Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
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
\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
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
T_\rho P^\infty(M) = C^\infty_0(M)
T_\rho^* P^\infty(M) \simeq C^\infty(M)/\mathbb{R}
\dot \rho + \mathrm{div}(\rho\nabla S) = 0
(\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
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
Symmetric by change of variables
\mu_0 = \rho_0 dx, \quad \mu_1 = \rho_1 dx
Optimal transport
\mu_0
\mu_1
\eta_*\mu_0
(L^2)
\min_{\eta_*\mu_0=\mu_1} \int_{\mathbb{R}^n} \left|\eta(x)-x \right|^2 d\mu_0
\mathbb{R}^n
Monge
Geodesic completeness failure
P^\infty(M)
\rho_0
\rho(t)
breakdown
\dot\rho_0
Fluid formulation
\dot S + u\cdot\nabla S = 0
\dot \rho + \mathrm{div}(\rho u) = 0
u=\nabla S
Transport of \(\rho\, dx\) and \(S\) by \(u\)
\dot S + \mathcal{L}_u S = 0
\dot \rho\, dx + \mathcal{L}_u(\rho\, dx) = 0
Inviscid Burgers'
\dot u + \nabla_u u = 0
(\, \dot u + u u_x = 0 \,)
Shock formation
u
x
Riemannian submersion viewpoint
\mathrm{Diff}(M)
P^\infty(M)
\mathrm{Id}
\rho_0
\rho_1
\pi(\eta)=\eta_*(\rho_0 dx)
Moser 1965:
Principal bundle
\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
Induces Otto metric
{\mathcal{G}}_\rho(\dot\rho,\dot\rho) \Rightarrow d_W^2(\mu_0,\mu_1)
\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 \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)/\mathrm{Diff}_{\mu_0}\simeq P^\infty(M) \times \mathfrak{X}^*
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 \rho + \mathrm{div}(\rho u) = 0
u=\nabla S
\displaystyle\mathcal G_\rho(\dot\rho,\dot\rho) = \int_M S\dot\rho \, dx
(1-\Delta)^{k+1}u=\rho\nabla S
Main result
\dot S + u\cdot\nabla S = 0
\dot \rho + \mathrm{div}(\rho u) = 0
(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)
P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
\mathrm{Id}
\rho_0
\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
\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)
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
\mathrm{id}
u_0
\eta(1)
\dot\eta(1)
u_1 = \dot\eta(1)\circ\eta(1)^{-1}
u_1
\eta(2)
Banach to Fréchet
\mathrm{Diff}^{k+1}(M)
P^k(M)\simeq \mathrm{Diff}^{k+1}(M)/\mathrm{Diff}^{k+1}_{\mu_0}
\mathrm{Id}
\rho_0
\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}
\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
Global results for geodesic equations on densities
By Klas Modin
Global results for geodesic equations on densities
Presentation given 2017-07 at the CRiSP Workshop in Trondheim.
