Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Wasserstein-Otto geometry
Klas Modin
Riemannian principal bundles
E
E
E/H≃B
E/H\simeq B
Hor
\mathrm{Hor}
E
E
↩H
\hookleftarrow H
↓
\downarrow
B
B
π
\pi
Invariant Riemannian metric on E
⇒ π Riemannian submersion
Riemannian principal bundles
E
E
E/H≃B
E/H\simeq B
Hor
\mathrm{Hor}
E
E
↩H
\hookleftarrow H
↓
\downarrow
B
B
π
\pi
Invariant Riemannian metric on E
⇒ π Riemannian submersion
Riemannian principal bundles
G
G
G/H≃B
G/H\simeq B
e
e
Hor
\mathrm{Hor}
G
G
↩H
\hookleftarrow H
↓
\downarrow
B
B
π
\pi
left co-sets [g]=g⋅H
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
Hor
\mathrm{Hor}
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
Gb0
G_{b_0}
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
horizontal flow
\text{horizontal flow}
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
vertical flow
\text{vertical flow}
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
g∈G
g\in G
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
b1=π(g)=g⋅b0
b_1 = \pi(g) = g\cdot b_0
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
γ(1)∈G
\gamma(1)\in G
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
b1=π(g)=g⋅b0
b_1 = \pi(g) = g\cdot b_0
γ′(t)∈Hor
\gamma'(t)\in \mathrm{Hor}
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
g=γ(1)(γ(1)−1g)
g = \gamma(1)(\gamma(1)^{-1}g)
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
b1=π(g)=g⋅b0
b_1 = \pi(g) = g\cdot b_0
γ′(t)∈Hor
\gamma'(t)\in \mathrm{Hor}
klasklas∈Gb0
\overbrace{\phantom{klasklas}}^{\in G_{b_0}}
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
g=γ(1)h
g = \gamma(1)h
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
b1=π(g)=g⋅b0
b_1 = \pi(g) = g\cdot b_0
γ′(t)∈Hor
\gamma'(t)\in \mathrm{Hor}
Riemannian principal bundles
G
G
G/Gb0≃B
G/G_{b_0}\simeq B
e
e
g=kh
g = k h
G
G
↩Gb0
\hookleftarrow G_{b_0}
↓
\downarrow
B
B
π(g)=g⋅b0
\pi(g) = g\cdot b_0
left co-sets [g]=g⋅Gb0
Semi-invariant Riemannian metric on G
⇒ π Riemannian submersion
b0
b_0
b1=π(g)=g⋅b0
b_1 = \pi(g) = g\cdot b_0
K
K
polar cone
Optimal mass transport (OMT)
μ0
\mu_0
μ1
\mu_1
φ∗μ0
\varphi_*\mu_0
φ∗μ0=μ1min∫MdM2(φ(x),x)μ0
\displaystyle\min_{\varphi*\mu_0=\mu_1} \int_{M} d_M^2(\varphi(x),x ) \mu_0
M
M
Monge problem, L2 version
Optimal mass transport (OMT)
μ0
\mu_0
μ1
\mu_1
φ∗μ0
\varphi_*\mu_0
φ∗μ0=μ1min∫Rn∣φ(x)−x∣2μ0
\displaystyle\min_{\varphi*\mu_0=\mu_1} \int_{\mathbb{R}^n} \lvert \varphi(x)-x \rvert^2 \mu_0
Rn
\mathbb{R}^n
Monge problem, L2 version
J(φ)klasklkllkklklas
\underbrace{\phantom{klasklkllkklklas}}_{J(\varphi)}
Riemannian structure of OMT
Diff(M)
\mathrm{Diff}(M)
Dens(M)≃Diff(M)/Diffμ0(M)
\mathrm{Dens}(M)\simeq \mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)
Id
\mathrm{Id}
μ0
\mu_0
μ1
\mu_1
π(φ)=φ∗μ0
\pi(\varphi)=\varphi_*\mu_0
Riemannian metric
Gφ(φ˙,φ˙)=∫M∣φ˙∣2μ0
\displaystyle\mathcal{G}_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu_0
Induced metric
Gμ(μ˙,μ˙)=∫M∣∇θ∣2μ
\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) = \int_M \lvert \nabla \theta\rvert^2 \mu
[Benamou & Brenier (2000), Otto (2001)]
