numerical experiments suggest that the proposed algorithm can indeed lead to advantages in practice.
To use block methods effectively, one needs to obtain a suitable partition of the rows (and or columns). Popa constructs such partitions by creating orthogonal blocks @cite_20 @cite_43 @cite_26 , whereas Needell and Tropp promote the use of row pavings to construct the partition @cite_29 .
@cite_43 @cite_29 @cite_26 @cite_20
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Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s.t. some objective function is optimized. High-quality partitions are important for many applications, whose objective functions are often NP-hard to optimize. Most state-of-the-art graph partitioning libraries use a variant of the Kernighan-Lin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called Bubble-FOS C [H. Meyerhenke, B. Monien, S. Schamberger, Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid, in: Proceedings of the 20th IEEE International Parallel and Distributed Processing Symposium, IPDPS'06, IEEE Computer Society, 2006, p. 57 (CD)], which optimizes partition shapes by a diffusive mechanism. However, it is too slow for practical use, despite its high solution quality. In this paper, besides proving that Bubble-FOS C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a different diffusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with Bubble-FOS C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph partitioning heuristic DibaP. Compared to Bubble-FOS C, DibaP shows a considerable acceleration, while retaining the positive properties of the slower algorithm. Experiments with popular benchmark graphs show that DibaP computes consistently better results than the state-of-the-art libraries METIS and JOSTLE. Moreover, with our new algorithm, we have improved the best known edge-cut values for a significant number of partitionings of six widely used benchmark graphs. In this paper we present an algorithm which, for a given (sparse) matrix, constructs a partition of its set of row-indices, such that each subset of this partition (except the last one obtained) contains indices which correspond to mutually orthogonal rows. We then use such decompositions in some classes of block-projections methods, previously extended by the author to general inconsistent linear least-squares problems. Numerical experiments on an inconsistent and rank-deficient least-squares model problem are described in the last section of the paper. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. Abstract We extend the direct approach for blockmodeling one-mode data to two-mode data. The key idea in this development is that the rows and columns are partitioned simultaneously but in different ways. Many (but not all) of the generalized block types can be mobilized in blockmodeling two-mode network data. These methods were applied to some ‘voting’ data from the 2000–2001 term of the Supreme Court and to the classic Deep South data on women attending events. The obtained partitions are easy to interpret and compelling. The insight that rows and columns can be partitioned in different ways can be applied also to one-mode data. This is illustrated by a partition of a journal-to-journal citation network where journals are viewed simultaneously as both producers and consumers of scientific knowledge.