Barycenter in Wasserstein space existence and consistency

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14309

Résumé

We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.

Barycenter in Wasserstein space existence and consistency

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application/pdf Barycenter in Wasserstein space existence and consistency Thibaut Le Gouic, Jean-Michel Loubes
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We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.
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We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.

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Barycenter in Wasserstein spaces: existence and consistency Thibaut Le Gouic and Jean-Michel Loubes* Institut de Math´ematiques de Marseille ´Ecole Centrale Marseille Institut Math´ematique de Toulouse* October 29th 2015 1 / 23 Barycenter in Wasserstein spaces Barycenter The barycenter of a set {xi }1≤i≤J of Rd for J points endowed with weights (λi )1≤i≤J is defined as 1≤i≤J λi xi . It is characterized by being the minimizer of x → 1≤i≤J λi x − xi 2 . 2 / 23 Barycenter in Wasserstein spaces Barycenter The barycenter of a set {xi }1≤i≤J of Rd for J points endowed with weights (λi )1≤i≤J is defined as 1≤i≤J λi xi . It is characterized by being the minimizer of x → 1≤i≤J λi x − xi 2 . Replace (Rd , . ) by a metric space (E, d), and minimize x → 1≤i≤J λi d(x, xi )2 . 2 / 23 Barycenter in Wasserstein spaces Barycenter Likewise, given a random variable/vector of law µ on Rd , its expectation EX is characterized by being the minimizer of x → E X − x 2 . 3 / 23 Barycenter in Wasserstein spaces Barycenter Likewise, given a random variable/vector of law µ on Rd , its expectation EX is characterized by being the minimizer of x → E X − x 2 . → extension to a metric space (it summarizes the information staying in a geodesic space) 3 / 23 Barycenter in Wasserstein spaces Barycenter Definition (p-barycenter) Given a probability measure µ on a geodesic space (E, d), the set arg min x ∈ E; d(x, y)p dµ(y) , is called the set of p-barycenters of µ. 4 / 23 Barycenter in Wasserstein spaces Barycenter Definition (p-barycenter) Given a probability measure µ on a geodesic space (E, d), the set arg min x ∈ E; d(x, y)p dµ(y) , is called the set of p-barycenters of µ. Existence ? 4 / 23 1 Geodesic space 2 Wasserstein space 3 Applications 5 / 23 Barycenter in Wasserstein spaces Geodesic space Definition (Geodesic space) A complete metric space (E, d) is said to be geodesic if for all x, y ∈ E, there exists z ∈ E such that 1 2 d(x, y) = d(x, z) = d(z, y). 6 / 23 Barycenter in Wasserstein spaces Geodesic space Definition (Geodesic space) A complete metric space (E, d) is said to be geodesic if for all x, y ∈ E, there exists z ∈ E such that 1 2 d(x, y) = d(x, z) = d(z, y). Include many spaces (vectorial normed spaces, compact manifolds, ...), 6 / 23 Barycenter in Wasserstein spaces Geodesic space Proposition (Existence) The p-barycenter of any probability measure on a locally compact geodesic space, with finite moments of order p, exists. 7 / 23 Barycenter in Wasserstein spaces Geodesic space Proposition (Existence) The p-barycenter of any probability measure on a locally compact geodesic space, with finite moments of order p, exists. Not unique e.g. the sphere Non positively curved space → unique barycenter, 1-Lipschitz on 2-Wasserstein space. 