Quadratic convergence for scaling of matrices

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    4 Citations (Scopus)

    Abstract

    Matrix scaling is an operation on nonnegative matrices with nonzero permanent. It multiplies the rows and columns of a matrix with positive factors such that the resulting matrix is (approximately) doubly stochastic. Scaling is useful at a preprocessing stage to make certain numerical computations more stable. Linial, Samorodnitsky and Wigderson have developed a strongly polynomial time algorithm for scaling. Furthermore, these authors have proposed to use this algorithm to approximate permanents in deterministic polynomial time. They have noticed an intriguing possibility to attack the notorious parallel matching problem. If scaling could be done efficiently in parallel, then it would approximate the permanent sufficiently well to solve the bipartite matching problem. As a first step towards this goal, we propose a scaling algorithm that is conjectured to run much faster than any previous scaling algorithm. It is shown that this algorithm converges quadratically for strictly scalable matrices. We interpret this as a hint that the algorithm might always be fast. All previously known approaches to matrix scaling can result in linear convergence at best.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics
    EditorsL. Arge, G.F. Italiano, R. Sedgewick
    Pages216-223
    Number of pages8
    StatePublished - 2004
    EventProceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithms and Combinatorics - New Orleans, LA, United States
    Duration: Jan 10 2004Jan 10 2004

    Other

    OtherProceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithms and Combinatorics
    CountryUnited States
    CityNew Orleans, LA
    Period1/10/041/10/04

    Fingerprint

    Polynomials

    All Science Journal Classification (ASJC) codes

    • Engineering(all)

    Cite this

    Furer, M. (2004). Quadratic convergence for scaling of matrices. In L. Arge, G. F. Italiano, & R. Sedgewick (Eds.), Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics (pp. 216-223)
    Furer, Martin. / Quadratic convergence for scaling of matrices. Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics. editor / L. Arge ; G.F. Italiano ; R. Sedgewick. 2004. pp. 216-223
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    title = "Quadratic convergence for scaling of matrices",
    abstract = "Matrix scaling is an operation on nonnegative matrices with nonzero permanent. It multiplies the rows and columns of a matrix with positive factors such that the resulting matrix is (approximately) doubly stochastic. Scaling is useful at a preprocessing stage to make certain numerical computations more stable. Linial, Samorodnitsky and Wigderson have developed a strongly polynomial time algorithm for scaling. Furthermore, these authors have proposed to use this algorithm to approximate permanents in deterministic polynomial time. They have noticed an intriguing possibility to attack the notorious parallel matching problem. If scaling could be done efficiently in parallel, then it would approximate the permanent sufficiently well to solve the bipartite matching problem. As a first step towards this goal, we propose a scaling algorithm that is conjectured to run much faster than any previous scaling algorithm. It is shown that this algorithm converges quadratically for strictly scalable matrices. We interpret this as a hint that the algorithm might always be fast. All previously known approaches to matrix scaling can result in linear convergence at best.",
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    Furer, M 2004, Quadratic convergence for scaling of matrices. in L Arge, GF Italiano & R Sedgewick (eds), Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics. pp. 216-223, Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithms and Combinatorics, New Orleans, LA, United States, 1/10/04.

    Quadratic convergence for scaling of matrices. / Furer, Martin.

    Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics. ed. / L. Arge; G.F. Italiano; R. Sedgewick. 2004. p. 216-223.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

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    N2 - Matrix scaling is an operation on nonnegative matrices with nonzero permanent. It multiplies the rows and columns of a matrix with positive factors such that the resulting matrix is (approximately) doubly stochastic. Scaling is useful at a preprocessing stage to make certain numerical computations more stable. Linial, Samorodnitsky and Wigderson have developed a strongly polynomial time algorithm for scaling. Furthermore, these authors have proposed to use this algorithm to approximate permanents in deterministic polynomial time. They have noticed an intriguing possibility to attack the notorious parallel matching problem. If scaling could be done efficiently in parallel, then it would approximate the permanent sufficiently well to solve the bipartite matching problem. As a first step towards this goal, we propose a scaling algorithm that is conjectured to run much faster than any previous scaling algorithm. It is shown that this algorithm converges quadratically for strictly scalable matrices. We interpret this as a hint that the algorithm might always be fast. All previously known approaches to matrix scaling can result in linear convergence at best.

    AB - Matrix scaling is an operation on nonnegative matrices with nonzero permanent. It multiplies the rows and columns of a matrix with positive factors such that the resulting matrix is (approximately) doubly stochastic. Scaling is useful at a preprocessing stage to make certain numerical computations more stable. Linial, Samorodnitsky and Wigderson have developed a strongly polynomial time algorithm for scaling. Furthermore, these authors have proposed to use this algorithm to approximate permanents in deterministic polynomial time. They have noticed an intriguing possibility to attack the notorious parallel matching problem. If scaling could be done efficiently in parallel, then it would approximate the permanent sufficiently well to solve the bipartite matching problem. As a first step towards this goal, we propose a scaling algorithm that is conjectured to run much faster than any previous scaling algorithm. It is shown that this algorithm converges quadratically for strictly scalable matrices. We interpret this as a hint that the algorithm might always be fast. All previously known approaches to matrix scaling can result in linear convergence at best.

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    Furer M. Quadratic convergence for scaling of matrices. In Arge L, Italiano GF, Sedgewick R, editors, Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algoritms and Combinatorics. 2004. p. 216-223