Wasserstein barycenters over Riemannian manifolds Brendan Pass - - PowerPoint PPT Presentation

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

# sets Rn 2 | A+B

2 |1/n ≥ 1 2[|A|1/n + |B|1/n]

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Motivation: Brunn-Minkowski inequality

# sets Rn M Riem. mfld., Ric ≥ 0 2 | A+B

2 |1/n ≥ 1 2[|A|1/n + |B|1/n]

vol(bc(A, B))1/n ≥ 1

2[vol(A)1/n + vol(B)1/n]

Cordero-Erausquin-McCann Schmuckenschlaeger ’01

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Motivation: Brunn-Minkowski inequality

# sets Rn M Riem. mfld., Ric ≥ 0 2 | A+B

2 |1/n ≥ 1 2[|A|1/n + |B|1/n]

vol(bc(A, B))1/n ≥ 1

2[vol(A)1/n + vol(B)1/n]

Cordero-Erausquin-McCann Schmuckenschlaeger ’01 m |

m

i=1 Ai

m

|1/n ≥ 1

m

m

i=1 |Ai|1/n

???? Proof: induction

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Motivation: Brunn-Minkowski inequality

# sets Rn M Riem. mfld., Ric ≥ 0 2 | A+B

2 |1/n ≥ 1 2[|A|1/n + |B|1/n]

vol(bc(A, B))1/n ≥ 1

2[vol(A)1/n + vol(B)1/n]

Cordero-Erausquin-McCann Schmuckenschlaeger ’01 m |

m

i=1 Ai

m

|1/n ≥ 1

m

m

i=1 |Ai|1/n

???? 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 y ∈ X →

• X

d2(x, y)dΩ(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 y ∈ X →

• X

d2(x, y)dΩ(x). When X = Rn, a unique barycenter exists and coincides with ¯ x =

• X xdΩ(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 y ∈ X →

• X

d2(x, y)dΩ(x). When X = Rn, a unique barycenter exists and coincides with ¯ x =

• X xdΩ(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 y ∈ X →

• X

d2(x, y)dΩ(x). When X = Rn, a unique barycenter exists and coincides with ¯ x =

• X xdΩ(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)δx0 + tδx1, the barycenter is γ(t), where γ is a geodesic joining x0 and x1.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Barycenters on Wasserstein space

Let (X, d) = (P(M), W2) 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), W2) 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 ν →

• P(M)

W 2

2 (ν, µ)dΩ(µ)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Barycenters on Wasserstein space

Let (X, d) = (P(M), W2) 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 ν →

• P(M)

W 2

2 (ν, µ)dΩ(µ)

When Ω = (1 − t)δµ0 + tδµ1 barycenters coincide with displacement interpolants.

If µ0 is absolutely continuous with respect to volume, µt 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), W2) 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 ν →

• P(M)

W 2

2 (ν, µ)dΩ(µ)

When Ω = (1 − t)δµ0 + tδµ1 barycenters coincide with displacement interpolants.

If µ0 is absolutely continuous with respect to volume, µt is unique McCann ’01 and absolutely continuous Cordero-Erausquin-McCann-Schmuckenschlaeger ’01.

When M = Rn, and Ω = m

i=1 λiδµi has finite support,

barycenters were studied by Agueh-Carlier ’10

If µ1 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

• nly 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

• nly defined for absolutely continuous measures.

Theorem (Kim-P ’14) Let Ω ∈ P(P(M)) be a probability measure on Wasserstein space

• ver a compact Riemannian manifold M.

If Ω(Pac(M)) > 0, the barycenter ¯ µ of Ω is unique. If Ω(PL∞(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 ∈ Pac(M). Therefore, ν →

• P(M)

W 2

2 (ν, µ)dΩ(µ)

is strictly convex if Ω(Pac(M)) ≥ 0. Absolute continuity: Approximate Ω by a finitely supported measure m

i=1 λiδµi.

Adapt an argument of Figalli-Juillet ’08: approximate each δi, for i = 2, ...m by a discrete measure. Show that we have T#¯ µ = µ1, 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(PL∞(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 αλ(y) := det[−D2

yz

• z=¯

xc(y, z)]

det[

• M D2

zz

• z=¯

xc(x, z)dλ(x)].

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 αλ(y) := det[−D2

yz

• z=¯

xc(y, z)]

det[

• M D2

zz

• z=¯

xc(x, z)dλ(x)].