7 / 23 1 Geodesic space 2 Wasserstein space 3 Applications 8 / 23 Barycenter in Wasserstein spaces Wasserstein metric Definition (Wasserstein metric) Let µ and ν be two probability measures on a metric space (E, d) and p ≥ 1. The p-Wasserstein distance between µ and ν is defined as W p p (µ, ν) = inf π∈Γ(µ,ν) dE (x, y)p dπ(x, y), where Γ(µ, ν) is the set of all probability measures on E × E with marginals µ and ν. 9 / 23 Barycenter in Wasserstein spaces Wasserstein metric Definition (Wasserstein metric) Let µ and ν be two probability measures on a metric space (E, d) and p ≥ 1. The p-Wasserstein distance between µ and ν is defined as W p p (µ, ν) = inf π∈Γ(µ,ν) dE (x, y)p dπ(x, y), where Γ(µ, ν) is the set of all probability measures on E × E with marginals µ and ν. Defined for any measure for which moments of order p are finite : Ed(X, x0)p < ∞ (denote this set Pp(E)), It is a metric on Pp(E) ; (Pp(E), Wp) is called the Wasserstein space, The topology of this metric is the weak convergence topology and convergence of moments of order p. 9 / 23 Barycenter in Wasserstein spaces Wasserstein metric The Wasserstein space of a complete geodesic space is a complete geodesic space. (Pp(E), Wp) is locally compact ⇔ (E, d) is compact. (E, d) ⊂ (Pp(E), Wp) isometrically. Existence of the barycenter on (Pp(E), Wp) ? 10 / 23 Barycenter in Wasserstein spaces Measurable barycenter application Definition (Measurable barycenter application) Let (E, d) be a geodesic space. (E, d) is said to admit measurable barycenter applications if for any J ≥ 1 and any weights (λj )1≤j≤J, there exists a measurable application T from EJ to E such that for all (x1, ..., xJ) ∈ EJ, min x∈E J j=1 λj d(x, xj )p = J j=1 λj d(T(x1, ..., xJ), xj )p . 11 / 23 Barycenter in Wasserstein spaces Measurable barycenter application Definition (Measurable barycenter application) Let (E, d) be a geodesic space. (E, d) is said to admit measurable barycenter applications if for any J ≥ 1 and any weights (λj )1≤j≤J, there exists a measurable application T from EJ to E such that for all (x1, ..., xJ) ∈ EJ, min x∈E J j=1 λj d(x, xj )p = J j=1 λj d(T(x1, ..., xJ), xj )p . Locally compact geodesic spaces admit measurable barycenter applications. 11 / 23 Barycenter in Wasserstein spaces Existence of barycenter Theorem (Existence of barycenter) Let (E, d) be a geodesic space that admits measurable barycenter applications. Then any probability measure P on (Pp(E), Wp) has a barycenter. 12 / 23 Barycenter in Wasserstein spaces Existence of barycenter Theorem (Existence of barycenter) Let (E, d) be a geodesic space that admits measurable barycenter applications. Then any probability measure P on (Pp(E), Wp) has a barycenter. Barycenter is not unique e.g. : E = Rd with P = 1 2δµ1 + 1 2δµ2 , µ1 = 1 2δ(−1,−1) + 1 2δ(1,1) and µ2 = 1 2δ(1,−1) + δ(−1,1) 12 / 23 Barycenter in Wasserstein spaces Existence of barycenter Theorem (Existence of barycenter) Let (E, d) be a geodesic space that admits measurable barycenter applications. Then any probability measure P on (Pp(E), Wp) has a barycenter. Barycenter is not unique e.g. : E = Rd with P = 1 2δµ1 + 1 2δµ2 , µ1 = 1 2δ(−1,−1) + 1 2δ(1,1) and µ2 = 1 2δ(1,−1) + δ(−1,1) Consistency of the barycenter ? 12 / 23 Barycenter in Wasserstein spaces 3 steps for existence 1 Multimarginal problem 2 Weak consistency 3 Approximation by finitely supported measures 13 / 23 Barycenter in Wasserstein spaces Push forward Definition (Push forward) Given a measure ν on E and an measurable application T : E → (F, F), the push forward of ν by T is given by T#ν(A) = ν T−1 (A) , ∀A ∈ F. Probabilist version : X is a r.v. on (Ω, A, P), then PX = X#P. 14 / 23 Barycenter in Wasserstein spaces Multimarginal problem Theorem (Barycenter and multi-marginal problem [Agueh and Carlier, 2011]) Let (E, d) be a complete separable geodesic space, p ≥ 1 and J ∈ N∗. Given (µi )1≤i≤J ∈ Pp(E)J and weights (λi )1≤i≤J, there exists a measure γ ∈ Γ(µ1, ..., µJ) minimizing ˆγ → inf x∈E 1≤i≤J λi d(xi , x)p dˆγ(x1, ..., xJ). If (E, d) admits a measurable barycenter application T : EJ → E then the measure ν = T#γ is a barycenter of (µi )1≤i≤J If T is unique, ν is of the form ν = T#γ. 15 / 23 Barycenter in Wasserstein spaces Weak consistency Theorem (Weak consistency of the barycenter) Let (E, d) be a geodesic space that admits measurable barycenter. Take (Pj )j≥1 ⊂ Pp(E) converging to P ∈ Pp(E). Take any barycenter µj of Pj . Then the sequence (µj )j≥1 is (weakly) tight and any limit point is a barycenter of P. 16 / 23 Barycenter in Wasserstein spaces Approximation by finitely supported measure Proposition (Approximation by finitely supported measure) For any measure P on Pp(E) there exists a sequence of finitely supported measures (Pj )j≥1 ⊂ Pp(E) such that Wp(Pj , P) → 0 as j → ∞. 