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Abstract

Permutations are ubiquitous in many real-world problems, such as voting, ranking, and data asso- ciation. Representing uncertainty over permutations is challenging, since there are n! possibilities, and typical compact and factorized probability distribution representations, such as graphical mod- els, 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 per- mutations compactly. We present Kronecker conditioning, a novel approach for maintaining and updating these distributions directly in the Fourier domain, allowing for polynomial time bandlim- ited 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

Number of pages: 74

Text Reference

Jonathan Huang, Carlos Ernesto Guestrin, and Leonidas Guibas, "Fourier Theoretic Probabilistic Inference over Permutations ," Journal of Machine Learning (JMLR), Vol. 10, May, 2009, pp. 997-1070.

BibTeX Reference

@article{Huang_2009_6385,
   author = "Jonathan Huang and Carlos Ernesto Guestrin and Leonidas Guibas",
   title = "Fourier Theoretic Probabilistic Inference over Permutations ",
   journal = "Journal of Machine Learning (JMLR)",
   pages = "997-1070",
   publisher = "Journal of Machine Learning ",
   month = "May",
   year = "2009",
   volume = "10",
}

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