Efficient Inference forDistributions on Permutations

Permutations are ubiquitous in many real world problems, such as voting, rankings and data association. Representing uncertainty over permutations is challenging, since there are $n!$ possibilities, and typical compact 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 such distributions compactly. We present \emph{Kronecker conditioning}, a new general, efficient approach for maintaining these distributions directly in the Fourier domain. Low order Fourier-based approximations can lead to functions that do not correspond to valid distributions. To address this problem, we present an efficient quadratic program defined directly in the Fourier domain to project the approximation onto the polytope of legal marginal distributions. We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting.