Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices
The iterative proportional fitting procedure (IPFP), introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Thus, given a rectangular non-negative matrix X0 and two positive marginals a and b, the algorithm generates a sequence of matrices (Xn)n≥0 starting at X0, supposed to converge to a biproportional fitting, that is, to a matrix Y whose marginals are a and b and of the form Y = D1X0D2, for some diagonal matrices D1 and D2 with positive diagonal entries.
When a biproportional fitting does exist, it is unique and the sequence (Xn)n≥0 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 X0, the sequence (Xn)n≥0 converges to the unique matrix whose marginals are a and b and which can be written as a limit of matrices of the form D1X0D2.
In the opposite case (when there exists no matrix with marginals a and b whose support is included in the support of X0), the sequence (Xn)n≥0 diverges but both subsequences (X2n)n≥0 and (X2n+1)n≥0 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 ⋯M2M1 of stochastic matrices Mn, with diagonal entries Mn(i, i) bounded away from 0 and with bounded ratios Mn(j, i)∕Mn(i, j). This theorem generalizes Lorenz’ stabilization theorem. We also provide an alternative proof of Touric and Nedić’s theorem on backward infinite products of doubly-stochastic 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 (Mn⋯M1)n≥0, but also its finite variation.
KeywordsInfinite products of stochastic matrices Contingency matrices Distributions with given marginals Iterative proportional fitting Relative entropy I-divergence
Subject Classifications15B51 62H17 62B10 68W40
We thank A. Coquio, D. Piau, G. Geenens, F. Pukelsheim and the referee for their careful reading and their useful remarks.
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