Lipschitz Changes of Variables between Perturbations · PDF fileLipschitz Changes of Variables between Perturbations of Log-concave Measures Yash Jhaveri University of Texas at Austin

Lipschitz Changes of Variables between Perturbations ofLog-concave Measures

Yash Jhaveri

University of Texas at Austin

Prairie Analysis SeminarKansas State University

September 25, 2015

Joint work with Maria Colombo and Alessio Figalli.

Yash Jhaveri (UT Austin) Lipschitz Changes of Variables September 25, 2015 1 / 16

An Old Result

Theorem (Caffarelli)

Let Q be a quadratic polynomial with D2Q > 0 and V be a convexfunction such that e−Q , e−Q−V ∈ P(Rn). Then, there exists a1-Lipschitz change of variables T taking e−Q to e−Q−V .


A New Result

Theorem (Colombo, Figalli, -)

Let V be a convex function with V (0) = inf V , R > 0, q ∈ Cc(BR), and cq ∈ Rbe such that e−V , e−V+cq−q ∈ P(Rn). Assume that λ Id ≤ D2V ≤ Λ Id andD2q ≥ −λq Id for 0 < λ,Λ, λq <∞ in the sense of distributions. Then, thereexists a change of variables T taking e−V to e−V+cq−q that satisfies

‖∇T‖L∞(Rn) ≤ C with C = C (R, λ,Λ, λq) > 0


Existence: The Optimal Transport Problem

- The map T is the unique solution of the Monge problem forquadratic cost:

min

{∫Rn

|x − T (x)|2 dµ(x) : T#µ = ν

},

where µ = e−V (x) dx , ν = e−V (y)+cq−q(y) dy ∈ P(Rn).


Brenier’s Theorem

Theorem (Brenier)

Let µ, ν ∈ P(Rn) such that µ = f (x) dx and∫Rn

|x |2 dµ(x) +

∫Rn

|y |2 dν(y) <∞.

Then, there exists a unique, optimal transport T taking µ to ν.Moreover, there is a convex function φ : Rn → R such that

T = ∇φ.


Regularity: OT and the Monge-Ampere Equation

- Let µ = f (x) dx , ν = g(y) dy ∈ P(Rn), ϕ ∈ C∞c (Rn), andassume that the OT map T taking f to g is a smoothdiffeomorphism.

Then,∫ϕ(T (x))f (x) dx

T#µ=ν=

∫ϕ(y)g(y) dy

y=T (x)=

∫ϕ(T (x))g(T (x))| det∇T (x)| dx .

- Since ϕ ∈ C∞c (Rn) was arbitrary,

f (x) = g(T (x))| det∇T (x)|.

- Recalling Brenier, if g > 0,

detD2φ =f

g(∇φ).



- Let µ = f (x) dx , ν = g(y) dy ∈ P(Rn), ϕ ∈ C∞c (Rn), andassume that the OT map T taking f to g is a smoothdiffeomorphism. Then,∫

ϕ(T (x))f (x) dxT#µ=ν

=

∫ϕ(y)g(y) dy

y=T (x)=

∫ϕ(T (x))g(T (x))| det∇T (x)| dx .


f (x) = g(T (x))| det∇T (x)|.


detD2φ =f

g(∇φ).





=

∫ϕ(y)g(y) dy

y=T (x)=

∫ϕ(T (x))g(T (x))| det∇T (x)| dx .


f (x) = g(T (x))| det∇T (x)|.


detD2φ =f

g(∇φ).





=

∫ϕ(y)g(y) dy

y=T (x)=

∫ϕ(T (x))g(T (x))| det∇T (x)| dx .


f (x) = g(T (x))| det∇T (x)|.


detD2φ =f

g(∇φ).


Some Heuristics

- Since T = ∇φ and φ is convex, we know that φαα ≥ 0 for everydirection α.

- So, we just need to find an upper bound on φαα.

- Let’s assume that φαα attains a global maximum at some pointx0 ∈ Rn, and without loss of generality, assume that this happensin the e1-direction.


Some Heuristics





Some Heuristics





Some Heuristics

- Since our potential φ solves a MA equation,

log detD2φ = −V + V (∇φ)− cq + q(∇φ).

- Differentiating the left side twice in the e1-direction andevaluating at x0,

∂11(LHS) = − tr([

[D2φ]−1D2φ1

]2)+ tr

([D2φ]−1D2φ11

)≤ 0

- On the right side,

∂11(RHS) = −V11 + 〈∇V (∇φ),∇φ11〉+ 〈D2V (∇φ)∇φ1,∇φ1〉+ 〈∇q(∇φ),∇φ11〉+ 〈D2q(∇φ)∇φ1,∇φ1〉

≥ −Λ + (λ− λq)|∇φ1(x0)|2.

- Hence, if λ− λq > 0, then |∇φ1(x0)| ≤√

Λ/(λ− λq).


Some Heuristics




∂11(LHS) = − tr([

[D2φ]−1D2φ1

]2)+ tr

([D2φ]−1D2φ11

)≤ 0



≥ −Λ + (λ− λq)|∇φ1(x0)|2.


Λ/(λ− λq).


