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Carnegie Mellon Robotics Institute

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Abstract

Permutations are ubiquitous in many real-world problems, such as voting, ranking, and data association. Representing uncertainty over permutations is challenging, since there are $n!$ possibilities, and typical compact and factorized probability distribution representations, such as graphical models, cannot capture the mutual exclusivity constraints associated with permutations. In this paper, we use the ``low-frequency'' terms of a Fourier decomposition to represent distributions over permutations compactly. We present \emph{Kronecker conditioning}, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlimited approximations. Low order Fourier-based approximations, however, may lead to functions that do not correspond to valid distributions. To address this problem, we present a quadratic program defined directly in the Fourier domain for projecting the approximation onto a relaxation of the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camera-based multi-person tracking scenario.

Notes

Text Reference

Jonathan Huang, Carlos Ernesto Guestrin, Leonidas Guibas, Jonathan Huang, Carlos Ernesto Guestrin, and Leonidas Guibas, "Inference for Distributions over the Permutation Group," tech. report CMU-ML-08-108, Machine Learning Department, Carnegie Mellon University, May, 2008

BibTeX Reference

@techreport{Huang_2008_6329,
   author = "Jonathan Huang and Carlos Ernesto Guestrin and Leonidas Guibas and Jonathan Huang and Carlos Ernesto Guestrin and Leonidas Guibas",
   title = "Inference for Distributions over the Permutation Group",
   booktitle = "",
   institution = "Machine Learning Department",
   month = "May",
   year = "2008",
   number= "CMU-ML-08-108",
   address= "Pittsburgh, PA",
}

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