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Many matching, tracking, sorting, and ranking problems require probabilisticreasoning about possible permutations, a set that grows factorially withdimension. Combinatorial optimization algorithms may enable efficient pointestimation, but fully Bayesian inference poses a severe challenge in thishigh-dimensional, discrete space. To surmount this challenge, we start with theusual step of relaxing a discrete set (here, of permutation matrices) to itsconvex hull, which here is the Birkhoff polytope: the set of alldoubly-stochastic matrices. We then introduce two novel transformations: first,an invertible and differentiable stick-breaking procedure that mapsunconstrained space to the Birkhoff polytope; second, a map that rounds pointstoward the vertices of the polytope. Both transformations include a temperatureparameter that, in the limit, concentrates the densities on permutationmatrices. We then exploit these transformations and reparameterizationgradients to introduce variational inference over permutation matrices, and wedemonstrate its utility in a series of experiments.

Reparameterizing the Birkhoff Polytope for Variational Permutation Inference