Iterated proportional fitting procedure and infinite products of stochastic matrices

Abstract

The iterative proportional fitting procedure, introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Given a rectangular non-negative matrix $X_0$ and two positive marginals $a$ and $b$, the algorithm generates a sequence of matrices $(X_n)$ starting at $X_0$, supposed to converge to a biproportional fitting, that is, to a matrix $Y$ whose marginals are $a$ and $b$ and of the form $Y=D_1X_0D_2$, for some diagonal matrices $D_1$ and $D_2$ with positive diagonal entries. When a biproportional fitting does exist, it is unique and the sequence $(X_n)$ converges to it at an at least geometric rate. More generally, when there exists some matrix with marginal $a$ and $b$ and with support included in the support of $X_0$, the sequence $(X_n)$ converges to the unique matrix whose marginals are $a$ and $b$ and which can be written as a limit of matrices of the form $D_1X_0D_2$. In the opposite case, the sequence $(X_n)$ diverges but both subsequences $(X_{2n})$ and $(X_{2n+1})$ converge. In the present paper, we use a new method to prove again these results and determine the two limit-points in the case of divergence. Our proof relies on a new convergence theorem for backward infinite products $\cdots M_2M_1$ of stochatic matrices $M_n$, with diagonal entries $M_n(i,i)$ bounded away from $0$ and with bounded ratios $M_n(j,i)/M_n(i,j)$. This theorem generalizes Lorenz' stabilization theorem. We also provide an alternative proof of Touric and Nedi\'c's theorem on backward infinite products of doubly-stochatic matrices, with diagonal entries bounded away from $0$. In both situations, we improve slightly the conclusion, since we establish not only the convergence of the sequence $(M_n \cdots M_1)$, but also its finite variation.