A novel notion of barycenter for probability distributions based on optimal weak mass transport
Dec 6, 2021
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Elsa Cazelles
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Felipe Tobar
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Joaquin Fontbona
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We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass Gozlan et. al. (2017), Backhoff-Veraguas et. al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging in a geometric way the input distributions, as the Wasserstein barycenter based on classic optimal transport does, weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our computation approaches are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass Gozlan et. al. (2017), Backhoff-Veraguas et. al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging in a geometric way the input distributions, as the Wasserstein barycenter based on classic optimal transp…
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NeurIPS 2021
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About NeurIPS 2021
Neural Information Processing Systems (NeurIPS) is a multi-track machine learning and computational neuroscience conference that includes invited talks, demonstrations, symposia and oral and poster presentations of refereed papers. Following the conference, there are workshops which provide a less formal setting.
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