Invariance: η∈Diffμ0(M)
Gφ(φ˙,φ˙)=Gφ∘η(φ˙∘η,φ˙∘η)
\displaystyle\mathcal{G}_\varphi(\dot\varphi,\dot\varphi) = \mathcal{G}_{\varphi\circ\eta}(\dot\varphi\circ\eta,\dot\varphi\circ\eta)
ρ˙+div(ρ∇θ)=0,ρ=μ/dx
\dot\rho + \operatorname{div}(\rho \nabla\theta) = 0, \; \rho = \mu/dx
Geodesics on Diff(Rn)
Geodesic equation:
φ¨=0⇒φ(t)=Id+tv0
\ddot\varphi = 0 \Rightarrow \varphi(t) = \mathrm{Id} + t\,v_0
Id
\mathrm{Id}
μ0
\mu_0
K
K
μ1
\mu_1
HorId=∇C∞(Rn)
\mathrm{Hor}_{\mathrm{Id}} = \nabla C^\infty(\mathbb{R}^n)
Easy to prove:
Polar cone K is isomorphic to strictly convex smooth functions via ϕ↦∇ϕ
Hard to prove:
Polar cone K a section of principal bundle
Geodesics on Diff(Rn)
Geodesic equation:
φ¨=0⇒φ(t)=Id+tv0
\ddot\varphi = 0 \Rightarrow \varphi(t) = \mathrm{Id} + t\,v_0
Id
\mathrm{Id}
μ0
\mu_0
K
K
μ1
\mu_1
φ=∇ϕ∘η,η∈Diffμ0(Rn)
\varphi = \nabla\phi\circ\eta, \; \eta\in \operatorname{Diff}_{\mu_0}(\mathbb{R}^n)
Easy to prove:
Polar cone K is isomorphic to strictly convex smooth functions via ϕ↦∇ϕ
Hard to prove:
Polar cone K a section of principal bundle
Brenier's decomposition of transport maps
Geodesic distance on Diff(Rn)
Geodesic curve:
φ(t)=(1−t)φ0+tφ1
\varphi(t) = (1-t)\varphi_0 + t \varphi_1
Id
\mathrm{Id}
μ0
\mu_0
K
K
μ1
\mu_1
dist(φ0,φ1)2=∫01Gφ(t)(φ˙(t),φ˙(t))dt
\mathrm{dist}(\varphi_0,\varphi_1)^2 = \int_0^1 \mathcal{G}_{\varphi(t)}(\dot\varphi(t),\dot\varphi(t)) dt
=∫Rn∣φ1(x)−φ0(t)∣2
= \int_{\mathbb{R}^n} \lvert \varphi_1(x)-\varphi_0(t) \rvert^2
=∫01∫Rn∣φ˙(t)∣2μ0dt
= \int_0^1\int_{\mathbb{R}^n} \lvert \dot\varphi(t)\rvert^2\mu_0 dt
In particular:
J(φ)=dist(Id,φ)2
J(\varphi) = \mathrm{dist}(\mathrm{Id},\varphi)^2
Monge-Ampere equation on Rn
Geodesic curve:
φ(t)=∇(∣x∣2/2+tf)
\varphi(t) = \nabla( |x|^2/2 + t f )
Id
\mathrm{Id}
μ0
\mu_0
K
K
μ1
\mu_1
(∇ϕ)∗μ0=μ1
(\nabla\phi)_*\mu_0 = \mu_1
In particular:
J(φ)=dist(Id,φ)2
J(\varphi) = \mathrm{dist}(\mathrm{Id},\varphi)^2
∇ϕklaklklsklkasl
\underbrace{\phantom{klaklklsklkasl}}_{\nabla\phi}
⇒det(∇2ϕ)=ρ1∘∇ϕρ0
\displaystyle \Rightarrow \operatorname{det}(\nabla^2\phi) = \frac{\rho_0}{\rho_1\circ \nabla\phi}
Monge-Ampere equation on Rn
Geodesic curve:
φ(t)=∇(∣x∣2/2+tf)
\varphi(t) = \nabla( |x|^2/2 + t f )
Id
\mathrm{Id}
μ0
\mu_0
K
K
μ1
\mu_1
(∇ϕ)∗μ0=μ1
(\nabla\phi)_*\mu_0 = \mu_1
In particular:
J(φ)=dist(Id,φ)2
J(\varphi) = \mathrm{dist}(\mathrm{Id},\varphi)^2
∇ϕklaklklsklkasl
\underbrace{\phantom{klaklklsklkasl}}_{\nabla\phi}
⇒det(∇2ϕ)=ρ1∘∇ϕρ0
\displaystyle \Rightarrow \operatorname{det}(\nabla^2\phi) = \frac{\rho_0}{\rho_1\circ \nabla\phi}
Linear optimal mass transport
Trivial observation: φ0(x)=A0x, φ1(x)=A1x linear diffeomorphisms ⇒ geodesic consists of linear diffeomorphisms
Consequence: GL(n) is totally geodesic subgroup of Diff(Rn)
Corresponding subspace of densities (statistical submanifold): multivariate Gaussians with zero mean
ρ(x)=(2π)ndet(Σ)1exp(−21x⊤Σ−1x)
\displaystyle \rho(x) = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(\Sigma)}}\mathrm{exp}(-\frac{1}{2}x^\top \Sigma^{-1}x)
Σ∈P(n)
\Sigma \in P(n)
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
π(A)=AΣ0A⊤
\pi(A)=A\Sigma_0 A^\top
