• Bernoulli
  • Volume 25, Number 2 (2019), 932-976.

Fréchet means and Procrustes analysis in Wasserstein space

Yoav Zemel and Victor M. Panaretos

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We consider two statistical problems at the intersection of functional and non-Euclidean data analysis: the determination of a Fréchet 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 Fréchet 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.

Article information

Bernoulli, Volume 25, Number 2 (2019), 932-976.

Received: January 2017
Revised: November 2017
First available in Project Euclid: 6 March 2019

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functional data analysis manifold statistics Monge–Kantorovich problem multimarginal transportation optimal transportation phase variation point process random measure registration shape theory warping


Zemel, Yoav; Panaretos, Victor M. Fréchet means and Procrustes analysis in Wasserstein space. Bernoulli 25 (2019), no. 2, 932--976. doi:10.3150/17-BEJ1009.

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Supplemental materials

  • Fréchet means and Procrustes analysis in Wasserstein space. The online supplement contains more details on the examples, additional technical material, as well as those proofs that were omitted from the main paper.