Efficient Inference for Distributions 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 efficiently capture the mutual exclusivity con- straints associated with permutations. In this paper, we use the “low-frequency” terms of a Fourier decomposition to represent such distributions compactly. We present Kronecker conditioning, a general and 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 a relaxed form of the marginal polytope. We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting.