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

\dot S + H(x,\nabla S) = 0

\displaystyle H(q,p) = |p|^2, \quad (q,p)\in 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

\dot S + |\nabla S|^2 = 0

P^\infty(M) = \{\rho\in C^\infty(M)\mid \rho(x)>0, \int_M \rho dx = 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

\dot S + |\nabla S|^2 = 0

T_\rho P^\infty(M) = C^\infty_0(M)

T_\rho P^\infty(M) = C^\infty_0(M)

T_\rho^* P^\infty(M) \simeq C^\infty(M)/\mathbb{R}

T_\rho^* P^\infty(M) \simeq C^\infty(M)/\mathbb{R}

\dot \rho + \mathrm{div}(\rho\nabla S) = 0

\dot \rho + \mathrm{div}(\rho\nabla S) = 0

(\rho,S)\in T^*P^\infty(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

\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

d_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

\mu_0 = \rho_0 dx, \quad \mu_1 = \rho_1 dx

Geodesic completeness failure

P^\infty(M)

P^\infty(M)

\rho_0

\rho_0

\rho(t)

\rho(t)

breakdown

\dot\rho_0

\dot\rho_0

Fluid formulation

\dot S + u\cdot\nabla S = 0

\dot S + u\cdot\nabla S = 0

\dot \rho + \mathrm{div}(\rho u) = 0

\dot \rho + \mathrm{div}(\rho u) = 0

u=\nabla S

u=\nabla S

Transport of $\rho\, dx$ and $S$ by $u$

\dot S + \mathcal{L}_u S = 0

\dot S + \mathcal{L}_u S = 0

\dot \rho\, dx + \mathcal{L}_u(\rho\, dx) = 0

\dot \rho\, dx + \mathcal{L}_u(\rho\, dx) = 0

Inviscid Burgers'

\dot u + \nabla_u u = 0

\dot u + \nabla_u u = 0

(\, \dot u + u u_x = 0 \,)

(\, \dot u + u u_x = 0 \,)

Shock formation

u

u

x

x

Riemannian submersion viewpoint

\mathrm{Diff}(M)

\mathrm{Diff}(M)

P^\infty(M)

P^\infty(M)

\mathrm{Id}

\mathrm{Id}

\rho_0

\rho_0

\rho_1

\rho_1

\pi(\eta)=\eta_*(\rho_0 dx)

\pi(\eta)=\eta_*(\rho_0 dx)

Moser 1965:

Principal bundle

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

\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)

{\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)

\mathrm{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

\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)

\displaystyle \nabla_{\dot\eta(x)}\dot\eta(x) = 0 \qquad (\ddot\eta = 0)

Geodesic equation

Symplectic reduction viewpoint

T^*\mathrm{Diff}(M)

T^*\mathrm{Diff}(M)

T^*\mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}\simeq P^\infty(M) \times \mathfrak{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^*\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

\dot S + u\cdot\nabla S = 0

\dot \rho + \mathrm{div}(\rho u) = 0

\dot \rho + \mathrm{div}(\rho u) = 0

u=\nabla S

u=\nabla S

\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx

\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx

(1-\Delta)^{k+1}u=\rho\nabla S

(1-\Delta)^{k+1}u=\rho\nabla S

Main result

\dot S + u\cdot\nabla S = 0

\dot S + u\cdot\nabla S = 0

\dot \rho + \mathrm{div}(\rho u) = 0

\dot \rho + \mathrm{div}(\rho u) = 0

(1-\Delta)^{k+1}u=\rho\nabla 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)

\mathrm{Diff}^{k+1}(M)

P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}

P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}

\mathrm{Id}

\mathrm{Id}

\rho_0

\rho_0

\pi(\eta)=\eta_*(\rho_0 dx)

\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-\Delta)^{k+1} u

\dot m + \nabla_u m + m\,\mathrm{div}(u) + (\nabla u)^\top 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)

\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

\varepsilon

\mathrm{id}

\mathrm{id}

u_0

u_0

\eta(1)

\eta(1)

\dot\eta(1)

\dot\eta(1)

u_1 = \dot\eta(1)\circ\eta(1)^{-1}

u_1 = \dot\eta(1)\circ\eta(1)^{-1}

u_1

u_1

\eta(2)

\eta(2)

Banach to Fréchet

\mathrm{Diff}^{k+1}(M)

\mathrm{Diff}^{k+1}(M)

P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}

P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}

\mathrm{Id}

\mathrm{Id}

\rho_0

\rho_0

\pi(\eta)=\eta_*(\rho_0 dx)

\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^\infty(M)\simeq \mathrm{Diff}(M)/\mathrm{Diff}_{\mu_0}

\mathrm{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

Global results for geodesic equations on densities

Klas Modin

Joint work with

Outline

Hamilton–Jacobi PDE

Hamilton–Jacobi PDE

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

Induced Riemannian distance = $L^2$ -Wasserstein distance

Optimal transport

Geodesic completeness failure

Fluid formulation

Shock formation

Riemannian submersion viewpoint

Symplectic reduction viewpoint

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

Main result

Infinite-dimensional manifold setting

Ebin–Marsden analysis for local results

Local to global result

Banach to Fréchet

Outlook

THANKS!