Statistical ranking using the $l^{1}$-norm  on graphs

Braxton Osting; Jérôme Darbon; Stanley Osher

doi:10.3934/ipi.2013.7.907

Article Contents

2013, Volume 7, Issue 3: 907-926. Doi: 10.3934/ipi.2013.7.907

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Statistical ranking using the $l^{1}$-norm on graphs

1.
Department of Mathematics, University of California, Los Angeles 90095
2.
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555

Received: December 31, 2011

Revised: December 31, 2012

Published: July 2013

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Abstract

We consider the problem of establishing a statistical ranking for a set of alternatives from a dataset which consists of an (inconsistent and incomplete) set of quantitative pairwise comparisons of the alternatives. If we consider the directed graph where vertices represent the alternatives and the pairwise comparison data is a function on the arcs, then the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential optimally agrees with the pairwise comparisons. Potentials, optimal in the $l^{2}$-norm sense, can be found by solving a least-squares problem on the digraph and, recently, the residual has been interpreted using the Hodge decomposition (Jiang et. al., 2010). In this work, we consider an $l^{1}$-norm formulation of the statistical ranking problem. We describe a fast graph-cut approach for finding $\epsilon$-optimal solutions, which has been used successfully in image processing and computer vision problems. Applying this method to several datasets, we demonstrate its efficacy at finding solutions with sparse residual.

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Mathematics Subject Classification: 62F07, 65F10, 05C20, 58A14, 05C85.

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