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

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Definitions SR structure on Ds(M) Geodesics Reachability

Sub-Riemannian structures on groups of diffeomorphisms

S. Arguillère
E. Trélat1
1Univ. 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

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Definitions SR structure on Ds(M) Geodesics Reachability

Definitions

(M, g) smooth oriented Riemannian manifold of dimension d, of bounded geometry Definition Ds(M) = Hs

0(M, M) ∩ Diff1(M)

→ connected component of e = idM in the space of diffeomorphisms of class Hs on M. We assume that s > d/2 + 1. Then : Ds(M) is an Hilbert manifold, and is an open subset of Hs(M, M) Ds(M) is a topological group for the composition (ϕ, ψ) → ϕ ◦ ψ right composition is smooth, but left composition is only continuous “ Ds(M) is not a Lie group, but D∞(M) = \

s>d/2+1

Ds(M) is a ILH Lie group ”

Omori Ebin Marsden Eichorn Schmid Kriegl Michor...

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Γs(TM) = set of vector fields of class Hs on M. We identify : Γs(TM) ≃ TeDs(M) ≃ set of right-invariant vector fields X on Ds(M) (satisfying X(ϕ) = X(e) ◦ ϕ ∀ϕ) Curves on Ds(M) : ϕ(·) ∈ H1(0, 1; Ds(M)) X(·) = ˙ ϕ(·) ◦ ϕ(·)−1 ∈ L2(0, 1; Γs(TM)) : time-dependent right-invariant vector field called logarithmic velocity of ϕ(·), and we have ˙ ϕ(t) = X(t) ◦ ϕ(t) for a.e. t ∈ [0, 1]. ∀ϕ0 ∈ Ds(M) fixed : X(·) ∈ L2(0, 1; Γs(TM)) bijective correspondence

← →

ϕ(·) ∈ H1(0, 1; Ds(M)) such that ϕ(0) = ϕ0

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Sub-Riemannian structure on Ds(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 Ds(M),

i.e., a right-invariant subbundle H of TDs(M), endowed with a right-invariant Riemannian metric

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Sub-Riemannian structure on Ds(M)

s > d/2 + 1, k ∈ I N (He, ·, ·) : (arbitrary) Hilbert space of vector fields of class Hs+k on M with continuous inclusion He ֒ → Γs+k(TM)

In practice : He 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 Hs ⊂ TDs(M) : ∀ϕ ∈ Ds(M) Hs

ϕ = RϕHe = He ◦ ϕ

endowed with the metric X, Yϕ = X ◦ ϕ−1, Y ◦ ϕ−1

Note that Hs is parametrized by Ds(M) × He, with the mapping (ϕ, X) → X ◦ ϕ (of class Ck )

→ “strong" right-invariant sub-Riemannian structure (Ds(M), Hs, , )

= “weak", for which He not complete. Example : He = volume-preserving vector fields (Arnold, Ebin Marsden). See also Grong Markina Vasil’ev.

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Examples

Two typical situations motivate the use of SR geometry in imaging and shape analysis :

1

either He is a dense horizontal distribution, defined e.g. with an exponential reproducing kernel

Bauer Bruveris Michor Mumford Gay-Balmaz Trouvé Vialard Younes ... 2

r He is a set of vector fields that are horizontal for a given SR structure on M :

image tracking with missing data, reconstruction of corrupted images with hypoelliptic Laplacians

Boscain Gauthier Rossi Sachkov Grong Markina Vasil’ev...

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Examples

Example 1 : He = Γs+k(TM) endowed with X, Y =

s+k

X

i=1

Z

M

gx (∇i X(x), ∇i Y(x)) dxg

→ dense horizontal distribution : dense subset of vector fields of class Hs+k in Hs

For k = 0 : Riemannian structure on Ds(M) Bauer Bruveris Michor Mumford Gay-Balmaz Trouvé Vialard Younes ...

Example 2 : let (M, ∆, g) be a SR structure. We take He = n X ∈ Γs+k(TM) | ∀x ∈ M X(x) ∈ ∆x

set of all horizontal vector fields on M of class Hs+k (endowed with Hs+k norm)

Boscain Gauthier Rossi Sachkov Khesin Lee Grong Markina Vasil’ev...

