Global results for geodesic equations on densities

Klas Modin

Joint work with

Sarang Joshi

University of Utah

Martin Bauer

Florida State University

Outline

  • Riemannian geometry of optimal mass transport
  • Geodesic completeness issues
  • Main result
  • Ebin–Marsden-type analysis
  • From local to global
  • From Banach to Fréchet manifolds
  • Outlook

Hamilton–Jacobi PDE

\dot S + H(x,\nabla S) = 0
S˙+H(x,∇S)=0\dot S + H(x,\nabla S) = 0
\displaystyle H(q,p) = |p|^2, \quad (q,p)\in T^*M
H(q,p)=∣p∣2,(q,p)∈T∗M\displaystyle H(q,p) = |p|^2, \quad (q,p)\in T^*M

On Riemannian manifold \(M\) consider

Hamilton–Jacobi PDE

\dot S + |\nabla S|^2 = 0
S˙+∣∇S∣2=0\dot S + |\nabla S|^2 = 0
P^\infty(M) = \{\rho\in C^\infty(M)\mid \rho(x)>0, \int_M \rho dx = 1 \}
P∞(M)={ρ∈C∞(M)∣ρ(x)>0,∫Mρdx=1}P^\infty(M) = \{\rho\in C^\infty(M)\mid \rho(x)>0, \int_M \rho dx = 1 \}

On Riemannian manifold \(M\) consider

Geodesic equation on probability density functions

How?

\(H^{-1}\) geodesics on \(P^\infty(M)\)

\dot S + |\nabla S|^2 = 0
S˙+∣∇S∣2=0\dot S + |\nabla S|^2 = 0
T_\rho P^\infty(M) = C^\infty_0(M)
TρP∞(M)=C0∞(M)T_\rho P^\infty(M) = C^\infty_0(M)
T_\rho^* P^\infty(M) \simeq C^\infty(M)/\mathbb{R}
Tρ∗P∞(M)≃C∞(M)/RT_\rho^* P^\infty(M) \simeq C^\infty(M)/\mathbb{R}
\dot \rho + \mathrm{div}(\rho\nabla S) = 0
ρ˙+div(ρ∇S)=0\dot \rho + \mathrm{div}(\rho\nabla S) = 0
(\rho,S)\in T^*P^\infty(M)
(ρ,S)∈T∗P∞(M)(\rho,S)\in T^*P^\infty(M)

Hamiltonian form of geodesic equation for Otto metric

\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx
Gρ(ρ˙,ρ˙)=∫MSρ˙dx\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx

Induced Riemannian distance = \(L^2\)-Wasserstein distance

d_W^2(\mu_0,\mu_1) = \inf_{\eta_*\mu_0=\mu_1} \int_{M} \mathrm{dist}^2(\eta(x),x) d\mu_0
dW2(μ0,μ1)=infη∗μ0=μ1∫Mdist2(η(x),x)dμ0d_W^2(\mu_0,\mu_1) = \inf_{\eta_*\mu_0=\mu_1} \int_{M} \mathrm{dist}^2(\eta(x),x) d\mu_0

Symmetric by change of variables

\mu_0 = \rho_0 dx, \quad \mu_1 = \rho_1 dx
μ0=ρ0dx,μ1=ρ1dx\mu_0 = \rho_0 dx, \quad \mu_1 = \rho_1 dx

Optimal transport

\mu_0
μ0\mu_0
\mu_1
μ1\mu_1
\eta_*\mu_0
η∗μ0\eta_*\mu_0
(L^2)
(L2)(L^2)
\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
minη∗μ0=μ1∫Rn∣η(x)−x∣2dμ0\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
\mathbb{R}^n
Rn\mathbb{R}^n

Monge

Geodesic completeness failure

P^\infty(M)
P∞(M)P^\infty(M)
\rho_0
ρ0\rho_0
\rho(t)
ρ(t)\rho(t)

breakdown

\dot\rho_0
ρ˙0\dot\rho_0

Fluid formulation

\dot S + u\cdot\nabla S = 0
S˙+u⋅∇S=0\dot S + u\cdot\nabla S = 0
\dot \rho + \mathrm{div}(\rho u) = 0
ρ˙+div(ρu)=0\dot \rho + \mathrm{div}(\rho u) = 0
u=\nabla S
u=∇Su=\nabla S

