Iterative proportional scaling revisited: a modern optimization perspective

Count data are ubiquitous in modern statistical applications. Such data are often cross-classified into contingency tables, where iterative proportional scaling (IPS) can be applied as a standard tool (Fienberg and Meyer, 2006). IPS was firstly introduced by Deming and Stephan (1940) to adjust a contingency table to obey prescribed column and row marginals, the problem of which is referred to as matrix raking nowadays. In general, IPS can be applied to Kullback-Leibler (KL) divergence minimization with linear constraints, and Poisson log-linear model fitting on multi-way tables (Ireland and Kullback, 1968; Bishop et al., 1975), and there is an interesting 1 duality between these two types of problems (Good, 1963; Csiszár, 1975). Theoretical studies regarding the convergence properties of IPS undergo a long history and we refer to Pukelsheim (2014) for a comprehensive survey.