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K
K
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
π(A)=AΣ0A⊤
\pi(A)=A\Sigma_0 A^\top
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
π(P)=PΣ0P=Σ1
\pi(P)=P\Sigma_0 P = \Sigma_1
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Monge-Ampere equation:
P
P
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
A=PQ
A = PQ
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Factorization theorem:
A
A
P
P
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
B˙=−Pr∇GJ(B),B(0)=A
\dot B = -\mathrm{Pr}\nabla_{\mathcal G}J(B), \; B(0) = A
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Vertical gradient flow:
A
A
Bundle structure
GL(n)
GL(n)
P(n)≃GL(n)/O(n,Σ0)
P(n)\simeq GL(n)/O(n,\Sigma_0)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
B˙=ΩB,B(0)=A
\dot B = \Omega B, \; B(0) = A
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Vertical gradient flow:
A
A
Σ1Ω+ΩΣ1=2Σ1(B−1−B−⊤)
\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
B˙=ΩB,B(0)=A
\dot B = \Omega B, \; B(0) = A
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Vertical gradient flow:
A
A
Σ1Ω+ΩΣ1=2Σ1(B−1−B−⊤)
\Sigma_1 \Omega + \Omega\Sigma_1 = 2\Sigma_1 (B^{-1}-B^{-\top})
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
Σ˙=Pr∇GˉHΣ1(Σ)
\dot \Sigma = \mathrm{Pr}\nabla_{\bar{\mathcal G}}H_{\Sigma_1}(\Sigma)
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
Σ(t)
\Sigma(t)
HΣ1(Σ)=−21tr(Σ1−1Σ)+21logdet(Σ1−1Σ)
\displaystyle H_{\Sigma_1}(\Sigma) = -\frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}\Sigma) + \frac{1}{2}\log\det(\Sigma_1^{-1}\Sigma)
Relative entropy
(Kullback-Leibler)
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
Σ˙=2I−Σ1−1Σ−ΣΣ1−1
\dot \Sigma = 2I - \Sigma_1^{-1}\Sigma - \Sigma\Sigma_1^{-1}
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
Σ(t)
\Sigma(t)
HΣ1(Σ)=−21tr(Σ1−1Σ)+21logdet(Σ1−1Σ)
\displaystyle H_{\Sigma_1}(\Sigma) = -\frac{1}{2}\mathrm{tr}(\Sigma_1^{-1}\Sigma) + \frac{1}{2}\log\det(\Sigma_1^{-1}\Sigma)
Relative entropy
(Kullback-Leibler)
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
P˙=P−1Σ0−1−Σ1−1P+V
\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
P(t)
P(t)
F(P)=HΣ1(PΣ0P)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on K for
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
P˙=P−1Σ0−1−Σ1−1P+V
\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
P(t)
P(t)
F(P)=HΣ1(PΣ0P)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on K for
Hessian of F(P) strictly positive on K ⇒ unique limit!
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
P˙=P−1Σ0−1−Σ1−1P+V
\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
P(t)
P(t)
F(P)=HΣ1(PΣ0P)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on K for
Hessian of F(P) strictly positive on K ⇒ unique limit!
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
P˙=P−1Σ0−1−Σ1−1P+V
\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
P(t)
P(t)
F(P)=HΣ1(PΣ0P)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on K for
Hessian of F(P) strictly positive on K ⇒ unique limit!
