A fast algorithm for matrix balancing

Knight, Philip and Ruiz, Daniel (2013) A fast algorithm for matrix balancing. IMA Journal of Numerical Analysis, 33 (3). pp. 1029-1047. ISSN 0272-4979

[img]
Preview
PDF (A fast algorithm for matrix balancing)
fastbalance.pdf
Preprint

Download (223kB)| Preview

    Abstract

    As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. 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 can give quadratic convergence at low cost.