Fréchet means and Procrustes analysis in Wasserstein space

By Yoav Zemel, Victor M. Panaretos. Bernoulli, 2017

129 citations recorded by the source corpus.

We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fr\'echet mean in the Wasserstein space of multivariate distributions; and the optimal registration of deformed random measures and point processes. We elucidate how the two problems are linked, each being in a sense dual to the other. We first study the finite sample version of the problem in the continuum. Exploiting the tangent bundle structure of Wasserstein space, we deduce the Fr\'echet mean via gradient descent. We show that this is equivalent to a Procrustes analysis for the registration maps, thus only requiring successive solutions to pairwise optimal coupling problems. We then study the population version of the problem, focussing on inference and stability: in practice, the data are i.i.d. realisations from a law on Wasserstein space, and indeed their observation is discrete, where one observes a proxy finite sample or point process. We construct regularised nonparametric estimators, and prove their consistency for the population mean, and uniform consistency for the population Procrustes registration maps.

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/9499485e3d679bbce121bfa0bbcf0df58d112f80

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Citations in the corpus

Wasserstein Regression (2020, methodology): “…we submitted this article, a preprint reporting independently conducted but related work on autoregressive modeling of distributional time series was posted by Zhang, Kokoszka, and Petersen (2021), where a simplified version of the distributional autoregressive model in Equation (29) was studied.”
Bayesian Learning with Wasserstein Barycenters (2018, methodology): “1 illustrates the fact that many Bayesian estimators, including the classic model average estimator, correspond to finding a so-called Fréchet mean [Panaretos and Zemel, 2017], or barycenter, under a suitable metric/divergence on probability measures.”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, background): “…has been among the most active domains of mathematical analysis, with applications in various fields, from fluid mechanics to economics (see Galichon (2016)), learning, and statistics (Carlier et al. (2016); Panaretos and Zemel (2016, 2018); Álvarez et al. (2018) and del Barrio et al. (2018)).”