Bounds for the Green Energy on SO (3) Damir Ferizovi c joint work - - PowerPoint PPT Presentation

Bounds for the Green Energy on SO(3)

Damir Ferizovi´ c

joint work with Carlos Beltr´ an (Universidad de Cantabria)

Institute of Analysis and Number Theory Graz University of Technology

December 21, 2018

Green Functions

Definition

A Green function G(x, y) for a linear differential operator L is given as the distributional solution to LxG(x, y) = δ0(x − y);

• r put differently, if we want to solve

Lu(x) = f (x) we set u(x) =

• f (y)G(x, y) dy.

*It follows that Lu(x) =

• f (y)LxG(x, y) dy =
• f (y)δ0(x − y) dy = f (x).

f-Energies of N-Point Sets

Definition

Given a non-empty set X, N ∈ N and a function f : X × X → R ∪ {±∞}; the (discrete) f -energy of X is given by E(f , N) = inf

{x1,...,xN}⊂X N

• j=k

f (xj, xk).

f-Energies of N-Point Sets

Definition

Given a non-empty set X, N ∈ N and a function f : X × X → R ∪ {±∞}; the (discrete) f -energy of X is given by E(f , N) = inf

{x1,...,xN}⊂X N

• j=k

f (xj, xk).

Example (Riesz potential)

Regard the unit sphere S2 ⊂ R3 and for s > 0, let f (x, y) = 1 x − ys . For some s < 0 the problem also makes sense (Fejes-Toth potential). The case s = 0 sometimes refers to the logarithmic potential.

Why Care?

Theorem (Beltr´ an, Corral, Del Cray)

Let M be a compact Riemannian manifold of dimension n > 1 and let G be the (normalized) Green function for its Laplace-Beltrami operator. The unique probability measure minimizing the continuous Green energy I G[µ] =

• M

G(x, y) dµ(x) dµ(y), is the uniform measure on M. Moreover, for each N > 1, let w∗

N = {x1, . . . , xN} be a set of minimizers

for the Green energy, then 1 N

• x∈w∗

N

δx

∗

⇀ λ. *“Discrete and Continuous Green Energy on Compact Manifolds” by C. Beltrn, N. Corral, and J. G. Criado Del Rey, (2017).

Lemma

The Green function for the Laplace-Beltrami operator on SO(3) is G(α, β) =

• π − ω(α−1β)
• cot

ω(α−1β) 2

• − 1.

Enter Determinantal Point Processes

We will have for any measurable function f : M × M → [0, ∞], E  

i=j

f (xi, xj)   =

• M

f (x, y)

• KH(x, x)KH(y, y) − |KH(x, y)|2

dµ(x) dµ(y), where H ⊆ L2(M) is any N–dimensional subspace in the set of square–integrable functions and KH is the projection kernel onto H.

Let’s Start Simple

A simple point process on a locally compact polish space Λ with reference measure µ is a positive Radon measure χ =

• j=1

δxj, with xj = xs for j = s. One usually identifies χ with a discrete subset of Λ.

Let’s Start Simple

A simple point process on a locally compact polish space Λ with reference measure µ is a positive Radon measure χ =

• j=1

δxj, with xj = xs for j = s. One usually identifies χ with a discrete subset of Λ. The joint intensities of χ w.r.t. µ, if they exist, are functions ρk : Λk → [0, ∞) for k > 0, such that for pairwise disjoint {Ds}k

s=1 ⊂ Λ

E k

• s=1

χ(Ds)

• =
• D1×...×Dk

ρk(y1, . . . , yk) dµ(y1) . . . dµ(yk), and ρk(y1, . . . , yk) = 0 in case yj = ys for some j = s.

Putting Determinant into Determinantal Point Processes

A simple point process is determinantal with kernel K, iff for every k ∈ N and all yj’s ρk(y1, . . . , yk) = det

• K(yj, ys)
• 1≤j,s≤k.

If the kernel is a projection kernel, then one speaks of a determinantal projection process. Hence if K(x, y) =

N

• j=1

φj(x)¯ φj(y) for some orthonormal system of φj’s, then E

• χ(Λ)
• =
• Λ

K(y, y) dµ(y) =

N

• j=1
• Λ

|φj(y)|2 dµ(y) = N.

