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 constraints associated with permutations. In this talk, we use the ''low-frequency'' terms of a Fourier decomposition to represent such distributions compactly. We first describe how the two standard probabilistic inference operations, conditioning and marginalization, can be performed entirely in the Fourier domain in terms of these low frequency components, without ever enumeration $n!$ terms. We also describe a novel approach for adaptively picking the complexity of this representation in order control the resulting approximation error. We demonstrate the effectiveness of our approach on a real camera-based multi-people tracking setting.

This presentation is joint work with Jon Huang and Leo Guibas.