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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

Horizontal curves

On the strong right-invariant SR structure (Ds(M), Hs, ·, ·) : Definition Horizontal curve : ϕ(·) ∈ H1(0, 1; Ds(M)) | ˙ ϕ(t) ∈ Hs

ϕ(t)

a.e. ⇔ X(·) = ˙ ϕ(·) ◦ ϕ(·)−1 ∈ L2(0, 1; He) (horizontal vector field) Let ϕ0 ∈ Ds(M). For k 1 : Ωϕ0 = {horizontal curves ϕ(·) ∈ H1(0, 1; Ds(M)) | ϕ(0) = ϕ0}. → Ck submanifold of H1(0, 1; Ds(M)) End-point mapping : endϕ0 : Ωϕ0 − → Ds(M) ϕ(·) − → ϕ(1) (of class Ck)

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Definitions SR structure on Ds(M) Geodesics Reachability

Singular curves

Given ϕ0 and ϕ1 in Ds(M) : Ωϕ0,ϕ1 = end−1

ϕ0 ({ϕ1})

(set of horizontal curves steering ϕ0 to ϕ1) → need not be a manifold Definition For k 1 : an horizontal curve ϕ(·) ∈ H1(0, 1; Ds(M)) is singular if ∃Pϕ1 ∈ T ∗

ϕ1Ds(M) \ {0} | (d endϕ0(ϕ(·)))∗.Pϕ1 = 0

⇔ codimTϕ1 Ds(M)(Range(d endϕ0(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 He = set of horizontal vector fields of class Hs+k on M

Then any autonomous vector field X such that X ◦ γ(·) = ˙ γ(·) generates a singular curve of diffeomorphisms.

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Definitions SR structure on Ds(M) Geodesics Reachability

Three possibilities :

1

Range(d endϕ0(ϕ(·))) = Tϕ1Ds(M)

→ ϕ(·) regular, Ωϕ0,ϕ1 local manifold 2

codimTϕ1 Ds(M)(Range(d endϕ0(ϕ(·)))) > 0

singular curve 3

Range(d endϕ0(ϕ(·))) proper dense subset of Tϕ1Ds(M)

The first possibility never occurs because He is never closed in Γs(TM)

(k 1)

The third possibility is due to infinite dimension
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Sub-Riemannian structures on groups of diffeomorphisms

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Definitions SR structure on Ds(M) Geodesics Reachability

SR distance

Definition Given a horizontal curve ϕ(·) ∈ H1(0, 1; Ds(M)), with logarithmic velocity X(·) = ˙ ϕ(·) ◦ ϕ(·)−1 ∈ L2(0, 1; He), we define the length and the action : L(ϕ(·)) = Z 1 p X(t), X(t) dt and A(ϕ(·)) = 1 2 Z 1 X(t), X(t) dt. Then : dSR(ϕ0, ϕ1) = inf{L(ϕ(·)) | ϕ(·) ∈ Ωϕ0,ϕ1}

ϕ(·) is said to be minimizing if dSR(ϕ(0), ϕ(1)) = L(ϕ(·)).
Minimizing the length = minimizing the action.

Theorem dSR is a right-invariant distance

(dSR(ϕ0, ϕ1) = 0 ⇒ ϕ0 = ϕ1)

∀ϕ0, ϕ1 ∈ Ds(M) s.t. dSR(ϕ0, ϕ1) < +∞, ∃ϕ(·) minimizer steering ϕ0 to ϕ1. (Ds(M), dSR) is a complete metric space.

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

A geodesic ϕ(·) ∈ H1(0, 1; Ds(M)) is a critical point of the action A|Ωϕ(0),ϕ(1). Any minimizing horizontal curve is a geodesic. Objective : derive first-order conditions for geodesics (Pontryagin maximum principle)

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Preliminary discussion : Lagrange multipliers Let ϕ(·) ∈ Ωϕ0,ϕ1 be a minimizer. Then ϕ(·) is a geodesic, and X(·) = ˙ ϕ(·) ◦ ϕ(·)−1 is a critical point of Fϕ0 : L2(0, 1; He) − → Ds(M) × I R X(·) − → (endϕ0(ϕX (·)), A(ϕX (·))) There are 2 cases : First case : codimTϕ1 Ds(M)×I

