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

## Optimal transport

\mu_0
$\mu_0$
\mu_1
$\mu_1$
\eta_*\mu_0
$\eta_*\mu_0$
(L^2)
$(L^2)$
\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
$\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0$
\mathbb{R}^n
$\mathbb{R}^n$

Monge

## 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

By Klas Modin

# Global results for geodesic equations on densities

Presentation given 2017-07 at the CRiSP Workshop in Trondheim.

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