Universal lower bounds for the energy of spherical codes: lifting - - PowerPoint PPT Presentation

Universal lower bounds for the energy of spherical codes: lifting the Levenshtein framework

• P. Boyvalenkov
• P. Dragnev

D.Hardin

• E. Saff
• M. Stoyanova

Optimal Point Configurations and Orthogonal Polynomials 2017 Castro Urdiales, Spain

Linear Programming Bounds: Notation

◮ Sn−1: unit sphere in Rn ◮ Spherical Code: A finite set C ⊂ Sn−1 with cardinality |C| ◮ r2 = |x − y|2 = 2 − 2x, y = 2 − 2t. ◮ Interaction potential h : [−1, 1) → R ◮ Riesz s-potential: h(t) = (2 − 2t)−s/2 = |x − y|−s ◮ The h-energy of a spherical code C:

E(n, h; C) :=

• x,y∈C,y=x

h(x, y), where t = x, y denotes Euclidean inner product of x and y.

◮ E(n, h; N) = min{Eh(C) | C ⊂ Sn−1, |C| = N}. ◮ Absolutely monotone h: h(k)(t) ≥ 0 for all t ∈ [−1, 1) and

all k ≥ 0.

Spherical Harmonics

◮ Harm(k): homogeneous harmonic polynomials in n variables

• f degree k restricted to Sn−1 with

rk := dim Harm(k) = k + n − 3 n − 2 2k + n − 2 k

• .

◮ Spherical harmonics (degree k): {Ykj(x) : j = 1, 2, . . . , rk}

• rthonormal basis of Harm(k) with respect to surface

measure on Sn−1.

Gegenbauer polynomials

◮ The Gegenbauer polynomials and spherical harmonics can be

defined through the Addition Formula: P(n)

k (t) := 1

rk

rk

• j=1

Ykj(x)Ykj(y), t = x, y, x, y ∈ Sn−1.

◮ {P(n) k (t)}∞ k=0 is orthogonal with respect to the weight

(1 − t2)(n−3)/2 on [−1, 1] and normalized so that P(n)

k (1) = 1.

Spherical Designs

◮ The k-th moment of a spherical code CSn−1 is

Mk(C) :=

• x,y∈C

P(n)

k (x, y) = 1

rk

rk

• j=1
• x∈C
• y∈C

Ykj(x)Ykj(y) = 1 rk

rk

• j=1
• x∈C

Ykj(x) 2 ≥ 0.

◮ Mk(C) = 0 if and only if x∈C Y (x) = 0 for all

Y ∈ Harm(k).

◮ If Mk(C) = 0 for 1 ≤ k ≤ τ, then C is called a spherical

τ-design and

• Sn−1 p(y) dσn(y) = 1

N

• x∈C

p(x), ∀ polys p of deg at most τ.

‘Good’ potentials for lower bounds

Suppose f : [−1, 1] → R is of the form f (t) =

∞

• k=0

fkP(n)

k (t),

fk ≥ 0 for all k ≥ 1. (1) f (1) = ∞

k=0 fk < ∞ =

⇒ convergence is absolute and uniform. Then: E(n, C; f ) =

• x,y∈C

f (x, y) − f (1)N =

∞

• k=0

fk

• x,y∈C

P(n)

k (x, y) − f (1)N

≥ f0N2 − f (1)N = N2

• f0 − f (1)

N

• .

‘Good’ potentials for lower bounds

Suppose f : [−1, 1] → R is of the form f (t) =

∞

• k=0

fkP(n)

k (t),

fk ≥ 0 for all k ≥ 1. (1) f (1) = ∞

k=0 fk < ∞ =

⇒ convergence is absolute and uniform. Then: E(n, C; f ) =

• x,y∈C

f (x, y) − f (1)N =

∞

• k=0

fkMk(C) − f (1)N ≥ f0N2 − f (1)N = N2

• f0 − f (1)

N

• .

Let An,h := {f : f (t) ≤ h(t), t ∈ [−1, 1), fk ≥ 0, k = 1, 2, . . . }.

Thm (Delsarte-Yudin LP Bound)

For any C ⊂ Sn−1 with |C| = N E(n, h; C) ≥ N2(f0 − f (1) N ). (2) C satisfies E(n, h; C) = E(n, f ; C) = N2(f0 − f (1)

N ) ⇐

⇒ (a) f (t) = h(t) for t ∈ {x, y : x = y, x, y ∈ C}, and (b) for all k ≥ 1, either fk = 0 or Mk(C) = 0.

Example: n-Simplex on Sn−1

Let C be N = n + 1 points on Sn−1 forming a regular simplex. Then there is only one inner product α0 = x, y for x = y ∈ C.

◮ The first degree Gegenbauer polynomial P(n) 1 (t) = t. ◮ M1(C) = x,y∈Cx, y = | x∈C x|2 = 0.

