Approximate Birkhoff-von-Neumann decomposition: a differentiable approach

Andrés Hoyos-Idrobo

Keywords: Birkhoff-von-Neumann decomposition, doubly stochastic matrices, Riemannian optimization, Fairness exposure in ranking

Abstract: The Birkhoff-von-Neumann (BvN) decomposition is a standard tool used to draw permutation matrices from a doubly stochastic (DS) matrix. The BvN decomposition represents such a DS matrix as a convex combination of several permutation matrices. Currently, most algorithms to compute the BvN decomposition employ either greedy strategies or custom-made heuristics. In this paper, we present a novel differentiable approach to approximate the BvN decomposition. Our algorithm builds upon recent advances in Riemannian optimization on Birkhoff polytopes. We offer an empirical evaluation of this approach in the fairness of exposure in rankings, where we show that the outcome of our method behaves similarly to greedy algorithms. Our approach is an excellent addition to existing methods for sampling from DS matrices, such as sampling from a Gumbel-Sinkhorn distribution. However, our approach is better suited for applications where the latency in prediction time is a constraint. Indeed, we can generally precompute an approximated BvN decomposition offline. Then, we select a permutation matrix at random with probability proportional to its coefficient. Finally, we provide an implementation of our method.

One-sentence Summary: We propose a differentiable method to approximate the Birkhoff-von-Neumann decomposition of a doubly stochastic matrix.

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