Spherical Designs and Determinantal Point Processes Masatake HIRAO ( - - PowerPoint PPT Presentation

Spherical Designs and Determinantal Point Processes

Masatake HIRAO (平尾 将剛)

(Aichi Prefectural University, Japan) ——————————————— 2015 April 22 Workshop on Spherical Design and Numerical Analysis@SJTU

• 1. Motivation

Problem in Geostatistics We want to investigate, e.g., temperature, GH gas, magnetic field, ... Where do we have to observe these data?

via Wikipedia

To challenge this optimal sampling problem, we study by combining

• “exact-type” cubature formula (Numerical analysis)
• Euclidean/spherical or combinatorial design

(Combinatorics)

• Hirao-Sawa-Jimbo(’15), Sawa-Hirao(15+)

Today, instead of the two above techniques, we focus on

• Determinantal point process(DPP)

(Probability theory)

• Quasi-Monte Carlo(QMC) design (“robust-type” cubature formula)

• I believe that good approximation points are also good observation

points for many statistical methods (e.g., regression analysis, kernel methods, ...).

• 1. Motivation
• 2. Spherical design and problem
• 3. Determinatal point process
• 4. QMC design and main results

Spherical design

X ⊂ Sd, |X| < ∞ Definition A set X is called a spherical t-design of Sd, if the following equation holds 1 |X| ∑

x∈X

f(x) = ∫

x∈Sd f(x) dν(x),

∀f ∈ Pt(Sd), where ν is the normalized surface meas. of Sd.

• Note. Spherical designs give a good approximation for other spaces.
• Ex. (spherical design of S2)

1 8

8

∑

i=1

f(xi) = 1 |S2| ∫

S2 f(x) dν(x),

∀f ∈ P3(R3).

1 N

N

∑

i=1

f(xi) = 1 |S2| ∫

S2 f(x) dν(x),

∀f ∈ Pt(R3). ⋆ t = 2 N = 4 ⋆ t = 3 N = 8 N = 6 ⋆ t = 5 N = 20 N = 12 Theorem (Bondarenko et al., ’11). Given t, there always exists a spherical t-design of Sd with N ≍ td (← lower bound order)

Regression analysis on Sd & our problems

f1(x), . . . , fm(x): a basis of Pe(Sd) (m := dim(Pe(Sd))) We consider a full regression model of degree e: Y (x) = θ1f1(x) + · · · + θmfm(x) + ϵ(x), x ∈ Sd. θ1, . . . , θm: unknown true parameters, ϵ(x): a random error.

• Ex. {Yl,k | k = 1, . . . , Z(d, l)}: a basis of Harml(Sd).

Y (x) =

e

∑

l=0 Z(d,l)

∑

k=1

θk,lYk,l(x) + ϵ(x).

⋆ A function f ∈ L2(Sd) admits the Fourier expansion f(x) =

∞

∑

l=0 Z(d,l)

∑

k=1

ˆ fk,lYk,l(x), ˆ fk,l := ∫

Sd f(x)Yk,l(x)dν(x).

Thus, we want to apply to higher degrees and higher dimension cases.

Theorem (e.g., Neumaier-Seidel, ’92). An experimental design Sd is optimal of degree e, iff it is a spherical 2e-design (with repeated points allows).

• Note. In order to estimate θ1, . . . , θm,

a set of spherical design is “good” observation points. Problem How we can construct a spherical t-design with sample size N ≍ td for large t?

• It is not easy (for me) to construct such a design. Thus, we try to

construct “approximate” spherical t-design by using DPP. Namely, we use random scattered points on Sd.

• 3. Determinantal point process(DPP)

Macchi (1975): position of an electron in quantum mechanics. . . . Peres-Vir´ ag(2005), Krishnapur (2009), Hough et al.(2009) . . .

• Example. Poisson and Ginibre point process
• Note. R ⊂ Rn or Cn

Q := Q(R): Z≥0-valued Radon meas., i.e., Q ∋ ξ = ∑

i δxi

(xi ∈ R)

• point process: Q-valued random variable (ex. Poisson, Cox, Hard-

Core, ...)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Poisson point process

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Ginibre point process

DPP → there is an effective repulsion among particles.

• Example. Ginibre point proc. with finite particles
• 30

20 10 10 20 30 30 20 10 10 20 30

Ginibre: n 400

• 30

20 10 10 20 30 30 20 10 10 20 30

Ginibre: n 900

Figure: 400-particles(left) and 900-particles(right) of Ginibre point processes

Poisson and Spherical ensemble

Figure: 100-particles of Poisson(left) and spherical ensemble(right)

Figure: 500-particles of Poisson(left) and Spherical ensemble(right)

Spherical ensemble as DPP

Definition Determinantal point process (DPP) is a point process having deteminantal correlation functions ρk(z1, . . . , zk) = det [ K(zi, zj) ]n

i,j=1,

for some specified function K(z, w).

