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Accumulation points of the iterative proportional fitting procedure

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Abstract

The asymptotic behavior of the iterative proportional fitting procedure (IPF procedure) is analyzed comprehensively. Given a nonnegative matrix as well as row and column marginals the IPF procedure generates a sequence of matrices, called the IPF sequence, by alternately fitting rows and columns to match their respective marginals. We prove that the IPF sequence has at most two accumulation points. They originate as the limits of the even-step subsequence, and of the odd-step subsequence. The well-known IPF convergence criteria are then retrieved easily. Our proof is based on Csiszár’s and Tusnády’s (Stat Decis Suppl Issue 1:205–237, 1984) results on the interplay of the I-divergence geometry and alternating minimization procedures.

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Acknowledgments

We are very grateful to our advisor Friedrich Pukelsheim for continuing support. We thank our friend and colleague Kai-Friederike Oelbermann from our Augsburg group for her valuable remarks when discussing the work. We are thankful to Fero Matúš and Ludger Rüschendorf for their assistance with the pertinent literature. We thank an anonymous referee for valuable remarks on an earlier version of this paper, which in particular helped us condense Sects. 2 and 3.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, 86135 , Augsburg, Germany

    Christoph Gietl & Fabian P. Reffel

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  1. Christoph Gietl
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  2. Fabian P. Reffel
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Correspondence to Fabian P. Reffel.

Additional information

This research has been supported by the Elite Network of Bavaria through its graduate program TopMath and by the TUM Graduate School through its Thematic Graduate Center TopMath.

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Gietl, C., Reffel, F.P. Accumulation points of the iterative proportional fitting procedure. Metrika 76, 783–798 (2013). https://doi.org/10.1007/s00184-012-0415-7

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  • DOI: https://doi.org/10.1007/s00184-012-0415-7

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