Putting Determinant into Determinantal Point Processes

A simple point process is determinantal with kernel K, iff for every k ∈ N and all yj’s ρk(y1, . . . , yk) = det

• K(yj, ys)
• 1≤j,s≤k.

If the kernel is a projection kernel, then one speaks of a determinantal projection process. Hence if K(x, y) =

N

• j=1

φj(x)¯ φj(y) for some orthonormal system of φj’s, then E

• χ(Λ)
• =
• Λ

K(y, y) dµ(y) =

N

• j=1
• Λ

|φj(y)|2 dµ(y) = N. It follows from the Macchi–Soshnikov theorem that a simple point process with N points, associated to the projection on a finite subspace exists in Λ.

Class Functions and Integrals on SO(3)

Definition (Rotation Angle Distance)

For α, β ∈ SO(3), we set ω(α−1β) = arccos Trace(α−1β) − 1 2

• .

Class Functions and Integrals on SO(3)

Definition (Rotation Angle Distance)

For α, β ∈ SO(3), we set ω(α−1β) = arccos Trace(α−1β) − 1 2

• .

Lemma

If we are given a function f ∈ L1( SO(3)) such that we can find ˜ f ∈ L1([0, π]) with f (x) = ˜ f (ω(x)), then

• SO(3)

f (x) dµ(x) = 2 π π ˜ f (t) sin2 t 2

• dt.

*“Surface Spline Approximation on SO(3)” by T. Hangelbroek, D. Schmid;

• Appl. Comput. Harmon. Anal. Volume 31, Issue 2, 169-184 (2011).

Eigen- Values and Vectors for the Laplacian on SO(3)

Lemma

The eigenvalues of ∆ in SO(3) are λℓ = ℓ(ℓ + 1) for ℓ ≥ 0. Moreover, if Hℓ is the eigenspace associated to λℓ, then the dimension of Hℓ is (2ℓ + 1)2 and an orthonormal basis of Hℓ is given by √ 2ℓ + 1Dℓ

m,n where

−ℓ ≤ m, n ≤ ℓ and Dℓ

m,n are Wigner’s D–functions.

It is known that

l

• m=−l

l

• n=−l

Dl

m,n(α)Dl m,n(β) = U2l

• cos

ω(α−1β)

2

• ,

where U2l(x) is the Chebyshev polynomial of second kind.

Calculating the Projection Kernel for SO(3)

Thus a projection kernel on the space HL = ⊕L

ℓ=1Hℓ is given by

K(α, β) =

L

• l=0

(2l + 1)

l

• m=−l

l

• n=−l

Dl

m,n(α)Dl m,n(β)

=

L

• l=0

(2l + 1) U2l

• cos

ω(α−1β)

2

• = d

dx

L

• l=0

T2l+1(x)

• cos(...)

= d dx 1 2U2L+1(x)

• cos(...)

= C(2)

2L

• cos

ω(α−1β)

2

• .

Here, C(2)

2L , L ≥ 0, is the sequence of Gegenbauer (ultraspherical)

polynomials.

Series Representation of Green’s Functions

Theorem

Given a compact Riemannian manifold (M, g), then a system of

• rthonormal eigenfunctions {φk}∞

k=1 of the Laplacian on M with

corresponding eigenvalues {λk}∞

k=1 forms a basis for the Hilbert space

L2( SO(3)); “the” Green function is given by G(x, y) =

• k≥1

φk(x)φk(y) λk .

Series Representation of Green’s Functions

Theorem

Given a compact Riemannian manifold (M, g), then a system of

• rthonormal eigenfunctions {φk}∞

k=1 of the Laplacian on M with

corresponding eigenvalues {λk}∞

k=1 forms a basis for the Hilbert space

L2( SO(3)); “the” Green function is given by G(x, y) =

• k≥1

φk(x)φk(y) λk .

Lemma

The Green function for the Laplace-Beltrami operator on SO(3) can be written in terms of the metric ω: G(α, β) =

• π − ω(α−1β)
• cot

ω(α−1β) 2

• − 1.