• J B Hough, M Krishnapur, Y Peres, and B Virag, Zeros of Gaussian Analytic Functions and

Determinantal Point Processes. American Mathematical Society, Providence, RI. 2004

• Ex. An, Bn: indep. n × n random matrices

with i.i.d. standard complex Gaussian entries. {λ1, . . . , λn} ⊂ C: the set of eigenvalues of A−1

n Bn.

Figure: 100-eigenvalues of a matrix A−1

100B100

• Krishnapur(’09) showed that these eigenvalues is a DPP:

For any continuous, compactly supported fun. F : Ck → C, E ∑

i1,...,ik pairwise distinct

F(λi1, . . . , λik), = ∫

Ck F(z1, . . . , zk)ρ(n) k

(z1, . . . , zk) dµ(z1) · · · dµ(zk) dµ(z) :=

n π(1+|z|2)n+1dz (dz: the Lebesgue meas. on C)

Deteminantal correlation fun.: ρ(n)

k

(z1, . . . , zk) = det [ K(n)(zi, zj) ]n

i,j=1 = det

[ (1 + zi ¯ zj)n−1]n

i,j=1.

g: the stereographic proj. from the north pole onto {(t1, t2, 0); t1, t2 ∈ R}. ⇒ Pi = g−1(λi) ⇒ Prop.

1 nX (n) := 1 n

∑n

i=1 δPi converges to the uniform meas. on S2.

• Lem. 2-point correlation function is given as the follows:

ˆ ρ(n)

2

(p, q) = ( n 4π)2(1 − (|p − q|2 4 )n−1), ∀p, q ∈ S2. Thus, for any F we have E ∑

i̸=j

F(Pj, Pi) = ∫

S2

∫

S2 F(p, q)ˆ

ρ(n)

2

(p, q) dν(p)dν(q).

⋆ By using this formula, we can calculate, e.g., Riesz s-energy F(x, y) = ∥x − y∥s.

Note: {ξi}: i.i.d. standard complex Gaussian random variables. fn(z) :=

n

∑

i=0

√(n i ) ξizi, n = 1, 2, . . . Then, the behavior of zeros of fn(z) is the same as spherical ensem- ble(Krishnapur,’09).

D Armentano, C Beltran, M Shub, Minimizing the discrete logarithmic energy on the sphere: the role of random polynomials, Trans. AMS 2009

Peres-Vir´ ag(’05), Krishnapur(’09): DPP ↔ Random matrix, Random analytic fun.

• 4. QMC design and main results

Sobolev space Hs(Sd)

Brauchart-Saff-Sloan-Womersley(’14): Hs(Sd) (⊂ H0(Sd) = L2(Sd))

Note: Hs(Sd) is a reproducing kernel Hilbert space.

Brauchart-Dick-Saff-Sloan-Wang-Womersley(’14+): W s

p(Sd)

{Yl,k | k = 1, . . . , Z(d, l)}: a basis of Harml(Sd), λl := l(l + d − 1).

• For f ∈ Hs(Sd) (s > 0), it holds that

∞

∑

l=0 Z(d,l)

∑

k=1

(1 + λl)s| ˆ fl,k|2 < ∞, ˆ fl,k = ∫

Sd f(x)Yl,k(x) dν(x)

• ∥f∥Hs :=
• ∑∞

l=0

∑Z(d,l)

k=1

a(s)

l

• ˆ

fl,k

• 2
• 1/2

.

• (f, g)Hs := ∑∞

l=0

∑Z(d,l)

k=1 (a(s) l

)−1 ˆ fl,kˆ gl,k, a(s)

l

≍ (1 + λl)−s

Worst-case error and QMC design

XN := {x1, . . . , xN} ⊂ Sd; Q[XN](f) := 1

N

∑

x∈XN f(x);

I(f) := ∫

Sd f(x)dν(x).

wce(Q[XN]; Hs(Sd)) := supf∈Hs(Sd)

∥f∥H≤1

• Q[XN](f) − I(f)
• Note: As a consequence of the definition we have the following error

bound for any f ∈ Hs(Sd):

• Q[XN](f) − I(f)
• ≤ wce(Q[XN; Hs(Sd)])∥f∥H.

⋆ An analog of well-known Koksma-Hlawka inequality.

