Wasserstein barycenters over Riemannian manifolds Brendan Pass (joint work with Y.H. Kim (UBC)) University of Alberta Dec. 10, 2014 Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Motivation: Brunn-Minkowski inequality R n # sets 2 | 1 / n ≥ 1 2 [ | A | 1 / n + | B | 1 / n ] | A + B 2 Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Motivation: Brunn-Minkowski inequality R n # sets M Riem. mfld., Ric ≥ 0 2 | 1 / n ≥ 1 2 [ | A | 1 / n + | B | 1 / n ] | A + B vol ( bc ( A , B )) 1 / n 2 2 [ vol ( A ) 1 / n + vol ( B ) 1 / n ] ≥ 1 Cordero-Erausquin-McCann Schmuckenschlaeger ’01 Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Motivation: Brunn-Minkowski inequality R n # sets M Riem. mfld., Ric ≥ 0 2 | 1 / n ≥ 1 2 [ | A | 1 / n + | B | 1 / n ] | A + B vol ( bc ( A , B )) 1 / n 2 2 [ vol ( A ) 1 / n + vol ( B ) 1 / n ] ≥ 1 Cordero-Erausquin-McCann Schmuckenschlaeger ’01 � m | 1 / n ≥ 1 i =1 A i � m i =1 | A i | 1 / n m | ???? m m Proof: induction Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Motivation: Brunn-Minkowski inequality R n # sets M Riem. mfld., Ric ≥ 0 2 | 1 / n ≥ 1 2 [ | A | 1 / n + | B | 1 / n ] | A + B vol ( bc ( A , B )) 1 / n 2 2 [ vol ( A ) 1 / n + vol ( B ) 1 / n ] ≥ 1 Cordero-Erausquin-McCann Schmuckenschlaeger ’01 � m | 1 / n ≥ 1 i =1 A i � m i =1 | A i | 1 / n m | ???? m m Proof: induction | E ( A ) | 1 / n ≥ E ( | A | 1 / n ) ∞ ???? Vitale’s Random Brunn-Minkowski inequality ’90 Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on metric spaces Let ( X , d ) be a metric space and Ω a compactly supported probability measure on X . Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on metric spaces Let ( X , d ) be a metric space and Ω a compactly supported probability measure on X . A barycenter ¯ x of Ω is a minimizer of � d 2 ( x , y ) d Ω( x ) . y ∈ X �→ X Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on metric spaces Let ( X , d ) be a metric space and Ω a compactly supported probability measure on X . A barycenter ¯ x of Ω is a minimizer of � d 2 ( x , y ) d Ω( x ) . y ∈ X �→ X When X = R n , a unique barycenter exists and coincides with � ¯ x = X xd Ω( x ), the weighted average or mean. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on metric spaces Let ( X , d ) be a metric space and Ω a compactly supported probability measure on X . A barycenter ¯ x of Ω is a minimizer of � d 2 ( x , y ) d Ω( x ) . y ∈ X �→ X When X = R n , a unique barycenter exists and coincides with � ¯ x = X xd Ω( x ), the weighted average or mean. In general, existence of a barycenter is easy to establish. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on metric spaces Let ( X , d ) be a metric space and Ω a compactly supported probability measure on X . A barycenter ¯ x of Ω is a minimizer of � d 2 ( x , y ) d Ω( x ) . y ∈ X �→ X When X = R n , a unique barycenter exists and coincides with � ¯ x = X xd Ω( x ), the weighted average or mean. In general, existence of a barycenter is easy to establish. If X = M , a Riemannian manifold, uniqueness depends on sectional curvature. If Ω = (1 − t ) δ x 0 + t δ x 1 , the barycenter is γ ( t ), where γ is a geodesic joining x 0 and x 1 . Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on Wasserstein space Let ( X , d ) = ( P ( M ) , W 2 ) be the space of probability measures on a compact Riemannian manifold M , equipped with Wasserstein distance. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on Wasserstein space Let ( X , d ) = ( P ( M ) , W 2 ) be the space of probability measures on a compact Riemannian manifold M , equipped with Wasserstein distance. Barycenters give a way to interpolate between a (finite or infinite) family of measures. Given a measure Ω ∈ P ( P ( M )), a barycenter is a minimizer of � W 2 ν �→ 2 ( ν, µ ) d Ω( µ ) P ( M ) Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on Wasserstein space Let ( X , d ) = ( P ( M ) , W 2 ) be the space of probability measures on a compact Riemannian manifold M , equipped with Wasserstein distance. Barycenters give a way to interpolate between a (finite or infinite) family of measures. Given a measure Ω ∈ P ( P ( M )), a barycenter is a minimizer of � W 2 ν �→ 2 ( ν, µ ) d Ω( µ ) P ( M ) When Ω = (1 − t ) δ µ 0 + t δ µ 1 barycenters coincide with displacement interpolants. If µ 0 is absolutely continuous with respect to volume, µ t is unique McCann ’01 and absolutely continuous Cordero-Erausquin-McCann-Schmuckenschlaeger ’01. