Solving the data association problem in multi-object tracking by Fourier analysis on the symmetric group

In addition to modeling the position of individual targets, multi-object tracking must also address the combinatorial problem of matching objects to corresponding tracks. In general, maintaining a probability distribution over all n! possibilities is clearly infeasible, while just maintaining an n×n matrix of “first order marginals” is a very impoverished representation. In this work we explain how to harness the theory of harmonic analysis on the symmetric group to get a hierarchy of approximations of increasing fidelity to this problem. Importatantly, not only are such band-limited approximations theoretically well justifiable, but they also admit efficient observations updates based on some ideas from Clausen’s FFT for the symmetric group.