A fast algorithm for matrix balancing

作者：Knight, Philip A.^*; Ruiz, Daniel

来源：IMA Journal of Numerical Analysis, 2013, 33(3): 1029-1047.

DOI：10.1093/imanum/drs019

• matrix balancing
• Sinkhorn-Knopp algorithm
• doubly stochastic matrix
• conjugate gradient iteration

As long as a square non-negative matrix A has total support then it can be balanced, that is, we can find a diagonal scaling of A that has row and column sums equal to one. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper, we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration converges at much lower cost (in terms of matrix-vector products) for a broad range of matrices; and succeeds in cases where the Sinkhorn-Knopp algorithm fails.

• 出版日期2013-7