Biproportional scaling of matrices and the iterative proportional fitting procedure

A short proof is given of the necessary and sufficient conditions for the convergence of the Iterative Proportional Fitting procedure. The input consists of a nonnegative matrix and of positive target marginals for row sums and for column sums. The output is a sequence of scaled matrices to approximate the biproportional fit, that is, the scaling of the input matrix by means of row and column divisors in order to fit row and column sums to target marginals. Generally it is shown that certain structural properties of a biproportional scaling do not depend on the particular sequence used to approximate it. Specifically, the sequence that emerges from the Iterative Proportional Fitting procedure is analyzed by means of the L1-error that measures how current row and column sums compare to their target marginals. As a new result a formula for the limiting L1-error is obtained. The formula is in terms of partial sums of the target marginals, and easily yields the other well-known convergence characterizations.