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We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric.
On the completeness of a certain metric space with an application to Blaschke’s selection theorem
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Separate continuity and measurability
B. E. Johnson · 1969
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On expected figures and a strong law of large numbers for random elements in quasi-metric spaces
Herbert Ziezold · 1977
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Lebesgue’s first theorem
W. Rudin · 1981
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Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference
J. Dauxois, A. Pousse, and Y. Romain · 1982
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Variational Convergence for Functions and Operators
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On convergence of closed sets in a metric space and distance functions
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Polar factorization and monotone rearrangement of vector-valued functions
Y. Brenier · 1991
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An Introduction to Γ \Gamma -convergence
G. Dal Maso · 1993
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Large sample estimation and hypothesis testing
W. K. Newey and D. McFadden · 1994
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Consistency of minimizers and the SLLN for stochastic programs
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Smoothed functional principal components analysis by choice of norm
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Inference for density families using functional principal component analysis
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Large sample theory of intrinsic and extrinsic sample means on manifolds
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Topics in Optimal Transportation
C. Villani · 2003
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Gradient flows with metric and differentiable structures, and applications to the Wasserstein space
L. Ambrosio, N. Gigli, and G. Savaré · 2004
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Barycenters in the Wasserstein space
M. Agueh and G. Carlier · 2011
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Optimal transport and Ricci curvature: Wasserstein space over the interval
O. Chodosh · 2011
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Dimensionality reduction when data are density functions
P. Delicado · 2011
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Probability in Banach spaces: Isoperimetry and processes
M. Ledoux and M. Talagrand · 2011
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Functional density synchronization
Z. Zhang and H.-G. Müller · 2011
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Consistent estimation of a population barycenter in the Wasserstein space
J. Bigot and T. Klein · 2012
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Principal geodesic analysis for the study of nonlinear statistics of shape
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Functional data analysis
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Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations
S. Sommer, F. Lauze, S. Hauberg, and M. Nielsen · 2010
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