17 / 23 Barycenter in Wasserstein spaces 3 steps for existence 1 Multimarginal problem 2 Weak consistency 3 Approximation by finitely supported measures 18 / 23 Barycenter in Wasserstein spaces 3 steps for existence 1 Multimarginal problem → existence of barycenter for P finitely supported. 2 Weak consistency 3 Approximation by finitely supported measures 18 / 23 Barycenter in Wasserstein spaces 3 steps for existence 1 Multimarginal problem → existence of barycenter for P finitely supported. 2 Weak consistency → existence of barycenter for probabilities that can be approximated by measures with barycenters. 3 Approximation by finitely supported measures 18 / 23 Barycenter in Wasserstein spaces 3 steps for existence 1 Multimarginal problem → existence of barycenter for P finitely supported. 2 Weak consistency → existence of barycenter for probabilities that can be approximated by measures with barycenters. 3 Approximation by finitely supported measures → any probability can be approximated by a finitely supported probability measure. 18 / 23 Barycenter in Wasserstein spaces Consistency of the barycenter Theorem (Consistency of the barycenter) Let (E, d) be a geodesic space that admits measurable barycenter. Take (Pj )j≥1 ⊂ Pp(E) and P ∈ Pp(E). Take any barycenter µj of Pj . Then the sequence (µj )j≥1 is totally bounded in (Pp(E), Wp) and any limit point is a barycenter of P. 19 / 23 Barycenter in Wasserstein spaces Consistency of the barycenter Theorem (Consistency of the barycenter) Let (E, d) be a geodesic space that admits measurable barycenter. Take (Pj )j≥1 ⊂ Pp(E) and P ∈ Pp(E). Take any barycenter µj of Pj . Then the sequence (µj )j≥1 is totally bounded in (Pp(E), Wp) and any limit point is a barycenter of P. Imply continuity of barycenter when barycenter are unique. No rate of convergence (barycenter Lipschitz on (E, d) Lipschitz on Pp(E)). Imply compactness of the set of barycenters. 19 / 23 1 Geodesic space 2 Wasserstein space 3 Applications 20 / 23 Barycenter in Wasserstein spaces Statistical application : improvement of measures accuracy Take (µn i )1≤j≤J → µj when n → ∞ and weights (λj )1≤j≤J. Set µn B the barycenter of (µn i )1≤j≤J. Then, as n → ∞, µn B → µB. 21 / 23 Barycenter in Wasserstein spaces Statistical application : improvement of measures accuracy Take (µn i )1≤j≤J → µj when n → ∞ and weights (λj )1≤j≤J. Set µn B the barycenter of (µn i )1≤j≤J. Then, as n → ∞, µn B → µB. Texture mixing [Rabin et al., 2011] 21 / 23 Barycenter in Wasserstein spaces Statistical application : growing number of measures Take (µn)n≥1 such that 1 n n i=1 µi → P. Set µn B the barycenter of 1 n n i=1 δµi . Then, as n → ∞, µn B → µB 22 / 23 Barycenter in Wasserstein spaces Statistical application : growing number of measures Take (µn)n≥1 such that 1 n n i=1 µi → P. Set µn B the barycenter of 1 n n i=1 δµi . Then, as n → ∞, µn B → µB Average of template deformation [Bigot and Klein, 2012],[Agull´o-Antol´ın et al., 2015] 22 / 23 Agueh, M. and Carlier, G. (2011). Barycenters in the wasserstein space. SIAM Journal on Mathematical Analysis, 43(2) :904–924. Agull´o-Antol´ın, M., Cuesta-Albertos, J. A., Lescornel, H., and Loubes, J.-M. (2015). A parametric registration model for warped distributions with Wasserstein’s distance. J. Multivariate Anal., 135 :117–130. Bigot, J. and Klein, T. (2012). Consistent estimation of a population barycenter in the Wasserstein space. ArXiv e-prints. Rabin, J., Peyr´e, G., Delon, J., and Bernot, M. (2011). Wasserstein Barycenter and its Application to Texture Mixing. SSVM’11, pages 435–446. 23 / 23