Bundle structure
GL(n)
GL(n)
P(n)
P(n)
I
I
Σ0
\Sigma_0
Σ1
\Sigma_1
P˙=P−1Σ0−1−Σ1−1P+V
\dot P = P^{-1}\Sigma_0^{-1} - \Sigma_1^{-1}P + V
GA(A˙,A˙)=tr(Σ0A˙⊤A˙)
\mathcal G_A(\dot A,\dot A) = \mathrm{tr}(\Sigma_0 \dot A^\top \dot A)
GˉΣ(Σ˙,Σ˙)=tr(ΣSS)
\bar{\mathcal G}_\Sigma(\dot \Sigma,\dot \Sigma) = \mathrm{tr}(\Sigma SS)
Σ˙=SΣ+ΣS
\dot\Sigma = S\Sigma + \Sigma S
K=P(n)
K = P(n)
Horizontal gradient (heat) flow:
P(t)
P(t)
F(P)=HΣ1(PΣ0P)
\displaystyle F(P) = H_{\Sigma_1}(P\Sigma_0 P)
Lifted gradient flow on K for
Hessian of F(P) strictly positive on K ⇒ unique limit!
Wasserstein-Otto vs. Fisher-Rao
Dens(M)
\mathrm{Dens}(M)
TμDens(M)≃C0∞(M)
T_\mu\mathrm{Dens}(M)\simeq C^\infty_0(M)
Wasserstein
Fisher-Rao
Gμ(μ˙,μ˙)=∫Mμμ˙μμ˙μ
\displaystyle\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) = \int_{M} \frac{\dot\mu}{\mu}\frac{\dot\mu}{\mu}\mu
Gρdx(ρ˙dx,ρ˙dx)=∫M∣∇θ∣2ρ
\displaystyle\overline{\mathcal{G}}_{\rho dx}(\dot\rho dx,\dot\rho dx) = \int_{M} |\nabla\theta|^2\rho
ρ˙+div(ρ∇θ)=0
\displaystyle \dot\rho + \mathrm{div}(\rho \nabla\theta) = 0
Dependent on Riemannian structure of M
Independent of Riemannian structure of M⇒Diff(M)-invariance
ρ(x)=(2π)ndet(Σ)1exp(−21x⊤Σ−1x)
\displaystyle \rho(x) = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(\Sigma)}}\mathrm{exp}(-\frac{1}{2}x^\top \Sigma^{-1}x)
ρ(x)=(2π)ndet(W)exp(−21x⊤Wx)
\displaystyle \rho(x) = \sqrt{\frac{\mathrm{det}(W)}{(2\pi)^n}}\mathrm{exp}(-\frac{1}{2}x^\top W x)
Brockett flow
P(n)
P(n)
N
N
W1
W_1
D(n)
D(n)
HN(W) relative entropy functional
Functional F(Q)=HN(Q⊤W1Q) on O(n)
Orb(W1)
\mathrm{Orb}(W_1)
HN(W)=−21tr(NW−1)+21logdet(NW−1)
\displaystyle H_{N}(W) = -\frac{1}{2}\mathrm{tr}(N W^{-1}) + \frac{1}{2}\log\det(N W^{-1})
Relative entropy
Heat flow
Wasserstein-Otto metric
Gμ(μ˙,μ˙)=∫M∣∇θ∣2μ
\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) = \int_M \lvert \nabla \theta\rvert^2 \mu
ρ˙+div(ρ∇θ)=0,ρ=μ/dx
\dot\rho + \operatorname{div}(\rho \nabla\theta) = 0, \; \rho = \mu/dx
⇒ Riemannian gradient flow ρ˙=−∇GF(ρ)
ρ˙=div(ρ∇δρδF)
\dot\rho = \operatorname{div}(\rho \nabla\frac{\delta F}{\delta\rho})
Take F(ρ)=∫Mlog(ρ)ρ⇒δF=log(ρ)+1
⇒ρ˙=Δρ
\Rightarrow \; \dot\rho = \Delta\rho
IPM and Toda
same potential, different Riemannian metrics:
IPM: L2 on velocity (H1 on stream function)
TODA: H−1 on velocity (L2 on stream function)
gradients flows on Diffμ(S2)
gravity
low density
(light particles)
high density
(heavy particles)
Diffμ(M2)
\mathrm{Diff}_\mu(M^2)
Id
\mathrm{Id}
ρ0
\rho_0
ρ1
\rho_1
IPM results
Toda results
IPM vs Toda results
IPM vs Toda results
Summary
- IPM and Toda ⇒ Riemannian gradient flows on Diffμ(M) (or quantized on SO(n))
- same potential function
- different (right-invariant) Riemannian metrics
IPM: L2 Toda: H−1 - Stronger metric ⇒ more regular flow
IPM: ODE on Diffμs(M) (for s>2)
Toda: not ODE on Diffμs(M)
Wasserstein-Otto geometry Klas Modin slides.com/kmodin
Wasserstein-Otto geometry
By Klas Modin
Wasserstein-Otto geometry
Tutorial talk given 2023-11 in Banff.
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