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Register a new userEvaluating the Potential of a Dual Randomized Kaczmarz Solver for Laplacian Linear Systems
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Added by
Kevin Deweese
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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A new method for solving Laplacian linear systems proposed by Kelner et al. involves the random sampling and update of fundamental cycles in a graph. Kelner et al. proved asymptotic bounds on the complexity of this method but did not report experimental results. We seek to both evaluate the performance of this approach and to explore improvements to it in practice. We compare the performance of this method to other Laplacian solvers on a variety of real world graphs. We consider different ways to improve the performance of this method by exploring different ways of choosing the set of cycles and the sequence of updates, with the goal of providing more flexibility and potential parallelism. We propose a parallel model of the Kelner et al. method, for evaluating potential parallelism in terms of the span of edges updated at each iteration. We provide experimental results comparing the potential parallelism of the fundamental cycle basis and our extended cycle set. Our preliminary experiments show that choosing a non-fundamental set of cycles can save significant work compared to a fundamental cycle basis.
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Linear system solving is one of the main workhorses in applied mathematics. Recently, theoretical computer scientists have contributed sophisticated algorithms for solving linear systems with symmetric diagonally dominant matrices (a class to which Laplacian matrices belong) in provably nearly-linear time. While these algorithms are highly interesting from a theoretical perspective, there are no published results how they perform in practice. With this paper we address this gap. We provide the first implementation of the combinatorial solver by [Kelner et al., STOC 2013], which is particularly appealing for implementation due to its conceptual simplicity. The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning tree with the low-stretch property. The results of our comprehensive experimental study are ambivalent. They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity. One other aspect predicted by theory is confirmed by our findings, though: Spanning trees with lower stretch indeed reduce the solvers running time. Yet, simple spanning tree algorithms perform in practice better than those with a guaranteed low stretch.
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 $n$-sphere $S$ centered at the solution $x_*=A^{-1}b$. We show that these points are nicely distributed on $S$. Taking the average of several points will lead to an effective approximation to the solution. We will show how to choose these points efficiently. The numerical tests show that in practice this deterministic scheme converges much faster than we expected and can beat the (block) randomized Kaczmarz methods.
Matrices associated with graphs, such as the Laplacian, lead to numerous interesting graph problems expressed as linear systems. One field where Laplacian linear systems play a role is network analysis, e. g. for certain centrality measures that indicate if a node (or an edge) is important in the network. One such centrality measure is current-flow closeness. To allow network analysis workflows to profit from a fast Laplacian solver, we provide an implementation of the LAMG multigrid solver in the NetworKit package, facilitating the computation of current-flow closeness values or related quantities. Our main contribution consists of two algorithms that accelerate the current-flow computation for one node or a reasonably small node subset significantly. One sampling-based algorithm provides an unbiased estimation of the related electrical farness, the other one is based on the Johnson-Lindenstrauss transform. Our inexact algorithms lead to very accurate results in practice. Thanks to them one is now able to compute an estimation of current-flow closeness of one node on networks with tens of millions of nodes and edges within seconds or a few minutes. From a network analytical point of view, our experiments indicate that current-flow closeness can discriminate among different nodes significantly better than traditional shortest-path closeness and is also considerably more resistant to noise -- we thus show that two known drawbacks of shortest-path closeness are alleviated by the current-flow variant.
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 randomized Motzkin-Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices, and the blo
With a quite different way to determine the working rows, we propose a novel greedy Kaczmarz method for solving consistent linear systems. Convergence analysis of the new method is provided. Numerical experiments show that, for the same accuracy, our method outperforms the greedy randomized Kaczmarz method and the relaxed greedy randomized Kaczmarz method introduced recently by Bai and Wu [Z.Z. BAI AND W.T. WU, On greedy randomized Kaczmarz method for solving large sparse linear systems, SIAM J. Sci. Comput., 40 (2018), pp. A592--A606; Z.Z. BAI AND W.T. WU, On relaxed greedy randomized Kaczmarz methods for solving large sparse linear systems, Appl. Math. Lett., 83 (2018), pp. 21--26] in term of the computing time.
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