When λ = m

i=1 λiδxi has finite support

αλ(xj) = lim

r→0

vol(BC(λ, Br(xj))) vol(Bλjr(xj)) where BC(λ, Br(xj)) = ∪y∈Br(xj)BC(m

i=j λiδxi + λjδy). 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

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 αλ(y) := det[−D2

yz

• z=¯

xc(y, z)]

det[

• M D2

zz

• z=¯

xc(x, z)dλ(x)].

When λ = m

i=1 λiδxi has finite support

αλ(xj) = lim

r→0

vol(BC(λ, Br(xj))) vol(Bλjr(xj)) where BC(λ, Br(xj)) = ∪y∈Br(xj)BC(m

i=j λiδxi + λjδy). This

extends the volume distortion coefficients of Cordero-Erausquin-McCann-Schmuckenschlaeger ’01. If Ric ≥ 0, then αλ(y) ≥ 1.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Barycentric density volume control

Let Tµ be the optimal map from the barycenter d ¯ µ(x) = ¯ f (x)dvol(x) of Ω to dµ(x) = fµ(x)dvol(x). For each x ∈ spt(¯ µ), let λx := (µ → Tµ(x))#Ω =

• P(M)

δTµ(x)dΩ(µ)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Barycentric density volume control

Let Tµ be the optimal map from the barycenter d ¯ µ(x) = ¯ f (x)dvol(x) of Ω to dµ(x) = fµ(x)dvol(x). For each x ∈ spt(¯ µ), let λx := (µ → Tµ(x))#Ω =

• P(M)

δTµ(x)dΩ(µ) Then Proposition Under the assumptions in the uniqueness and absolute continuity theorem, we have: ¯ f (x) ≤ [

• P(M)

α1/n

λx (Tµ(x))

(fµ(Tµ(x)))1/n dΩ(µ)]−n

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Barycentric density volume control

Let Tµ be the optimal map from the barycenter d ¯ µ(x) = ¯ f (x)dvol(x) of Ω to dµ(x) = fµ(x)dvol(x). For each x ∈ spt(¯ µ), let λx := (µ → Tµ(x))#Ω =

• P(M)

δTµ(x)dΩ(µ) Then Proposition Under the assumptions in the uniqueness and absolute continuity theorem, we have: ¯ f (x) ≤ [

• P(M)

α1/n

λx (Tµ(x))

(fµ(Tµ(x)))1/n dΩ(µ)]−n In particular, if Ric ≥ 0, we have ||f ||L∞(M) ≤ (||fµ||L∞(M))L∞(Ω)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Distorted Wasserstein Jensen’s inequality

Define, for dµ(x) = f (x)dvol(x), U(µ) =

• M

U(f (x))dvol(x) where U : [0, ∞) → R is such that r → rnU(r−n) is convex non-increasing.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Distorted Wasserstein Jensen’s inequality

Define, for dµ(x) = f (x)dvol(x), U(µ) =

• M

U(f (x))dvol(x) where U : [0, ∞) → R is such that r → rnU(r−n) is convex non-increasing. Theorem (Distorted Jensen’s inequality) Under the previous assumptions on Ω: U(¯ µ) ≤

• P(M)
• M

U( fµ(x) αλT−1

µ (x)(x))αλT−1 µ (x)(x)dvol(x)dΩ(µ). Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Distorted Wasserstein Jensen’s inequality

Define, for dµ(x) = f (x)dvol(x), U(µ) =

• M

U(f (x))dvol(x) where U : [0, ∞) → R is such that r → rnU(r−n) is convex non-increasing. Theorem (Distorted Jensen’s inequality) Under the previous assumptions on Ω: U(¯ µ) ≤

• P(M)
• M

U( fµ(x) αλT−1

µ (x)(x))αλT−1 µ (x)(x)dvol(x)dΩ(µ).

For Ric ≥ 0, this implies U(¯ µ) ≤

• P(M)

U(µ)dΩ(µ).

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Remarks

This extends a convexity over barycenters result of Agueh-Carlier ’10 for finitely many measures on Rn.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Remarks

This extends a convexity over barycenters result of Agueh-Carlier ’10 for finitely many measures on Rn. At the same time, extends convexity results initiated in Cordero-Erausquin-Schmuckenschlaeger ’01; see Villani OTON, Chapter 17

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Remarks

This extends a convexity over barycenters result of Agueh-Carlier ’10 for finitely many measures on Rn. At the same time, extends convexity results initiated in Cordero-Erausquin-Schmuckenschlaeger ’01; see Villani OTON, Chapter 17 Geometric Jensen’s inequalities (using barycenters rather than linear averages) are known on, for examples, smooth manifolds and NPC spaces; see, for example, Emery-Mokobodzki ’91, Sturm ’03 and Kuwawe ’14.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Remarks

This extends a convexity over barycenters result of Agueh-Carlier ’10 for finitely many measures on Rn. At the same time, extends convexity results initiated in Cordero-Erausquin-Schmuckenschlaeger ’01; see Villani OTON, Chapter 17 Geometric Jensen’s inequalities (using barycenters rather than linear averages) are known on, for examples, smooth manifolds and NPC spaces; see, for example, Emery-Mokobodzki ’91, Sturm ’03 and Kuwawe ’14. Wasserstein space does not have upper sectional curvature bounds (Ambrosio-Gigli-Savare ’05).