An Upper Bound for the Green Energy

E  

i=j

f (xi, xj)   =

• M

f (x, y)

• KH(x, x)KH(y, y) − |KH(x, y)|2

dµ(x) dµ(y) =

• SO(3)2 G(α, β)
• C(2)

2L (1)

2 −

• C(2)

2L

• cos

ω(α−1β)

2

2 dµ(α) dµ(β) =

• SO(3)

G(α, 1)

• C(2)

2L (1)

2 −

• C(2)

2L

• cos

ω(α−1)

2

2 dµ(α) = − 2 π π

• (π − t) cot( t

2) − 1

• C(2)

2L

• cos

t

2

2 sin t 2 2 dt ...some technicalities occur... = −4 3 4 4

3 N 4 3 + O(N).

Handling Technicalities

Lemma

The Gegenbauer polynomials C(2)

n−2(x) satisfy

1 (x2 − 1)

• C(2)

n−2(x)

2 dx = O(n2 log(n)).

Lemma

The Gegenbauer polynomials C(2)

n−2(x) satisfy

1

• C(2)

n−2(x)

2 dx = n4 16 + O(n2 log(n)). Actually we have exact formulae.

A Lower Bound for the Green Energy

We define for α, β ∈ SO(3) and t > 0: Gt(α, β) =

∞

• l=1

e−l(l+1)·t 2l + 1 l(l + 1)U2l

• cos

ω(α−1β)

2

• .

A Lower Bound for the Green Energy

We define for α, β ∈ SO(3) and t > 0: Gt(α, β) =

∞

• l=1

e−l(l+1)·t 2l + 1 l(l + 1)U2l

• cos

ω(α−1β)

2

• .

Lemma (N. Elkies)

For all t > 0 and α = β we have G(α, β) ≥ Gt(α, β) − t.

A Lower Bound for the Green Energy

We define for α, β ∈ SO(3) and t > 0: Gt(α, β) =

∞

• l=1

e−l(l+1)·t 2l + 1 l(l + 1)U2l

• cos

ω(α−1β)

2

• .

Lemma (N. Elkies)

For all t > 0 and α = β we have G(α, β) ≥ Gt(α, β) − t. Then for any collection of distinct points {α1, . . . , αN} ⊂ SO(3)

N

• s=k

G(αs, αk) + N(N − 1)2t ≥

N

• s=k

G2t(αs, αk).

A Lower Bound for the Green Energy

N

• s=k

G2t(αs, αk) =

∞

• l=1

2l + 1 l(l + 1)

l

• m=−l

l

• n=−l

N

• s=k

e−l(l+1)·2tDl

m,n(αs)Dl m,n(αk) = ∞

• l=1

2l + 1 l2 + l

l

• m,n=−l
• N
• k=1

e−l(l+1)·tDl

m,n(αk)

• 2

−

N

• k=1

e−l(l+1)·2t

• Dl

m,n(αk)

• 2
• ≥ −

∞

• l=1

2l + 1 l(l + 1)

l

• m=−l

l

• n=−l

N

• k=1

e−l(l+1)·2t

• Dl

m,n(αk)

• 2

= −NG2t(α, α). A special choice for t yields

N

• s=k

G(αs, αk) ≥ −3 3 √πN

4 3 + O(N).

In Summary

Let L ≥ 0 and let HL ⊆ L2( SO(3)) be the span of the union of the eigenspaces of λ0, . . . , λL. Then, HL has dimension N = dim(HL) = 2L + 3 3

• = C(2)

2L (1) = (2L + 3)(L + 1)(2L + 1)

3 .

Theorem (Beltr´ an, F. (2018))

The minimal Green energy E(G, N) in SO(3) for N as above satisfies −3 3 √πN4/3 + O(N) ≤ E(G, N) ≤ −4 3 4 4/3 N4/3 + O(N). The whole exposition can also be thought of as a blueprint for the same set of questions on various spaces.

Thank you for your Time

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Functions and Determinantal Point Processes; American Mathematical Society, Providence, RI (2009).

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points on the sphere. Electron. J. Probab. 20 (2015), no. 23, 27 pp.

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invariant determinantal point processes in high dimensional spheres; J. Complexity 37, 76-109 (2016).

• J. Marzo and J. Ortega-Cerd: Expected Riesz energy of some determinantal

processes on flat tori; Constructive Approximation 47 (1), 75-88 (2018).

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Odd-Dimensional Spheres; arXive (2017).

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