Theorem (BSSW, ’14). Given s > d/2, there exists C(s, d) > 0 depending on Hs(Sd)-norm, s.t., for every N-points spherical t- design XN on Sd there holds wce(Q[XN]; Hs(Sd)) ≤ C(s, d) ts . Recalling Bondarenko et al.(’11): Given t, there always exists a spherical t-design of Sd with N ≍ td. (i.e., ts ≍ Ns/d) Definition Given s > d/2, a sequence (XN) of N-point configura- tions on Sd with N → ∞ is said to be a sequence of QMC designs for Hs(Sd) if there exists c(s, d) > 0, independent of N, such that wce(Q[XN]; Hs(Sd)) ≤ c(s, d) Ns/d .

Probabilistic approach (BSSW, ’14)

Theorem (BSSW,’14). Given s > d/2, E [ wce(Q[XN]; Hs(Sd))2]1/2 = b(s, d) N1/2 for some constant b(s, d) > 0, where the points x1, . . . , xN are independently and uniformly distributed on Sd.

Theorem (BSSW,’14). Let (DN) be a sequence of partitions of Sd into N equal area subsets Dj,N, j = 1, . . . , N, s.t., diamDj,N ≤ c/N1/d, where c is a positive constant independently of j and

• N. Let XN = {x1,N, . . . , xN,N}, where xj,N is chosen randomly

from Dj,N with respect to uniform measure on Dj,N. Then, for d/2 < s < d/2 + 1, β′ Ns/d ≤ E [ wce(Q[XN]; Hs(Sd))2]1/2 ≤ β Ns/d Note: Let d = 2. Given 1 < s < 2, β′ Ns/2 ≤ E [ wce(Q[XN]; Hs(S2))2]1/2 ≤ β Ns/2

Main small result

Claim Let d = 2. Given 1 < s < 2, E [ wce(Q[X (N)]; Hs(S2))2]1/2 = 2s−1B(s, N)1/2, where X (N) is the set of N-particle spherical ensemble, and B(s, N) is the Beta function. ⋆ This is a well-known result?? ⋆ Given s, by using Stirling’s formula, B(s, N)1/2 ∼ Γ(s)1/2

Ns/2 .

Sketch of Proof: By combining the following eqs, we obtain the desired result:

• BSSW(’14): for 1 < s < 2,

wce(Q[X (N); Hs(S2)])2 = 22s−1 2s − 1 N2 ∑

1≤i,j≤N

|Pj − Pi|2s−2

• Alishahi-Zamani(’14+): for 1 < s < 2,

1 N2 E ∑

1≤i,j≤N

|Pj − Pi|2s−2 = 22s−1 2s − 22s−2B(s, N)

Higher dimensional cases (d ≥ 3)

• We want to construct a DPP on Sd. However, I have not succeed

to construct a “good” analog of spherical ensemble on Sd. Example. R ⊂ Rd or Cd Theorem (e.g, Hough et al, ’04). Suppose {ϕk}n

k=1 is an orthonor-

mal set in L2(R). Then, there exists a DPP with kernel K(x, y) =

n

∑

k=1

ϕk(x)¯ ϕk(y). Thus, we set a correlation fun. as follows: for x, y ∈ Sd, KL(x·y) := KL(x, y) =

L

∑

l=0 Z(d,l)

∑

k=1

Yl,k(x)Yl,k(y) =

L

∑

l=0

Z(d, l)P (d)

l

(x·y).

⋆ KL(x) is an orthogonal polynomial of degree L w.r.t. ∫ 1

−1 · (1 − x)d/2(1 + x)d/2−1dx.

Remark

• 1

N

∑

x∈XN δx converges to the uniform meas. on Sd.

• The spherical cap L2-discrepancy of XN:

DC

L2(XN) ≤ cs,d

N1/2.

• For d/2 < s < d/2 + 1,

E[{wce(Q[XN]; Hs(Sd))}2] = 1 N2 ∫

Sd

∫

Sd KL (x · y)2 |x − y|2s−d dσ(x)dσ(y)

( ≤ DC

L2(XN) ≤ cs,d

N1/2 )

⋆ The above inequalities are not good.

Some references

• K Alishahi, M S Zamani, The spherical ensemble and uniform

distribution of points on the sphere, arXiv:1407.5832v1

• D Armentano, C Beltran, M Shub, Minimizing the discrete loga-

rithmic energy on the sphere: the role of random polynomials, Trans. AMS 2009

• J Brauchart, E Saff, I Sloan, R Womersley, Optimal order quasi-

Monte Carlo integration schemes on the sphere, Math Comp 2014

• J Brauchart, J Dick, E Saff, I Sloan, Y Wang, R Womersley,

Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces, arXiv:1407.831v

• J Brauchart, J Dick, L. Fang, Spatial low-discrepancy sequences,

spherical cone discrepancy, and applications in financial modeling, arXiv:1408.4609

• J B Hough, M Krishnapur, Y Peres, and B Virag, Zeros of Gaussian

Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence, RI. 2004