Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Invariant Riemannian metric on \(E\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
Invariant Riemannian metric on \(E\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot H \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
left co-sets \([g] = g\cdot G_{b_0} \)
Semi-invariant Riemannian metric on \(G\)
\(\Rightarrow\) \(\pi\) Riemannian submersion
polar cone
Monge problem, \(L^2\) version
Monge problem, \(L^2\) version
Riemannian metric
Induced metric
[Benamou & Brenier (2000), Otto (2001)]
Invariance: \(\eta\in\mathrm{Diff}_{\mu_0}(M)\)
Geodesic equation:
Easy to prove:
Polar cone \(K\) is isomorphic to strictly convex smooth functions via \(\phi \mapsto \nabla\phi\)
Hard to prove:
Polar cone \(K\) a section of principal bundle
Geodesic equation:
Easy to prove:
Polar cone \(K\) is isomorphic to strictly convex smooth functions via \(\phi \mapsto \nabla\phi\)
Hard to prove:
Polar cone \(K\) a section of principal bundle
Brenier's decomposition of transport maps
Geodesic curve:
In particular:
Geodesic curve:
In particular:
Geodesic curve:
In particular:
Trivial observation: \(\varphi_0(x) = A_0 x\), \(\varphi_1(x) = A_1 x\) linear diffeomorphisms \(\Rightarrow\) geodesic consists of linear diffeomorphisms
Consequence: \(GL(n)\) is totally geodesic subgroup of \(\operatorname{Diff}(\mathbb{R}^n)\)
Corresponding subspace of densities (statistical submanifold): multivariate Gaussians with zero mean
Monge-Ampere equation:
Factorization theorem:
Vertical gradient flow:
Vertical gradient flow:
Vertical gradient flow:
Horizontal gradient (heat) flow:
Relative entropy
(Kullback-Leibler)
Horizontal gradient (heat) flow:
Relative entropy
(Kullback-Leibler)
Horizontal gradient (heat) flow:
Lifted gradient flow on \(K\) for
Horizontal gradient (heat) flow:
Lifted gradient flow on \(K\) for
Hessian of \(F(P)\) strictly positive on \(K\) \(\Rightarrow\) unique limit!
Horizontal gradient (heat) flow:
Lifted gradient flow on \(K\) for
Hessian of \(F(P)\) strictly positive on \(K\) \(\Rightarrow\) unique limit!
Horizontal gradient (heat) flow:
Lifted gradient flow on \(K\) for
Hessian of \(F(P)\) strictly positive on \(K\) \(\Rightarrow\) unique limit!
Horizontal gradient (heat) flow:
Lifted gradient flow on \(K\) for
Hessian of \(F(P)\) strictly positive on \(K\) \(\Rightarrow\) unique limit!
Wasserstein
Fisher-Rao
Dependent on Riemannian structure of \(M\)
Independent of Riemannian structure of \(M \Rightarrow \mathrm{Diff}(M)\)-invariance
\(H_N(W)\) relative entropy functional
Functional \(F(Q) = H_N(Q^\top W_1 Q)\) on \(O(n)\)
Relative entropy
Wasserstein-Otto metric
\(\Rightarrow\) Riemannian gradient flow \(\dot\rho = -\nabla_{\overline{\mathcal G}}F(\rho)\)
Take \(F(\rho) = \int_M \log(\rho) \rho \Rightarrow \delta F = \log(\rho)+1\)
same potential, different Riemannian metrics:
IPM: \(L^2\) on velocity (\(H^{1}\) on stream function)
TODA: \(H^{-1}\) on velocity (\(L^2\) on stream function)
gradients flows on \(\mathrm{Diff}_\mu(S^2)\)
gravity
low density
(light particles)
high density
(heavy particles)
By Klas Modin
Tutorial talk given 2023-11 in Banff.
Mathematician at Chalmers University of Technology and the University of Gothenburg