## Directional Statistics on Permutations

* Sergey Plis, Stephen McCracken, Terran Lane, Vince Calhoun*; JMLR W&CP 15:600-608, 2011.

### Abstract

Distributions over permutations arise in applications ranging from multi-object tracking to ranking. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of entities (n!). The direct definition of a multinomial distribution over permutation space is impractical for all but a very small n. In this work we propose an embedding of all n! permutations for a given n in a surface of a hypersphere defined in R^((n-1)^2). As a result, we acquire the ability to define continuous distributions over a hypersphere with all the benefits of directional statistics. We provide polynomial time projections between the continuous hypersphere representation and the n!-element permutation space. The framework provides a way to use continuous directional probability densities and the methods developed thereof for establishing densities over permutations. As a demonstration of the benefits of the framework we derive an inference procedure for a state-space model over permutations. We demonstrate the approach with applications and comparisons to existing models.[pdf][supplementary]