Inference for Distributions over the Permutation Group

摘要

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 Kronecker conditioning, a new general and efficient approach for maintaining and updating these distributions directly in the Fourier domain. Low order Fourier-based approximations, however, may 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 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.