R(Range(dFϕ0(X(·)))) > 0

⇔ ker((dFϕ0(X(·)))∗) = {0} i.e., there exists a Lagrange multiplier (Pϕ1, p0) ∈ T ∗

ϕ1Ds(M) × I

R \ {(0, 0)} s.t. (d endϕ0(ϕ(·)))∗.Pϕ1 + p0dA(ϕ(·)) = 0 a) either p0 = 0 (normal case) → p0 = −1 → normal geodesics b) or p0 = 0 (abnormal case) → abnormal geodesics (and ϕ(·) singular curve)

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Preliminary discussion : Lagrange multipliers Let ϕ(·) ∈ Ωϕ0,ϕ1 be a minimizer. Then ϕ(·) is a geodesic, and X(·) = ˙ ϕ(·) ◦ ϕ(·)−1 is a critical point of Fϕ0 : L2(0, 1; He) − → Ds(M) × I R X(·) − → (endϕ0(ϕX (·)), A(ϕX (·))) There are 2 cases : Second case : Range(dFϕ0(X(·))) = Tϕ1Ds(M) × I R ⇔ ker((dFϕ0(X(·)))∗) = {0} and then there is no Lagrange multiplier, no PMP . This case is specific to infinite dimension. Such a ϕ(·) ∈ H1(0, 1; Ds(M)) is said to be elusive.

Remark : stronger topology of target space ⇒ larger dual space (see also Kurcyusz) ⇒ easy to exhibit elusive curves by playing with the order of regularity

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

KHe : H∗

e → He

= inverse of the operator X → X, ·

i.e. ∀α ∈ H∗

e

α(·) = KHe α, ·

KHϕ = dRϕ KHe dR∗

ϕ : T ∗ ϕDs(M) → TϕDs(M)

(Ck vector bundle morphism) Normal Hamiltonian h : T ∗Ds(M) → I R defined by h(ϕ, P) = 1

2 P(KHϕP) (cometric induced on T ∗Ds(M) by the SR structure) In local canonical coordinates : KHϕ P = X(ϕ, P) ◦ ϕ with X(ϕ, P) = KHe (dRϕ)∗.P and hence h(ϕ, P) = 1 2 P(X(ϕ, P) ◦ ϕ) = 1 2 X(ϕ, P), X(ϕ, P).

Normal geodesic equations ( ˙ ϕ(t), ˙ P(t)) = ∇ωh(ϕ(t), P(t)) a.e.

(symplectic gradient) (if s > d/2 + 1 and k 2 then there is a Ck normal flow)

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

KHe : H∗

e → He

= inverse of the operator X → X, ·

i.e. ∀α ∈ H∗

e

α(·) = KHe α, ·

KHϕ = dRϕ KHe dR∗

ϕ : T ∗ ϕDs(M) → TϕDs(M)

(Ck vector bundle morphism) Normal Hamiltonian h : T ∗Ds(M) → I R defined by h(ϕ, P) = 1

2 P(KHϕP) (cometric induced on T ∗Ds(M) by the SR structure) In local canonical coordinates : KHϕ P = X(ϕ, P) ◦ ϕ with X(ϕ, P) = KHe (dRϕ)∗.P and hence h(ϕ, P) = 1 2 P(X(ϕ, P) ◦ ϕ) = 1 2 X(ϕ, P), X(ϕ, P).

Normal geodesic equations ( ˙ ϕ(t), ˙ P(t)) = ∇ωh(ϕ(t), P(t)) a.e.

(symplectic gradient) In local canonical coordinates : ˙ ϕ(t) = KHϕ(t) P(t), ˙ P(t) = −(∂ϕKHϕ(t) P(t))∗.P(t).

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

KHe : H∗

e → He

= inverse of the operator X → X, ·

i.e. ∀α ∈ H∗

e

α(·) = KHe α, ·

KHϕ = dRϕ KHe dR∗

ϕ : T ∗ ϕDs(M) → TϕDs(M)

(Ck vector bundle morphism) Normal Hamiltonian h : T ∗Ds(M) → I R defined by h(ϕ, P) = 1

2 P(KHϕP) (cometric induced on T ∗Ds(M) by the SR structure) In local canonical coordinates : KHϕ P = X(ϕ, P) ◦ ϕ with X(ϕ, P) = KHe (dRϕ)∗.P and hence h(ϕ, P) = 1 2 P(X(ϕ, P) ◦ ϕ) = 1 2 X(ϕ, P), X(ϕ, P).

