Symmetric Iterative Proportional Fitting

Sven Kurras
2015 International Conference on Artificial Intelligence and Statistics  
Iterative Proportional Fitting (IPF) generates from an input matrix W a sequence of matrices that converges, under certain conditions, to a specific limit matrix W . This limit is the relative-entropy nearest solution to W among all matrices of prescribed row marginals r and column marginals c. We prove this known fact by a novel strategy that contributes a pure algorithmic intuition. Then we focus on the symmetric setting: W = W T and r = c. Since IPF inherently generates non-symmetric
more » ... , we introduce two symmetrized variants of IPF. We prove convergence for both of them. Further, we give a novel characterization for the existence of W in terms of expansion properties of the undirected weighted graph represented by W . Finally, we show how our results contribute to recent work in machine learning. INTRODUCTION Iterative Proportional Fitting (IPF) refers to an iterative algorithm whose origins date back to research on traffic networks in the 1930s. It was rediscovered in other fields, in several variants, and in a large variety of different names (for example as Sheleikhovskii's method, Kruithof's algorithm, Furness method, Sinkhorn-Knopp algorithm, or RAS method, just to name a few). Nowadays, IPF is well-known in machine learning and many other disciplines like statistics, optimization, matrix factorization, economics, or network theory. In particular it serves as a bridge that allows to transfer results and interpretations between these disciplines. IPF takes
dblp:conf/aistats/Kurras15 fatcat:g3gkerfnebdchmxt5ogtc4llae