Directional Statistics on Permutations
Sergey M. Plis, Terran Lane, Vince D. Calhoun

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 (). It makes the direct definition of a multinomial distribution over permutation space impractical for all but a very small . In this work we propose an embedding of all permutations for a given in a surface of a hypersphere defined in . 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 -element permutation space. The framework provides a way to use continuous directional probability densities…
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Taxonomy
TopicsBayesian Methods and Mixture Models · Data-Driven Disease Surveillance · Statistical Methods and Bayesian Inference
