2016/06/29 by Jean Brossard, Brossard, Jean, Christophe Leuridan +1
Engineering · Mathematics · #15B51 #62B10 #62H17 #68W40 #FOS: Mathematics #Mathematical Inequalities and Applications #Point processes and geometric inequalities #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1606.09126
openalex publication_date 2016/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 X0 and two positive marginals a and b, the algorithm generates a sequence of matrices (Xn) 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) 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) 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, the sequence (Xn) diverges but both subsequences (X2n) and (X2n+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 ⋯ M2M1 of stochatic 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-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 (Mn ⋯ M1), but also its finite variation.