### Abstract

Language | English |
---|---|

Pages | 261-275 |

Number of pages | 15 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 30 |

Issue number | 1 |

Early online date | 5 Mar 2008 |

DOIs | |

Publication status | Published - 2008 |

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### Keywords

- matrix balancing
- Sinkhorn-Knopp algorithm
- PageRank
- doubly stochastic matrix

### Cite this

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*SIAM Journal on Matrix Analysis and Applications*, vol. 30, no. 1, pp. 261-275. https://doi.org/10.1137/060659624

**The Sinkhorn-Knopp algorithm : convergence and applications.** / Knight, P.A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Sinkhorn-Knopp algorithm

T2 - SIAM Journal on Matrix Analysis and Applications

AU - Knight, P.A.

PY - 2008

Y1 - 2008

N2 - As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear, and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices. We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that, with an appropriate modi. cation, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

AB - As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear, and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices. We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that, with an appropriate modi. cation, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

KW - matrix balancing

KW - Sinkhorn-Knopp algorithm

KW - PageRank

KW - doubly stochastic matrix

U2 - 10.1137/060659624

DO - 10.1137/060659624

M3 - Article

VL - 30

SP - 261

EP - 275

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 1

ER -