Sub-Riemannian structures on groups of diffeomorphisms E. Trlat 1 - PowerPoint PPT Presentation

SR structure on D s ( M ) Definitions Geodesics Reachability Sub-Riemannian structures on groups of diffeomorphisms E. Trélat 1 S. Arguillère 1 Univ. Paris 6 (Labo. J.-L. Lions) et Institut Universitaire de France Infinite-dimensional Riemannian geometry with applications to image matching and shape analysis Wien, Jan. 2015 E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Definitions ( M , g ) smooth oriented Riemannian manifold of dimension d , of bounded geometry Definition D s ( M ) = H s 0 ( M , M ) ∩ Diff 1 ( M ) → connected component of e = id M in the space of diffeomorphisms of class H s on M . We assume that s > d / 2 + 1. Then : D s ( M ) is an Hilbert manifold, and is an open subset of H s ( M , M ) D s ( M ) is a topological group for the composition ( ϕ, ψ ) �→ ϕ ◦ ψ right composition is smooth, but left composition is only continuous “ ” D s ( M ) is not a Lie group, but D ∞ ( M ) = \ D s ( M ) is a ILH Lie group s > d / 2 + 1 Omori Ebin Marsden Eichorn Schmid Kriegl Michor... E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Γ s ( TM ) = set of vector fields of class H s on M . We identify : Γ s ( TM ) ≃ T e D s ( M ) ≃ set of right-invariant vector fields X on D s ( M ) (satisfying X ( ϕ ) = X ( e ) ◦ ϕ ∀ ϕ ) Curves on D s ( M ) : ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) ϕ ( · ) ◦ ϕ ( · ) − 1 ∈ L 2 ( 0 , 1 ; Γ s ( TM )) : time-dependent right-invariant vector X ( · ) = ˙ field called logarithmic velocity of ϕ ( · ) , and we have ϕ ( t ) = X ( t ) ◦ ϕ ( t ) for a.e. t ∈ [ 0 , 1 ] . ˙ ∀ ϕ 0 ∈ D s ( M ) fixed : bijective correspondence ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) ← → X ( · ) ∈ L 2 ( 0 , 1 ; Γ s ( TM )) such that ϕ ( 0 ) = ϕ 0 E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Sub-Riemannian structure on D s ( M ) Reminder : sub-Riemannian (SR) structure ( M , ∆ , h ) : M manifold ∆ ⊂ TM subbundle (horizontal distribution) h Riemannian metric on ∆ Objective : define a right-invariant SR structure on D s ( M ) , i.e., a right-invariant subbundle H of T D s ( M ) , endowed with a right-invariant Riemannian metric E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Sub-Riemannian structure on D s ( M ) s > d / 2 + 1 , k ∈ I N ( H e , �· , ·� ) : (arbitrary) Hilbert space of vector fields of class H s + k on M → Γ s + k ( TM ) with continuous inclusion H e ֒ In practice : H e is often defined by its kernel (as a RKHS, e.g. consisting of analytic vector fields) (example : heat kernel, in shape deformation analysis) Definition Subbundle H s ⊂ T D s ( M ) : ∀ ϕ ∈ D s ( M ) H s ϕ = R ϕ H e = H e ◦ ϕ endowed with the metric � X , Y � ϕ = � X ◦ ϕ − 1 , Y ◦ ϕ − 1 � Note that H s is parametrized by D s ( M ) × H e , with the mapping ( ϕ, X ) �→ X ◦ ϕ (of class C k ) → “strong" right-invariant sub-Riemannian structure ( D s ( M ) , H s , � , � ) � = “weak", for which H e not complete. Example : H e = volume-preserving vector fields (Arnold, Ebin Marsden). See also Grong Markina Vasil’ev. E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Examples Two typical situations motivate the use of SR geometry in imaging and shape analysis : 1 either H e is a dense horizontal distribution, defined e.g. with an exponential reproducing kernel Bauer Bruveris Michor Mumford Gay-Balmaz Trouvé Vialard Younes ... or H e is a set of vector fields that are horizontal for a given SR structure on M : 2 image tracking with missing data, reconstruction of corrupted images with hypoelliptic Laplacians Boscain Gauthier Rossi Sachkov Grong Markina Vasil’ev... E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Examples s + k Z g x ( ∇ i X ( x ) , ∇ i Y ( x )) d x g Example 1 : H e = Γ s + k ( TM ) endowed with � X , Y � = X M i = 1 → dense horizontal distribution : dense subset of vector fields of class H s + k in H s For k = 0 : Riemannian structure on D s ( M ) Bauer Bruveris Michor Mumford Gay-Balmaz Trouvé Vialard Younes ... Example 2 : let ( M , ∆ , g ) be a SR structure. We take n o X ∈ Γ s + k ( TM ) | ∀ x ∈ M H e = X ( x ) ∈ ∆ x set of all horizontal vector fields on M of class H s + k (endowed with H s + k norm) Boscain Gauthier Rossi Sachkov Khesin Lee Grong Markina Vasil’ev... E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Horizontal curves On the strong right-invariant SR structure ( D s ( M ) , H s , �· , ·� ) : Definition Horizontal curve : ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) ϕ ( t ) ∈ H s | ˙ a.e. ϕ ( t ) ϕ ( · ) ◦ ϕ ( · ) − 1 ∈ L 2 ( 0 , 1 ; H e ) ⇔ X ( · ) = ˙ (horizontal vector field) Let ϕ 0 ∈ D s ( M ) . For k � 1 : Ω ϕ 0 = { horizontal curves ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) | ϕ ( 0 ) = ϕ 0 } . → C k submanifold of H 1 ( 0 , 1 ; D s ( M )) D s ( M ) end ϕ 0 : Ω ϕ 0 − → (of class C k ) End-point mapping : ϕ ( · ) �− → ϕ ( 1 ) E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Singular curves Given ϕ 0 and ϕ 1 in D s ( M ) : Ω ϕ 0 ,ϕ 1 = end − 1 ϕ 0 ( { ϕ 1 } ) (set of horizontal curves steering ϕ 0 to ϕ 1 ) → need not be a manifold Definition For k � 1 : an horizontal curve ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) is singular if ∃ P ϕ 1 ∈ T ∗ ϕ 1 D s ( M ) \ { 0 } | ( d end ϕ 0 ( ϕ ( · ))) ∗ . P ϕ 1 = 0 ⇔ codim T ϕ 1 D s ( M ) ( Range ( d end ϕ 0 ( ϕ ( · )))) > 0 Examples of singular curves of diffeomorphisms : - take any finite-dimensional SR structure ( M , ∆ , h ) on which there exists a nontrivial singular curve γ ( · ) - take H e = set of horizontal vector fields of class H s + k on M Then any autonomous vector field X such that X ◦ γ ( · ) = ˙ γ ( · ) generates a singular curve of diffeomorphisms. E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Three possibilities : Range ( d end ϕ 0 ( ϕ ( · ))) = T ϕ 1 D s ( M ) 1 → ϕ ( · ) regular , Ω ϕ 0 ,ϕ 1 local manifold codim T ϕ 1 D s ( M ) ( Range ( d end ϕ 0 ( ϕ ( · )))) > 0 2 singular curve Range ( d end ϕ 0 ( ϕ ( · ))) proper dense subset of T ϕ 1 D s ( M ) 3 - The first possibility never occurs because H e is never closed in Γ s ( TM ) ( k � 1 ) - The third possibility is due to infinite dimension E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability SR distance Definition Given a horizontal curve ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) , with logarithmic velocity ϕ ( · ) ◦ ϕ ( · ) − 1 ∈ L 2 ( 0 , 1 ; H e ) , we define the length and the action : X ( · ) = ˙ Z 1 Z 1 A ( ϕ ( · )) = 1 p L ( ϕ ( · )) = � X ( t ) , X ( t ) � d t and � X ( t ) , X ( t ) � d t . 2 0 0 Then : d SR ( ϕ 0 , ϕ 1 ) = inf { L ( ϕ ( · )) | ϕ ( · ) ∈ Ω ϕ 0 ,ϕ 1 } - ϕ ( · ) is said to be minimizing if d SR ( ϕ ( 0 ) , ϕ ( 1 )) = L ( ϕ ( · )) . - Minimizing the length = minimizing the action. Theorem d SR is a right-invariant distance ( d SR ( ϕ 0 , ϕ 1 ) = 0 ⇒ ϕ 0 = ϕ 1 ) ∀ ϕ 0 , ϕ 1 ∈ D s ( M ) s.t. d SR ( ϕ 0 , ϕ 1 ) < + ∞ , ∃ ϕ ( · ) minimizer steering ϕ 0 to ϕ 1 . ( D s ( M ) , d SR ) is a complete metric space. E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Geodesics on D s ( M ) A geodesic ϕ ( · ) ∈ H 1 ( 0 , 1 ; D s ( M )) is a critical point of the action A | Ω ϕ ( 0 ) ,ϕ ( 1 ) . Any minimizing horizontal curve is a geodesic. Objective : derive first-order conditions for geodesics (Pontryagin maximum principle) E. Trélat Sub-Riemannian structures on groups of diffeomorphisms

SR structure on D s ( M ) Definitions Geodesics Reachability Geodesics on D s ( M ) Preliminary discussion : Lagrange multipliers ϕ ( · ) ◦ ϕ ( · ) − 1 is Let ϕ ( · ) ∈ Ω ϕ 0 ,ϕ 1 be a minimizer. Then ϕ ( · ) is a geodesic, and X ( · ) = ˙ a critical point of F ϕ 0 : L 2 ( 0 , 1 ; H e ) D s ( M ) × I − → R ( end ϕ 0 ( ϕ X ( · )) , A ( ϕ X ( · ))) X ( · ) �− → There are 2 cases : First case : ker (( d F ϕ 0 ( X ( · ))) ∗ ) � = { 0 } R ( Range ( d F ϕ 0 ( X ( · )))) > 0 ⇔ codim T ϕ 1 D s ( M ) × I i.e., there exists a Lagrange multiplier ( P ϕ 1 , p 0 ) ∈ T ∗ ϕ 1 D s ( M ) × I R \ { ( 0 , 0 ) } s.t. ( d end ϕ 0 ( ϕ ( · ))) ∗ . P ϕ 1 + p 0 d A ( ϕ ( · )) = 0 a) either p 0 � = 0 (normal case) → p 0 = − 1 → normal geodesics b) or p 0 = 0 (abnormal case) → abnormal geodesics (and ϕ ( · ) singular curve) E. Trélat Sub-Riemannian structures on groups of diffeomorphisms