1 Geometric Understand in Higher Dimension / 8 Juin 2017 / Coll` ege de France Quentin M´ erigot / Universit´ e Paris-Sud Computational geometry, optimal transport and applications Joint works with Thomas Gallou¨ et, Jun Kitagawa, Pedro Machado, Jocelyn Meyron, Jean-Marie Mirebeau, Boris Thibert
103
Embed
Computational geometry, optimal transport and · 1 Geometric Understand in Higher Dimension / 8 Juin 2017 / Coll ege de France Quentin M erigot / Universit e Paris-Sud Computational
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Geometric Understand in Higher Dimension / 8 Juin 2017 / College de France
Quentin Merigot / Universite Paris-Sud
Computational geometry, optimal transport and
applications
Joint works with Thomas Gallouet, Jun Kitagawa, Pedro Machado,Jocelyn Meyron, Jean-Marie Mirebeau, Boris Thibert
2
Overview
1. Optimal transport & Laguerre diagrams
2. First application: non-imaging optics
3. Second application: enforcing incompressibility
3
1. Optimal transport & Laguerre diagrams
4
Optimal transport
Data: ρ = prob density on X ν probability meas. on Y
YX
4
Optimal transport
Data: ρ = prob density on X ν probability meas. on Y
YX
Think of ρ, ν as describing piles of sand, made of many grains.
Assume that moving a grain with mass dm from x to y costs c(x, y)dm.
Optimal transport problem: what is the cheapest way of moving ρ to ν ?
4
Optimal transport
Data: ρ = prob density on X ν probability meas. on Y
T : X → Y YT−1(B)X
T is a transport map (written T#ρ = ν) if for all B ⊆ Y, ρ(T−1(B)) = ν(B)
B
4
Optimal transport
Data: ρ = prob density on X ν probability meas. on Y
T : X → Y YT−1(B)X
T is a transport map (written T#ρ = ν) if for all B ⊆ Y, ρ(T−1(B)) = ν(B)
B
Optimal transport problem: minimize∫Xc(x, T (x)) d ρ(x) where T#ρ = ν
4
Optimal transport
Data: ρ = prob density on X ν probability meas. on Y
T : X → Y YT−1(B)X
T is a transport map (written T#ρ = ν) if for all B ⊆ Y, ρ(T−1(B)) = ν(B)
B
Optimal transport problem: minimize∫Xc(x, T (x)) d ρ(x) where T#ρ = ν
Many applications:
computer graphics, machine learning, inverse problems, etc.
PDEs, functional inequalities, probabilities,
# articles containing ”OT”
5
Computational optimal transport
Hungarian algorithm
linear programming
Discrete source and targetαi βj
Sinkhorn/IPFP
5
Computational optimal transport
Hungarian algorithm
linear programming
Discrete source and targetαi βj
Source and target with density:
dynamic (Benamou-Brenier) formulation
finite-differences for Monge-Ampere
Sinkhorn/IPFP
5
Computational optimal transport
Hungarian algorithm
linear programming
Discrete source and targetαi βj
Source with density, discrete target:
Source and target with density:
dynamic (Benamou-Brenier) formulation
finite-differences for Monge-Ampere
Minkowski, Alexandrov, etc.
Sinkhorn/IPFP
5
Computational optimal transport
Hungarian algorithm
linear programming
Discrete source and targetαi βj
Source with density, discrete target:
Source and target with density:
dynamic (Benamou-Brenier) formulation
finite-differences for Monge-Ampere
Flexibility for the cost function but computationally expensive
Computationally efficient but restricted to ”geometric” cost functions.
Minkowski, Alexandrov, etc.
Sinkhorn/IPFP
5
Computational optimal transport
Hungarian algorithm
linear programming
Discrete source and targetαi βj
Source with density, discrete target:
Source and target with density:
dynamic (Benamou-Brenier) formulation
finite-differences for Monge-Ampere
Minkowski, Alexandrov, etc.
”semi-discrete optimal transport”
Sinkhorn/IPFP
6
Semi-discrete optimal transport
Data: ρ = prob density on X ν =∑y∈Y νyδy prob. on finite Y
Y
X
6
Semi-discrete optimal transport
Data: ρ = prob density on X ν =∑y∈Y νyδy prob. on finite Y
y
T : X → Y
Y
T−1(y)X
T is a transport map if for every y ∈ Y, ρ(T−1({y})) = νy (capacity constraint)
6
Semi-discrete optimal transport
Data: ρ = prob density on X ν =∑y∈Y νyδy prob. on finite Y
y
T : X → Y
Y
T−1(y)X
T is a transport map if for every y ∈ Y, ρ(T−1({y})) = νy (capacity constraint)
The set of transport maps is huge (⊆ measurable partitions of X) . . .
6
Semi-discrete optimal transport
Data: ρ = prob density on X ν =∑y∈Y νyδy prob. on finite Y
y
T : X → Y
Y
T−1(y)X
T is a transport map if for every y ∈ Y, ρ(T−1({y})) = νy (capacity constraint)
The set of transport maps is huge (⊆ measurable partitions of X) . . .
