Iterated proportional fitting algorithm and infinite products of stochastic matrices

Abstract : 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 X 0 and two positive marginals a and b, the algorithm generates a sequence of matrices (X n) n≥0 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 1 X 0 D 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) 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 X 0 , the sequence (X n) 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 D 1 X 0 D 2. In the opposite case (when there exists no matrix with marginals a and b whose support is included in the support of X 0), the sequence (X n) n≥0 diverges but both subsequences (X 2n) n≥0 and (X 2n+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 · · · M 2 M 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 Nedić'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 · · · M 1) n≥0 , but also its finite variation.
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Jean Brossard, Christophe Leuridan. Iterated proportional fitting algorithm and infinite products of stochastic matrices. Séminaire de Probabilités XLIX, 2215, Springer, pp.75--117, 2018, Lecture Notes in Mathematics. ⟨hal-01339038v3⟩

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