Normal geodesic equations ( ˙ ϕ(t), ˙ P(t)) = ∇ωh(ϕ(t), P(t)) a.e.

(symplectic gradient) If K is the reproducing kernel associated with He, then, for every x ∈ M, ∂t ϕ(t, x) = Z

M

K(ϕ(t, x), ϕ(t, y))P(t, y) dyg, ∂t P(t, x) = −P(t, x) Z

M

∂1K(ϕ(t, x), ϕ(t, y))P(t, y) dyg

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Momentum formulation Momentum map : µ : T ∗Ds(M) → Γ−s(T ∗M) (ϕ, P) → µ(ϕ, P) = (dRϕ)∗.P . Setting µ(t) = µ(ϕ(t), P(t)), we obtain ∂tµ(t) = ad∗

X(t)µ(t) = −LX(t)µ(t)

(with X(t) = KHeµ(t)) → sub-Riemannian version of the Euler-Arnol’d equation Moreover, we have µ(t) = ϕ(t)∗µ(0).

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Momentum formulation Momentum map : µ : T ∗Ds(M) → Γ−s(T ∗M) (ϕ, P) → µ(ϕ, P) = (dRϕ)∗.P . Setting µ(t) = µ(ϕ(t), P(t)), we obtain ∂tµ(t) = ad∗

X(t)µ(t) = −LX(t)µ(t)

(with X(t) = KHeµ(t)) → sub-Riemannian version of the Euler-Arnol’d equation Moreover, we have µ(t) = ϕ(t)∗µ(0). In I Rd, we obtain ∂tµ(t) = −(X.∇)µ − (div X)µ − (dX)∗µ SR-EPDiff

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Momentum formulation Momentum map : µ : T ∗Ds(M) → Γ−s(T ∗M) (ϕ, P) → µ(ϕ, P) = (dRϕ)∗.P . Setting µ(t) = µ(ϕ(t), P(t)), we obtain ∂tµ(t) = ad∗

X(t)µ(t) = −LX(t)µ(t)

(with X(t) = KHeµ(t)) → sub-Riemannian version of the Euler-Arnol’d equation Moreover, we have µ(t) = ϕ(t)∗µ(0). If He = Γs(TM), we have, equivalently, ∂tX(t) = ad⊤

X(t)X(t) (note that KHe ad∗

X = ad⊤ X KHe )

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Abnormal Hamiltonian H0 : T ∗Ds(M) × He → I R defined by H0(ϕ, P, X) = P(X ◦ ϕ) = P(dRϕ.X) Abnormal geodesic equations ( ˙ ϕ(t), ˙ P(t)) = ∇ωH0(ϕ(t), P(t), X(t)) ∂X H0(ϕ(t), P(t), X(t)) = (dRϕ(t))∗.P(t) = 0 a.e.

(if s > d/2 + 1 and k 1 : ϕ(·) singular ⇔ projection of an abnormal geodesic)

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Pontryagin maximum principle We assume that s > d/2 + 1 and k 1. Any geodesic ϕ(·) is : either normal, and then ∃P(·) A.C. on [0, 1], with P(t) ∈ T ∗

ϕ(t)Ds(M), satisfying

the normal geodesic equations ;

r singular, and then ∃P(·) A.C. on [0, 1], with P(t) ∈ T ∗

ϕ(t)Ds(M) \ {0},

satisfying the abnormal geodesic equations ;

r elusive.

Examples : in D(I Rd) : vector fields with classical Gaussian kernels, or with (less classical) sub-Riemannian kernels... shape spaces of landmarks (singular curves with Dirac momenta)

in Arguillère Trélat (2014)

More general shape deformation analysis (from the optimal control viewpoint) :

Arguillère Trélat Trouvé Younes (2014)

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Definitions SR structure on Ds(M) Geodesics Reachability

Geodesics on Ds(M)

Open questions Is the set of end-points of normal geodesics an open dense subset of Ds(M) ?

In finite dimension this is true (Rifford Trélat 2005, Agrachev 2008).

Are there generic results for singular curves on Ds(M) ?