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

k-Displacement convexity

We can also prove a k-convex version of Jensen’s inequality. Assume the functional U is k-displacement convex; that is, U(µt) ≤ (1 − t)U(µ0) + tU(µt) − k 2t(1 − t)W 2

2 (µ0, µ1).

for all displacement interpolants µt. Then we have U(¯ µ) ≤

• P(M)

U(µ)dΩ(µ) − k 2

• P(M)

W 2

2 (¯

µ, µ)dΩ(µ)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

k-Displacement convexity

We can also prove a k-convex version of Jensen’s inequality. Assume the functional U is k-displacement convex; that is, U(µt) ≤ (1 − t)U(µ0) + tU(µt) − k 2t(1 − t)W 2

2 (µ0, µ1).

for all displacement interpolants µt. Then we have U(¯ µ) ≤

• P(M)

U(µ)dΩ(µ) − k 2

• P(M)

W 2

2 (¯

µ, µ)dΩ(µ) It is unclear whether these two upper bounds on U(¯ µ) are comparable (the two marginal analogue is related to Villani OTON Open Problem 17.39 ).

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Curved random Brunn-Minkowski inequality

Let A : (P, Ω) → 2M be a random set on M. A selection of A is a random variable V : (P, Ω) → M such that V ∈ A almost surely. Define E(A) = {¯ x : ¯ x ∈ argmin

• P

d2(x, V (ω))dΩ(ω) : V is a selection of A.}

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Curved random Brunn-Minkowski inequality

Let A : (P, Ω) → 2M be a random set on M. A selection of A is a random variable V : (P, Ω) → M such that V ∈ A almost surely. Define E(A) = {¯ x : ¯ x ∈ argmin

• P

d2(x, V (ω))dΩ(ω) : V is a selection of A.} Theorem (Kim-P ’14) Let A be a random set on M, with Ric ≥ 0. Then (vol(E(A)))1/n ≥ E(vol(A)1/n)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Special cases

When M = Rn we get an optimal transport based proof of Vitale’s random Brunn-Minkowski inequality.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Special cases

When M = Rn we get an optimal transport based proof of Vitale’s random Brunn-Minkowski inequality. When law(A) = 1

2δA0 + 1 2δA1 we get the curved

Brunn-Minkowski inequality of Cordero-Erausquin-McCann-Schmuckenschlaeger ’01.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Special cases

When M = Rn we get an optimal transport based proof of Vitale’s random Brunn-Minkowski inequality. When law(A) = 1

2δA0 + 1 2δA1 we get the curved

Brunn-Minkowski inequality of Cordero-Erausquin-McCann-Schmuckenschlaeger ’01. When law(A) = m

i=1 λiδAi we get

vol(BC(A1, ...Am; λ1, ..., λm))1/n ≥

m

• i=1

λivol(Ai)1/n

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Sketch of proof

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Sketch of proof

Set µA =

1A vol(A) ( a random measure) and let ¯

µ be its barycenter.

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Sketch of proof

Set µA =

1A vol(A) ( a random measure) and let ¯

µ be its barycenter. Apply Wasserstein Jensen’s inequality to U(r) = −(r1−1/n − r).

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Sketch of proof

Set µA =

1A vol(A) ( a random measure) and let ¯

µ be its barycenter. Apply Wasserstein Jensen’s inequality to U(r) = −(r1−1/n − r). Straightforward calculations lead to vol(spt(¯ µ))1/n ≥ E(vol(A)1/n)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds

Sketch of proof

Set µA =

1A vol(A) ( a random measure) and let ¯

µ be its barycenter. Apply Wasserstein Jensen’s inequality to U(r) = −(r1−1/n − r). Straightforward calculations lead to vol(spt(¯ µ))1/n ≥ E(vol(A)1/n) For ¯ µ almost all x, show that x is a barycenter of λx. V (ω) = TµA(ω)(x) is then a selection of A with x as a barycenter, and so we have, spt(¯ µ) ⊆ E(A)

Brendan Pass (joint work with Y.H. Kim (UBC)) Wasserstein barycenters over Riemannian manifolds