Transport of \(\rho\, dx\) and \(S\) by \(u\)

\dot S + \mathcal{L}_u S = 0
S˙+LuS=0\dot S + \mathcal{L}_u S = 0
\dot \rho\, dx + \mathcal{L}_u(\rho\, dx) = 0
ρ˙dx+Lu(ρdx)=0\dot \rho\, dx + \mathcal{L}_u(\rho\, dx) = 0

Inviscid Burgers'

\dot u + \nabla_u u = 0
u˙+∇uu=0\dot u + \nabla_u u = 0
(\, \dot u + u u_x = 0 \,)
(u˙+uux=0)(\, \dot u + u u_x = 0 \,)

Shock formation

u
uu
x
xx

Riemannian submersion viewpoint

\mathrm{Diff}(M)
Diff(M)\mathrm{Diff}(M)
P^\infty(M)
P∞(M)P^\infty(M)
\mathrm{Id}
Id\mathrm{Id}
\rho_0
ρ0\rho_0
\rho_1
ρ1\rho_1
\pi(\eta)=\eta_*(\rho_0 dx)
π(η)=η∗(ρ0dx)\pi(\eta)=\eta_*(\rho_0 dx)

Moser 1965:

Principal bundle

\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)
Diff(M)/Diffμ0(M)\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}(M)

\(L^2\) metric on \(\mathrm{Diff}(M)\)

\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0
G‾η(η˙,η˙)=∫M∣η˙∣2μ0\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0

Induces Otto metric

{\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)
Gρ(ρ˙,ρ˙)⇒dW2(μ0,μ1){\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)
\mathrm{Hor}
Hor\mathrm{Hor}
\displaystyle\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0 = \int_{M}\varphi_*(\left\vert \dot\eta \right\vert^2 \mu_0) = \int_{M}\left\vert \dot\eta\circ\varphi^{-1} \right\vert^2 \varphi_*\mu_0
G‾η(η˙,η˙)=∫M∣η˙∣2μ0=∫Mφ∗(∣η˙∣2μ0)=∫M∣η˙∘φ−1∣2φ∗μ0\displaystyle\overline{\mathcal{G}}_\eta(\dot\eta,\dot\eta) = \int_{M}\left\vert \dot\eta \right\vert^2 \mu_0 = \int_{M}\varphi_*(\left\vert \dot\eta \right\vert^2 \mu_0) = \int_{M}\left\vert \dot\eta\circ\varphi^{-1} \right\vert^2 \varphi_*\mu_0
\displaystyle \nabla_{\dot\eta(x)}\dot\eta(x) = 0 \qquad (\ddot\eta = 0)
∇η˙(x)η˙(x)=0(η¨=0)\displaystyle \nabla_{\dot\eta(x)}\dot\eta(x) = 0 \qquad (\ddot\eta = 0)

Geodesic equation

Symplectic reduction viewpoint

T^*\mathrm{Diff}(M)
T∗Diff(M)T^*\mathrm{Diff}(M)
T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}\simeq P^\infty(M) \times \mathfrak{X}^*
T∗Diff(M)/Diffμ0≃P∞(M)×X∗T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}\simeq P^\infty(M) \times \mathfrak{X}^*
T^*\mathrm{Diff}(M)//\mathrm{Diff}_{\mu_0}\simeq T^*P^\infty(M)
T∗Diff(M)//Diffμ0≃T∗P∞(M)T^*\mathrm{Diff}(M)//\mathrm{Diff}_{\mu_0}\simeq T^*P^\infty(M)

Poisson reduction

Symplectic reduction

Higher order metrics on \(P^\infty(M)\)

\dot S + u\cdot\nabla S = 0
S˙+u⋅∇S=0\dot S + u\cdot\nabla S = 0
\dot \rho + \mathrm{div}(\rho u) = 0
ρ˙+div(ρu)=0\dot \rho + \mathrm{div}(\rho u) = 0
u=\nabla S
u=∇Su=\nabla S
\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx
Gρ(ρ˙,ρ˙)=∫MSρ˙dx\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx
(1-\Delta)^{k+1}u=\rho\nabla S
(1−Δ)k+1u=ρ∇S(1-\Delta)^{k+1}u=\rho\nabla S

Main result

\dot S + u\cdot\nabla S = 0
S˙+u⋅∇S=0\dot S + u\cdot\nabla S = 0
\dot \rho + \mathrm{div}(\rho u) = 0
ρ˙+div(ρu)=0\dot \rho + \mathrm{div}(\rho u) = 0
(1-\Delta)^{k+1}u=\rho\nabla S
(1−Δ)k+1u=ρ∇S(1-\Delta)^{k+1}u=\rho\nabla S

Theorem \(M\) closed Riemannian \(n\)-manifold and \(k>n/2\).

