Matthew J. Hirn École normale supérieure Analyse non ... · Isometric Lipschitz extensions Isometric extension property Two metric spaces (X;d X) and (Z;d Z) have the isometric

Minimal C1,1 Extensions


Matthew J. HirnÉcole normale supérieure

Analyse non-linéaire et EDPApril 15, 2014


Collaborators

(a) Ariel Herbert-Voss (Uni-versity of Utah)

(b) Erwan Le Gruyer (INSAde Rennes and IRMAR)

(c) Frederick McCollum(University of Arkansas)


Outline

The C1,1 Extension Problem

Absolutely Minimal Lipschitz Extensions (with E. Le Gruyer)

Algorithms (with A. Herbert-Voss and F. McCollum)



Outline






Definitions

I For a function F : Rd → Rn,

Lip(F) , supx,y∈Rd

x 6=y

|F(x)− F(y)||x− y|

.

I Cm(Rd): Space of functions F : Rd → R with continuousderivatives up to order m.

I C0,1(Rd) , {F ∈ C0(Rd) | Lip(F)



The Lipschitz extension problemC0,1 extension problem

What we are given:1. A close set E ⊂ Rd.2. A function f : E → R.

If possible, we want a function F ∈ C0,1(Rd) such that:1. Extension: F(a) = f (a) for each a ∈ E.2. Minimal: For all other extensions G ∈ C0,1(Rd),

Lip(F) ≤ Lip(G)

Theorem (Whitney; McShane; Kirszbraun; 1934)

If Lip(f )



The C1,1(Rd) extension problem

C1,1 extension problem

What we are given:1. A closed set E ⊂ Rd.2. A polynomial Pa ∈ P1 for each a ∈ E. We call the set

PE = {Pa | a ∈ E}

a 1-field.If possible, we want a function F ∈ C1,1(Rd) such that:

1. Extension: JaF = Pa for each a ∈ E.2. Minimal: For all other extensions G ∈ C1,1(Rd),

Lip(∇F) ≤ Lip(∇G)



Whitney’s Extension Theorem

Theorem (Whitney’s Extension Theorem for C1,1(Rd), 1934)Let E ⊂ Rd be a closed set with associated 1-field PE = {Pa | a ∈ E}.If there exists a constant M such that

1. |Pa(a)− Pb(a)| ≤ M|a− b|2, ∀ a, b ∈ E2. |∂Pa∂xi (a)−

∂Pb∂xi

(a)| ≤ M|a− b|, ∀ a, b ∈ E, i = 1, . . . , dthen there exists a function F ∈ C1,1(Rd) that extends the 1-field, i.e.,

JaF = Pa, ∀ a ∈ E

Remark: The theorem does not give the minimal value of Lip(∇F),even if one were to consider the infimum over all M satisfying theabove conditions.



The minimal value of Lip(∇F)DefinitionFor E ⊂ Rd with 1-field PE = {Pa | a ∈ E}, define:

A(a, b) ,|Pa(a)− Pb(a) + Pa(b)− Pb(b)|

|a− b|2, B(a, b) ,

|∇Pa −∇Pb||a− b|

Γ1(PE) , supa,b∈Ea 6=b

√A(a, b)2 + B(a, b)2 + A(a, b)

Theorem (Le Gruyer, 2009)

If Γ1(PE)



Lipschitz constant for 1-fields

I For F ∈ C1,1(Rd), define its associated 1-field as:

JF , {JaF | a ∈ Rd}

Note that Γ1(JF) = Lip(∇F).I If F is an extension of PE with Lip(∇F) = Γ1(PE), then the 1-field

JF extends the 1-field PE while preserving the value of Γ1, i.e.,Γ1(JF) = Γ1(PE).

I Thus the functional Γ1 can be thought of as a Lipschitz constantfor 1-fields that satisfies the isometric extension property.


Absolutely Minimal Lipschitz Extensions

Outline






Isometric Lipschitz extensions

Isometric extension property

Two metric spaces (X, dX) and (Z, dZ) have the isometric extensionproperty if for any function f : E → Z with Lip(f )



Non-uniqueness of isometric extensionsI f : E → RI E = {−1, 0, 1} ⊂ RI f (−1) = f (0) = 0, f (1) = 1



Absolutely minimal Lipschitz extensions

Absolutely minimal Lipschitz extension (Aronsson, 1967)

For f : E → Z, let F : X → Z be an isometric extension, i.e.,Lip(F) = Lip(f ). The function F is an absolutely minimal Lipschitzextension (AMLE) if for every open subset V ⊂ X \ E,

Lip(F|V) = Lip(F|∂V)

I The AMLE is the locally best Lipschitz extension of f .I Again we can extend the definition to functionals Φ other than

Lip, including Γ1.



AMLE ExampleI f : E → RI E = {−1, 0, 1} ⊂ RI f (−1) = f (0) = 0, f (1) = 1



Existence and uniqueness for Z = R

Existence of AMLEs when Z = R:I X = Rd: Aronsson, 1967I (X, dX) = length space: Mil’man, 1999

Uniqueness of AMLEs when Z = R:I X = Rd: Jensen, 1993I (X, dX) = length space: Peres, Schramm, Sheffield, Wilson, 2009



Relationship to PDEs

I The infinity Laplacian is defined as:

∆∞g ,d∑

i,j=1

∂2g∂xi∂xj

∂g∂xi

∂g∂xj

Theorem (Aronsson, 1967; Jensen, 1993)

Given f : E → R with Lip(f )



Tug of warI Two player, zero-sum, random turn game.I Given E ⊂ X, f : E → R with Lip(f ) 0.I Rules of the game:

I Fix a starting point x0 = x ∈ X \ E.I At the kth turn, the players toss a coin and the winner choses an xk

with dX(xk, xk−1) < ε.I The game ends when xk ∈ E.I The payoff is f (xk). Player I wants to maximize f (xk), while player II

wants to minimize f (xk).I For any starting point x ∈ X \ E, let Fε(x) be the expected payoff

optimized over all possible player I and player II strategies.