Some Heuristics




∂11(LHS) = − tr([

[D2φ]−1D2φ1

]2)+ tr

([D2φ]−1D2φ11

)≤ 0



≥ −Λ + (λ− λq)|∇φ1(x0)|2.


Λ/(λ− λq).


Some Heuristics




∂11(LHS) = − tr([

[D2φ]−1D2φ1

]2)+ tr

([D2φ]−1D2φ11

)≤ 0



≥ −Λ + (λ− λq)|∇φ1(x0)|2.


Λ/(λ− λq).


Some Heuristics

- If OT map takes x0 outside of BR , then the right side becomes,

∂11(RHS) = −V11 + 〈D2V (∇φ)∇φ1,∇φ1〉+ 0 + 0

≥ −Λ + λ|∇φ1(x0)|2.

- And so, |∇φ1(x0)| ≤√

Λ/λ.


Some Heuristics

- If OT map takes x0 outside of BR , then the right side becomes,

∂11(RHS) = −V11 + 〈D2V (∇φ)∇φ1,∇φ1〉+ 0 + 0

≥ −Λ + λ|∇φ1(x0)|2.

- And so, |∇φ1(x0)| ≤√

Λ/λ.


Where Does Our (Inverse) Transport Take Points?

- Question: Can we quantify how far points travel under our OT orits inverse?

- Answer: Yes.

Lemma

For any P ≥ R , there exists a constant P ′ = P ′(P , λ,Λ, λq) > 0 suchthat

T−1(BP) ⊆ BP′ .




- Answer: Yes.

Lemma


T−1(BP) ⊆ BP′ .




- Answer: Yes.

Lemma


T−1(BP) ⊆ BP′ .


A More Refined A Priori Estimate

Lemma

If h(x , α) := φαα(x)eψ(|∇φ(x)|) attains a maximum at some point(x0, α0) among all (x , α) ∈ Rn × Sn−1, then

h(x0, α0) ≤ C with C = C (R , λ,Λ, λq) > 0.

R P

ψ(t)


A More Refined A Priori Estimate

Lemma

If h(x , α) := φαα(x)eψ(|∇φ(x)|) attains a maximum at some point(x0, α0) among all (x , α) ∈ Rn × Sn−1, then

h(x0, α0) ≤ C with C = C (R , λ,Λ, λq) > 0.

R P

ψ(t)


Proof (Sketch) of Theorem: Two Cases

1: Our auxiliary function h attains a maximum at some point(x0, α0) ∈ Rn × Sn−1.

2: Our auxiliary function h is not guaranteed to achieve a maximumin Rn × Sn−1.










Proof (Sketch) of Theorem: Case 2

- First, we consider the approximating OT maps ∇φj = T j takinge−V to ecq,j−V

j+cq−q, where for j � R

V j(x) =

{V (x) x ∈ Bj(0)

+∞ x ∈ Bcj (0)

and cq,j > 0 are normalizing constants.



- Second, we use that the functions

hjε(x , α) := (φj(x + εα) + φj(x − εα)− 2φj(x))eψ(|∇φj (x)|)

achieve maxima outside BR for all ε� 1.

- Third, we formalize our original heuristics replacing pure secondderivatives with second order difference quotients to conclude thatφjαα ≤

√Λ/λ.

- Finally, we pass to the limit via stability of OT.







√Λ/λ.








√Λ/λ.



Other Theorems: 1-d and the Radially Symmetric Case


Let V : R→ R ∪ {∞} be a convex function and q be a boundedfunction such that e−V , e−V−q ∈ P(R). Then, the optimal transportT that takes e−V to e−V−q is Lipschitz and satisfies

‖ logT ′‖L∞(R) ≤ ‖q+‖L∞(R) + ‖q−‖L∞(R).


Let V : Rn → R ∪ {∞} be a convex, radially symmetric function andq be a bounded, radially symmetric function such thate−V , e−V−q ∈ P(Rn). Then, the optimal transport T that takes e−V

to e−V−q is Lipschitz: for a.e. x ∈ Rn

e−‖q+‖L∞(Rn)−‖q−‖L∞(Rn) Id ≤ ∇T (x) ≤ e‖q

+‖L∞(Rn)+‖q−‖L∞(Rn) Id .


Other Theorems: 1-d and the Radially Symmetric Case


Let V : R→ R ∪ {∞} be a convex function and q be a boundedfunction such that e−V , e−V−q ∈ P(R). Then, the optimal transportT that takes e−V to e−V−q is Lipschitz and satisfies

‖ logT ′‖L∞(R) ≤ ‖q+‖L∞(R) + ‖q−‖L∞(R).


Let V : Rn → R ∪ {∞} be a convex, radially symmetric function andq be a bounded, radially symmetric function such thate−V , e−V−q ∈ P(Rn). Then, the optimal transport T that takes e−V

to e−V−q is Lipschitz: for a.e. x ∈ Rn

e−‖q+‖L∞(Rn)−‖q−‖L∞(Rn) Id ≤ ∇T (x) ≤ e‖q

+‖L∞(Rn)+‖q−‖L∞(Rn) Id .


The End

Thank you.


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Lipschitz Changes of Variables between Perturbations · PDF fileLipschitz Changes of Variables between Perturbations of Log-concave Measures Yash Jhaveri University of Texas at Austin