In finite dimension : no minimizing singular curve for generic horizontal distributions of rank 3. (Agrachev Gauthier 2003, Chitour Jean Trélat 2006) Since dim He = +∞, we expect that the same result is true in Ds(M).

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Definitions SR structure on Ds(M) Geodesics Reachability

Reachability properties in Ds(M)

Objective : provide sufficient conditions on He ensuring approximate or exact reachability from e. Definition Reachable set from e = idM : R(e) = {ϕ ∈ Ds(M) | dSR(e, ϕ) < +∞}. → set of ϕ ∈ Ds(M) that can be connected from e by means of an horizontal curve ϕ(·) ∈ H1(0, 1; Ds(M)). ϕ ∈ Ds(M) is reachable from e if ϕ ∈ R(e), ϕ ∈ Ds(M) is approximately reachable from e if ϕ ∈ R(e).

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Definitions SR structure on Ds(M) Geodesics Reachability

Approximate reachability

Proposition If M is connected and He = Γs(TM), then R(e) = Ds(M)

(approximate reachability) (Dudnikov Samborski 1980, Heintze Liu 1999, Grong Markina Vasil’ev 2012)

Proposition Assume : M compact (

r

X

i=1

uiXi | u1, . . . , ur ∈ C∞(M) ) ⊂ He with X1, . . . , Xr smooth vector fields on M any two points x, y ∈ M can be connected by a ∆-horizontal smooth curve x(·)

n M, with ∆ = Span{X1, . . . , Xr}

(⇐ Lie(∆) = TM) Then D∞(M) ⊂ R(e) ⊂ R(e) = Ds(M)

(approximate reachability) (follows from Agrachev Caponigro 2009)

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Definitions SR structure on Ds(M) Geodesics Reachability

Exact reachability

M Riemannian compact, (X1, . . . , Xr) smooth vector fields on M, ∆ = Span{X1, . . . , Xr} ⊂ TM

Remark : a curve ϕ(·) is horizontal in Ds(M) if and only if any “particle" trajectory t → ϕ(t, x) is ∆-horizontal

Theorem Assume : (

r

X

i=1

uiXi | u1, . . . , ur ∈ Hs(M) ) ⊂ He any two points x, y ∈ M can be connected by a ∆-horizontal smooth curve x(·)

n M

(⇐ Lie(∆) = TM) Then R(e) = Ds(M)

(exact reachability) (straightforward extension from Agrachev Caponigro 2009) If Lie(∆) = TM, then Γs(TM) = linear combinations of brackets of Xi with coefficients in Hs(M). Actually :

If Lie(∆) = TM, then, moreover, the topology induced on Ds(M) by the SR distance dSR coincides with the intrinsic manifold topology of Ds(M).

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Definitions SR structure on Ds(M) Geodesics Reachability

Exact reachability

Actually, under the assumption that Lie(∆) = TM, we prove that : ∃C > 0, ∃U neighborhood of 0 in Hs(M, I Rm), ∃φ : U → Ds(M) C1-submersion, with φ(0) = e, such that dSR(e, φ(u1, . . . , um)) C

m

X

i=1

ui1/ji

s

→ “half of" the classical ball-box theorem in SR geometry. Open question Prove the converse inequality.

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Definitions SR structure on Ds(M) Geodesics Reachability

Applications of exact controllability

Corollary 1 If Lie(∆) = TM : exact controllability for controlled transport PDE’s ∂tµ(t) +

r

X

i=1

div(ui(t)µ(t)Xi) = 0 in spaces of Hs volume forms, with ui(t) ∈ Hs(M).

(because µ(t) = ϕ(t)∗µ(0))

Corollary 2 : Moser trick with horizontal flows If Lie(∆) = TM and s > d/2 + 1, then : Let µ0 = f0dxg, µ1 = f1dxg be Hs volume forms on M s.t. R

M f0 dxg =

R

M f1 dxg.

∃ϕ(·) ∈ H1(0, 1; Ds+1(M)) horizontal such that ϕ(0) = e and ϕ(1)∗µ0 = µ1. Let ω0, ω1 be Hs symplectic forms on M s.t. [ω0] = [ω1]. ∃ϕ(·) ∈ H1(0, 1; Ds+1(M)) horizontal such that ϕ(0) = e and ϕ(1)∗ω0 = ω1.