A fast algorithm for matrix balancing

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. Introduction.For at least 70 years, scientists in a wide variety of disciplines have attempted to transform square nonnegative matrices into doubly stochastic form by applying diagonal scalings. That is, given A ∈ R n×n , A ≥ 0, find diagonal matrices D 1 and D 2 so that P = D 1 AD 2 is doubly stochastic. Motivations for achieving this balance include interpreting economic data [1], preconditioning sparse matrices [16], understanding traffic circulation [14], assigning seats fairly after elections [3], matching protein samples [4] and ordering nodes in a graph [12]. In all of these applications, one of the main methods considered is SK 1 . This is an iterative process that attempts to find D 1 and D 2 by alternately normalising columns and rows in a sequence of matrices starting with A. Convergence conditions for this algorithm are well known: if A has total support 2 then the algorithm will converge linearly with asymptotic rate equal to the square of the subdominant singular value of P [22,23,12].Clearly, in some cases the convergence will be painfully slow. The principal aim of this paper is to derive some new algorithms for the matrix balancing problem with an eye on speed, especially for large systems. First we look at a simple Newton method for symmetric matrices, closely related to a method proposed by Khachiyan and Kalantari [11] for positive definite (but not necessarily nonnegative) matrices. We will show that as long as Newton's method produces a sequence of positive iterates, the Jacobians we generate will be positive semi-definite and that this is also true when we adapt the method to cope with nonsymmetric matrices.To apply Newton's method exactly we require a linear system solve at each step, and this is usually prohibitively expensive. We therefore look at iterative techniques for approximating the solution at each step. First we look at splitting methods and we see that SK is a member of this family of methods, as is the algorithm proposed by Livne and Golub in [16]. We give an asymptotic bound on the (linear) rate of convergence of these methods. For symmetric matrices we can get significant improvement *

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