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycenters on Wasserstein space Let ( X , d ) = ( P ( M ) , W 2 ) be the space of probability measures on a compact Riemannian manifold M , equipped with Wasserstein distance. Barycenters give a way to interpolate between a (finite or infinite) family of measures. Given a measure Ω ∈ P ( P ( M )), a barycenter is a minimizer of � W 2 ν �→ 2 ( ν, µ ) d Ω( µ ) P ( M ) When Ω = (1 − t ) δ µ 0 + t δ µ 1 barycenters coincide with displacement interpolants. If µ 0 is absolutely continuous with respect to volume, µ t is unique McCann ’01 and absolutely continuous Cordero-Erausquin-McCann-Schmuckenschlaeger ’01. When M = R n , and Ω = � m i =1 λ i δ µ i has finite support, barycenters were studied by Agueh-Carlier ’10 If µ 1 is absolutely continuous, then the barycenter is unique and absolutely continuous. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Uniqueness and absolute continuity of the barycenter Note: absolute continuity is important for studying displacement convexity, as many interesting displacement convex functionals are only defined for absolutely continuous measures. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Uniqueness and absolute continuity of the barycenter Note: absolute continuity is important for studying displacement convexity, as many interesting displacement convex functionals are only defined for absolutely continuous measures. Theorem (Kim-P ’14) Let Ω ∈ P ( P ( M )) be a probability measure on Wasserstein space over a compact Riemannian manifold M. If Ω( P ac ( M )) > 0 , the barycenter ¯ µ of Ω is unique. If Ω( P L ∞ ( M )) > 0 , then ¯ µ is absolutely continuous with respect to volume (and in fact has an L ∞ density) Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Sketch of proofs Uniqueness: The function ν �→ W 2 2 ( ν, µ ) is (linearly) convex, and the convexity is strict if µ ∈ P ac ( M ). Therefore, � W 2 ν �→ 2 ( ν, µ ) d Ω( µ ) P ( M ) is strictly convex if Ω( P ac ( M )) ≥ 0. Absolute continuity: Approximate Ω by a finitely supported measure � m i =1 λ i δ µ i . Adapt an argument of Figalli-Juillet ’08: approximate each δ i , for i = 2 , ... m by a discrete measure. Show that we have T # ¯ µ = µ 1 , where T is Lipschitz, with a constant only depending on M and λ . Pass to the limit: this proves absolute continuity when Ω is finitely supported. Under the Ω( P L ∞ ( M )) > 0, get similar bounds on ¯ µ , and pass to the limit. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycentric volume distortion coefficients Let λ ∈ P ( M ) be a measure on M with a unique barycenter ¯ x . Define the barycentric volume distortion coefficients at y ∈ M by det[ − D 2 � x c ( y , z )] yz � z =¯ α λ ( y ) := x c ( x , z ) d λ ( x )] . � M D 2 � det[ zz � z =¯ Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
Barycentric volume distortion coefficients Let λ ∈ P ( M ) be a measure on M with a unique barycenter ¯ x . Define the barycentric volume distortion coefficients at y ∈ M by det[ − D 2 � x c ( y , z )] yz � z =¯ α λ ( y ) := x c ( x , z ) d λ ( x )] . � M D 2 � det[ zz � z =¯ When λ = � m i =1 λ i δ x i has finite support vol ( BC ( λ, B r ( x j ))) α λ ( x j ) = lim vol ( B λ j r ( x j )) r → 0 where BC ( λ, B r ( x j )) = ∪ y ∈ B r ( x j ) BC ( � m i � = j λ i δ x i + λ j δ y ). This extends the volume distortion coefficients of Cordero-Erausquin-McCann-Schmuckenschlaeger ’01. Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds
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