1. For smooth initial data \(\rho_0,S_0\), there is a unique solution \(\rho\in C^\infty(\mathbb{R}\times M)\), \(S \in C^\infty(\mathbb{R}\times M)/\mathbb{R}\).

2. Solution depends smoothly on initial data.

Geodesic completeness!

Infinite-dimensional manifold setting

\mathrm{Diff}^{k+1}(M)
Diffk+1(M)\mathrm{Diff}^{k+1}(M)
P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
Pk(M)≃Diffk+1(M)/Diffμ0k+1P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
\mathrm{Id}
Id\mathrm{Id}
\rho_0
ρ0\rho_0
\pi(\eta)=\eta_*(\rho_0 dx)
π(η)=η∗(ρ0dx)\pi(\eta)=\eta_*(\rho_0 dx)

Option 1: Fréchet manifolds \(\rightarrow\) no fixed-point theorem

Option 2: Banach manifolds \(\rightarrow\) only topological group

only \(C^0\)-map

Ebin–Marsden analysis for local results

1. Fluid formulation (EPDiff equation)

m = (1-\Delta)^{k+1} u
m=(1−Δ)k+1um = (1-\Delta)^{k+1} u
\dot m + \nabla_u m + m\,\mathrm{div}(u) + (\nabla u)^\top m = 0
m˙+∇um+mdiv(u)+(∇u)⊤m=0\dot m + \nabla_u m + m\,\mathrm{div}(u) + (\nabla u)^\top m = 0

2. Write equation in \(\eta\in \mathrm{Diff}^{k+1}(M)\) and \(\dot\eta = u\circ\eta\in T_\eta\mathrm{Diff}^{k+1}(M)\)

\ddot\eta = F(\eta,\dot\eta)
η¨=F(η,η˙)\ddot\eta = F(\eta,\dot\eta)

3. Prove this is an ODE on \(T\mathrm{Diff}^{k+1}(M)\)

(\(F\) is smooth geodesic spray)

4. Picard iterations \(\rightarrow\) local existence and uniqueness

Local to global result

\varepsilon
ε\varepsilon
\mathrm{id}
id\mathrm{id}
u_0
u0u_0
\eta(1)
η(1)\eta(1)
\dot\eta(1)
η˙(1)\dot\eta(1)
u_1 = \dot\eta(1)\circ\eta(1)^{-1}
u1=η˙(1)∘η(1)−1u_1 = \dot\eta(1)\circ\eta(1)^{-1}
u_1
u1u_1
\eta(2)
η(2)\eta(2)

Banach to Fréchet

\mathrm{Diff}^{k+1}(M)
Diffk+1(M)\mathrm{Diff}^{k+1}(M)
P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
Pk(M)≃Diffk+1(M)/Diffμ0k+1P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
\mathrm{Id}
Id\mathrm{Id}
\rho_0
ρ0\rho_0
\pi(\eta)=\eta_*(\rho_0 dx)
π(η)=η∗(ρ0dx)\pi(\eta)=\eta_*(\rho_0 dx)

only \(C^0\)-map

Problem: \(H^{k+1}\) geodesics do not descend

Remedy:

1. \(H^{k+1}\) existence interval contained in \(H^{k+2}\) interval

2. Inverse limit topology \(k\to\infty\) gives Fréchet manifold \(\mathrm{Diff}(M)\)

P^\infty(M)\simeq \mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}
P∞(M)≃Diff(M)/Diffμ0P^\infty(M)\simeq \mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}
\mathrm{Diff}(M)
Diff(M)\mathrm{Diff}(M)

\(C^\infty\)-map

Outlook

  • Numerical methods for new high-order geodesic equations on \(P^\infty(M) \)
     
  • Applications in imaging and shape analysis (initial motivation)

THANKS!

Reference:

On Geodesic Completeness of Riemannian Metrics on Smooth Probability Densities , Calc. Var. PDE, 2017

doi:10.1007/s00526-017-1195-8