Directional Statistics on Permutations

TL;DR

This paper introduces a novel hypersphere embedding for permutations enabling continuous probability distributions and efficient inference, facilitating applications in multi-object tracking and ranking.

Contribution

It proposes a hypersphere embedding of permutations, allowing continuous distributions and polynomial-time projections, which simplifies modeling and inference over permutation spaces.

Findings

Efficient polynomial-time projection between permutation space and hypersphere.

Framework enables use of directional statistics for permutation distributions.

Demonstrated application in state-space models for permutation inference.

Abstract

Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of considered entities ($n!$). It makes the direct definition of a multinomial distribution over permutation space 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 $\mathbbm{R}^{(n-1)^2}$. As a result of the embedding, we acquire 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.