Theorem (Peres, Schramm, Sheffield, Wilson, 2009)

Let F(x) = limε→0 uε(x). If (X, dX) is a length space, then F is theAMLE of f .



Other applications of AMLEsI Image and surface inpainting/interpolation (various)

(d) Pyramid missing top (e) Inpainted pyramid

Figure : Caselles, Haro, Sapiro, Verdera

I Analysis of shapes of sandpiles (Aronsson, Evans, Wu, 1996)



Results for Z 6= R

Theorem (Naor and Sheffield, 2012)

AMLEs exist and are unique whenI (X, dX) is a locally compact length space.I (Z, dZ) is a metric tree.

Theorem (Sheffield and Smart, 2012)

Tight AMLEs exist and are unique when:I (X, dX) is a finite graph.I (Z, dZ) = Rn.

Also addresses (X, dX) = Rd and the complications that ensue.



Quasi-AMLEs

Theorem (H. and Le Gruyer, 2014)

I X = Rd

I Z = P1

I Γ1 which satisfies the isometric extension property

Let E ⊂ X and f : E → Z. Then there exists a quasi-AMLE F : X → Zsuch that F(a) = f (a) for all a ∈ E and

Γ1(F|V) = Γ1(F|∂V), ∀ open V ⊂ X \ E

“almost” holds.



Quasi-AMLEs


I X = Rd

I Z = P1

I Γ1 which satisfies the isometric extension property


|Γ1(F|V)− Γ1(F|∂V)| < ε, ∀ open V ⊂ X \ E

“almost” holds for all ε > 0.



Quasi-AMLEs


I X = metric space with some restrictionsI Z = complete metric spaceI Φ which satisfies the isometric extension property


|Φ(F|V)−Φ(F|∂V)| < ε ∀ open V ⊂ X \ E

“almost” holds for all ε > 0.


Algorithms

Outline





Algorithms

Overview of algorithm

Computing the minimal extension for a 1-field:I Inputs: The user inputs into the computer:

I A finite set E ⊂ Rd with N points.I The N function values fa ∈ R.I The N gradients Daf ∈ Rd.

I One time work: The computer then performs a seriescalculations.

I Query work: After finishing the one time work, the computerprompts the user for a query point x ∈ Rd \ E. The computerperforms a few more calculations, and returns F(x) and ∇F(x),i.e., JxF.


Algorithms

Related work

Algorithms for Cm(Rd):I Fefferman and Klartag, 2009: Computes an extension

F ∈ Cm(Rd) such that

c(m, d)M ≤ ‖F‖Cm(Rd) ≤ C(m, d)M

where M is the minimum possible norm. Requires O(N log N) onetime work, O(log N) query work, and O(N) storage.

I Fefferman, 2010: Computes an extension F ∈ Cm(Rd) such that

M ≤ ‖F‖Cm(Rd) ≤ (1 + ε)M

Requires solving a linear programming problem of sizeC(m, d)| log η|ε− 32 dN.


Algorithms

Our algorithm

Highlights of our algorithm for C1,1(Rd).I Computes an extension F with Lip(∇F) = Γ1(PE).I The one time work is proportional to the amount of work to

compute a convex hull in Rd+1.I The query work is O(log Cd+1(N)), where Cd+1(N) is the

complexity of the convex hull we compute.I All the pieces of the algorithm are based on algorithms from the

computational geometry that are practical and can beimplemented on an actual computer.


Algorithms

Computing Γ1

I First we must compute

Γ1(PE) = maxa,b∈E

√A(a, b)2 + B(a, b)2 + A(a, b)

I Straightforward to compute it in O(N2) operations (linear in d).I Less straightforward to compute it in O(N log N) operations

(exponential in d), to within an absolute constant. Uses a wellseparated pairs decomposition (Callahan and Kosaraju, 1995).


Algorithms

Wells’ construction

I A paper of Wells (1977) gives a construction of an interpolantwhich can be used as a roadmap for our algorithm. Much of theone-time work is devoted to encoding this roadmap.

I Each piece of the construction can be translated into anappropriate structure from computational geometry.

I The main pieces are:I Power diagramI Convex hullI Balanced tree encoding hyperplane cuts (Fuchs, Jones, Morari,

2010)

Proposition (Le Gruyer and Phan, 2014)

The 1-field associated to the construction of Wells is a an AMLEwhen E is finite.


Algorithms

Power diagram


Algorithms

Dual triangulation


Algorithms

Merging of the two


Algorithms

Extension

interpolantregions.movMedia File (video/quicktime)


Algorithms

Open questions

Cm,1(Rd)I Can we find formulas for Γm?

AMLE

I Quasi-AMLE→ AMLE?I For 1-fields:

I Correct formulation?I Relationship to PDEs?I Relationship to stochastic games?

Algorithm

I Improve dependence on N while maintaining the “nice” aspectsof the algorithm?

I Applications?


Algorithms

Thank you!

www.di.ens.fr/∼hirn

Code available at:http://csce.uark.edu/∼fmccollu

The C1,1 Extension ProblemAbsolutely Minimal Lipschitz Extensions (with E. Le Gruyer)Algorithms (with A. Herbert-Voss and F. McCollum)

Matthew J. Hirn École normale supérieure Analyse non ... · Isometric Lipschitz extensions Isometric extension property Two metric spaces (X;d X) and (Z;d Z) have the isometric

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