. . . but fortunately optimal maps form a much smaller (finite-dimensional) set.
7
Semi-discrete OT and Laguerre diagrams
ρ : X → R density of population
Y = location of bakeries
c(x, y) := ‖x− y‖2 cost of walking from x to y
7
Semi-discrete OT and Laguerre diagrams
ρ : X → R density of population
Y = location of bakeries
c(x, y) := ‖x− y‖2 cost of walking from x to y
Vor(y) = {x ∈ X;∀z ∈ Y, c(x, y) ≤ c(x, z)}
I If the price of bread is uniform, people go the closest bakery:
7
Semi-discrete OT and Laguerre diagrams
ρ : X → R density of population
Y = location of bakeries
c(x, y) := ‖x− y‖2 cost of walking from x to y
y0
Vor(y) = {x ∈ X;∀z ∈ Y, c(x, y) ≤ c(x, z)}
I If the price of bread is uniform, people go the closest bakery:
Minimizes total distance walked . . . but might exceed the capacity of bakery y0!
7
Semi-discrete OT and Laguerre diagrams
ρ : X → R density of population
Y = location of bakeries
c(x, y) := ‖x− y‖2 cost of walking from x to y
I If prices are given by ψ : Y → R, people make a compromise:
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Strong monotonicity of G):
(y, z) ∈ H ⇐⇒ Lzy > 0
I Consider the matrix (Lyz) :=(∂Gy
∂1z(ψ))
and the graph H:
. . . recall that Gy(ψ) = ρ(Lagy(ψ)) . . .
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Strong monotonicity of G):
(y, z) ∈ H ⇐⇒ Lzy > 0
I Consider the matrix (Lyz) :=(∂Gy
∂1z(ψ))
and the graph H:
I H = 1-skeleton of (Lagy(ψ) ∩ {ρ > 0})y∈Y
. . . recall that Gy(ψ) = ρ(Lagy(ψ)) . . .
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Strong monotonicity of G):
(y, z) ∈ H ⇐⇒ Lzy > 0
I Consider the matrix (Lyz) :=(∂Gy
∂1z(ψ))
and the graph H:
I If {ρ > 0} is connected and ψ ∈ Eε, then H is connected
I H = 1-skeleton of (Lagy(ψ) ∩ {ρ > 0})y∈Y
. . . recall that Gy(ψ) = ρ(Lagy(ψ)) . . .
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Strong monotonicity of G):
(y, z) ∈ H ⇐⇒ Lzy > 0
I Consider the matrix (Lyz) :=(∂Gy
∂1z(ψ))
and the graph H:
I If {ρ > 0} is connected and ψ ∈ Eε, then H is connected
I H = 1-skeleton of (Lagy(ψ) ∩ {ρ > 0})y∈Y
I L is the Laplacian of a connected graph =⇒ KerL = R · cst
. . . recall that Gy(ψ) = ρ(Lagy(ψ)) . . .
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Smoothness of G): Relies heavily on a convexity property of Laguerre cells:
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Smoothness of G): Relies heavily on a convexity property of Laguerre cells:
Loeper’s condition: there exists expcy : Rd → X diffeo. s.t. ∀ψ and y
Lagψ(y) ⊆ X[expcy]−1(Lagψ(y)) ⊆ Rd
is convexexpcy
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Smoothness of G): Relies heavily on a convexity property of Laguerre cells:
Loeper’s condition: there exists expcy : Rd → X diffeo. s.t. ∀ψ and y
Lagψ(y) ⊆ X[expcy]−1(Lagψ(y)) ⊆ Rd
is convexexpcy
−→ Restrictive condition, which fortunately is satisfied for the reflector problem.
18
Convergence of Damped Newton
Theorem: Let X be an hemisphere of S2. Assume that Y ⊂ S2 \X and that
ρ ∈ Cα(X) and {ρ > 0} is connected
Then, the damped Newton algorithm for SD-OT converges globally with linear rateand locally with rate (1 + α). special case of [Kitagawa, M., Thibert ’15]
(Smoothness of G): Relies heavily on a convexity property of Laguerre cells:
Loeper’s condition: there exists expcy : Rd → X diffeo. s.t. ∀ψ and y
Lagψ(y) ⊆ X[expcy]−1(Lagψ(y)) ⊆ Rd
is convexexpcy
−→ Restrictive condition, which fortunately is satisfied for the reflector problem.
−→ Loeper’s condition originates from regularity theory for OT...
19
3. Second application: enforcing incompressibilityJoint work with J.M. Mirebeau
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
[Brenier ’93]
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
−→ theory of generalized geodesics
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
−→ non-deterministic behavior of fluid particles. [Shnirelman ’94]
[Brenier ’93]
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
−→ theory of generalized geodesics
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
−→ non-deterministic behavior of fluid particles. [Shnirelman ’94]
[Brenier ’93]
I Discretization: N = number of particles, T = number of timesteps
At time i, the particles are at positions mi := (M1i , . . . ,M
Ni ) ∈ RNd.
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
−→ theory of generalized geodesics
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
−→ non-deterministic behavior of fluid particles. [Shnirelman ’94]
[Brenier ’93]
I Discretization: N = number of particles, T = number of timesteps
At time i, the particles are at positions mi := (M1i , . . . ,M
Ni ) ∈ RNd.
action incompressibilityboundary conditions
minm0,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ(‖m0 − s∗‖22 + ‖mT − s∗‖22 + ???
)
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
−→ theory of generalized geodesics
20
Geodesics between incompressible maps
Thm: Smooth solutions to Euler equations for incompressible fluids are geodesics in
SDiff = {volume-preserving diffeo. from X to X} ⊆ E := L2(X,Rd)[Arnold ’66]
I What about the minimizing geodesics between s∗, s∗ ∈ SDiff ?
−→ non-deterministic behavior of fluid particles. [Shnirelman ’94]
[Brenier ’93]
I Discretization: N = number of particles, T = number of timesteps
At time i, the particles are at positions mi := (M1i , . . . ,M
Ni ) ∈ RNd.
action incompressibilityboundary conditions
minm0,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ(‖m0 − s∗‖22 + ‖mT − s∗‖22 + ???
)
inf
{∫ 1
0
‖s′(t)‖2E d t | s : [0, 1]→ SDiff, s0 = s∗, s1 = s∗}
A point cloud cannot be exactly incompressible =⇒ penalization using optimal transport.
−→ theory of generalized geodesics
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 900, X = [0, 1]2.
m
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 900, X = [0, 1]2.
m
dS(m) ' 0, 031
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 900, X = [0, 1]2.
m
dS(m) ' 0, 031 −N2 ∇ d2S(m)
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 900, X = [0, 1]2.
m
dS(m) ' 0, 031 −N2 ∇ d2S(m) m− N
2 ∇ d2S(m)
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 740, X = [0, 1]2.
m
dS(m) ' 0, 14
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 740, X = [0, 1]2.
m
dS(m) ' 0, 14 −N2 ∇ d2S(m)
21
Distance to incompressible maps
where ρ is uniform on X and c(x, y) = ‖x− y‖2.
Definition: Given m = (M1, . . . ,MN ) ∈ RNd, we define
d2S(m) = min. transport cost between ρ and ν =
1
N
N∑k=1
δMk
Example: N = 740, X = [0, 1]2.
m
dS(m) ' 0, 14 −N2 ∇ d2S(m) m− N
2 ∇ d2S(m)
22
From particles to paths
I Time-discretization of geodesic with endpoints s∗, s∗ ∈ RNd
action incompressibilityboundary conditions
minm1,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ
(‖m0 − s∗‖22 + ‖mT − s∗‖22 +
T−1∑i=1
d2S(mi)
)
22
From particles to paths
I Given m = (m1, . . . ,mT ) ∈ RTNd, let γk ∈ C0([0, 1],Rd) be PL with γk(ti) = Mki
I Time-discretization of geodesic with endpoints s∗, s∗ ∈ RNd
action incompressibilityboundary conditions
minm1,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ
(‖m0 − s∗‖22 + ‖mT − s∗‖22 +
T−1∑i=1
d2S(mi)
)
t = 0 t = 1
γk
ti = iT
22
From particles to paths
I Given m = (m1, . . . ,mT ) ∈ RTNd, let γk ∈ C0([0, 1],Rd) be PL with γk(ti) = Mki
I Time-discretization of geodesic with endpoints s∗, s∗ ∈ RNd
action incompressibilityboundary conditions
minm1,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ
(‖m0 − s∗‖22 + ‖mT − s∗‖22 +
T−1∑i=1
d2S(mi)
)
t = 0 t = 1
γk
−→ One can associate to m a probability measure
ti = iT
µm := 1N
∑Nk=1 δγk ∈ Prob(C0([0, 1],Rd))
over the set of C0 paths:
22
From particles to paths
I Given m = (m1, . . . ,mT ) ∈ RTNd, let γk ∈ C0([0, 1],Rd) be PL with γk(ti) = Mki
I Time-discretization of geodesic with endpoints s∗, s∗ ∈ RNd
action incompressibilityboundary conditions
minm1,...,mT∈RNd
T
2
T−1∑i=0
‖mi+1 −mi‖22 + λ
(‖m0 − s∗‖22 + ‖mT − s∗‖22 +
T−1∑i=1
d2S(mi)
)
t = 0 t = 1
γk
−→ One can associate to m a probability measure
ti = iT
µm := 1N
∑Nk=1 δγk ∈ Prob(C0([0, 1],Rd))
over the set of C0 paths:
−→ Under suitable hypotheses, minimizers of thediscrete problem converge to a so-called
generalized minimizing geodesic,
µ ∈ Prob(C0([0, 1],Rd)).
23
Numerical result: Inversion of the DiskX = B(0, 1) ⊆ R2 (s∗, s