If h is convex and increasing then linear interpolant f (t) = h(α0) + h′(α0)(t + 1/n) has (a) f1 = h′(α0) ≥ 0 and (b) f (t) ≤ h(t) = ⇒ E(n, h; N = n + 1) = E(n, h; C) .

Coding Problem: Separation

Consider ∆(C) := minx=y∈C |x − y| Suppose f ∈ C[−1, 1] has nonnegative Gegenbauer coefficients fk ≥ 0 and that f (t) ≤ 0 for t ∈ [−1, t0) for some t0 ∈ (−1, 1). Let M = maxt f (t) and define h(t) =

• −1 ≤ t ≤ t0

M t0 < t ≤ 1 . Then:

◮ f ∈ A(n, h). ◮ E(n, C; h) > 0 ⇐

⇒ ∆(C) < cos(t0).

◮ f0 − f (1) N

> 0 = ⇒ N ≤ f (1)/f0 if there is any C with ∆(C) ≥ cos(t0) and |C| = N.

Linear program: Maximize D-Y lower bound

Maximizing Delsarte-Yudin lower bound is a linear programming problem. Max F(f ) := N2(f0 − f (1) N ), subject to f ∈ An,h. For a subspace Λ ⊂ C([−1, 1]), we consider W(n, N, Λ; h) := sup

f ∈Λ∩An,h

N2(f0 − f (1)/N). (3)

1/N-Quadrature Rules and Hermite Interpolation

◮ For a subspace Λ ⊂ C([−1, 1]) and N > 1, we say

{(αi, ρi)}k

i=1 is a 1/N-quadrature rule exact for Λ if

−1 ≤ αi < 1, ρi > 0 for i = 1, 2, . . . , k, and f0 = γn 1

−1

f (t)(1−t2)(n−3)/2dt = f (1) N +

k

• i=1

ρif (αi), (f ∈ Λ).

◮ For f ∈ Λ ∩ An,h,

f0 − f (1) N =

k

• i=1

ρif (αi) ≤

k

• i=1

ρih(αi), and so W(n, N, Λ; h) ≤

k

• i=1

ρih(αi). (4)

◮ If there is some f ∈ Λ ∩ An,h such that f (αi) = h(αi) for

i = 1, . . . , k, then equality holds in (4).

Sharp Codes

A spherical design C of degree m yields a quadrature rule that is exact for Λ = Πm (polynomials of degree m) with nodes {x, y | x = y ∈ C}.

Definition

A spherical code C ⊂ Sn−1 is sharp if there are m inner products between distinct points in it and C is a spherical (2m − 1)-design.

Theorem (Cohn and Kumar, 2006)

If C ⊂ Sn−1 is a sharp code, then C is universally optimal; i.e., C is h-energy optimal for any h that is absolutely monotone on [−1, 1]. Idea of proof: Show Hermite interpolant to h is in A(n, h).

Levenshtein Framework - 1/N-Quadrature Rule

◮ For every fixed (cardinality) N > D(n, 2k − 1)(the DGS

bound) there exist real numbers −1 ≤ α1 < α2 < · · · < αk < 1 and ρ1, ρ2, . . . , ρk, ρi > 0 for i = 1, 2, . . . , k, such that the equality f0 = f (1) N +

k

• i=1

ρif (αi) holds for every real polynomial f (t) of degree at most 2k − 1.

◮ The numbers αi, i = 1, 2, . . . , k, are the roots of the equation

Pk(t)Pk−1(s) − Pk(s)Pk−1(t) = 0, where s = αk, Pi(t) = P(n−1)/2,(n−3)/2

i

(t) is a Jacobi polynomial.

Universal Lower Bound (ULB)

ULB Theorem - (BDHSS, 2016)

Let h be a fixed absolutely monotone potential, n and N be fixed, and N ≥ D(n, 2k − 1). Then the Levenshtein nodes {αi} provide the bounds E(n, N, h) ≥ N2

k

• i=1

ρih(αi). The Hermite interpolants at these nodes are the optimal polynomials which solve the finite LP in the class Pτ ∩ An,h.

Improvement of ULB and Test Functions

Test functions (Boyvalenkov, Danev, Boumova, ‘96) Qj(n, αk) := 1 N +

k

• i=1

ρiP(n)

j

(αi).

Subspace ULB Improvement Theorem (BDHSS, 2016)

Let {(αi, ρi)}k

i=1 be a 1/N-quadrature rule that is exact for a

subspace Λ ⊂ C([−1, 1]) and such that equality holds in (4), namely W(n, N, Λ; h) = N2

k

• i=1

ρih(αi). Suppose Λ′ = Λ span {P(n)

j

: j ∈ I} for some index set I ⊂ N. If Q(n)

j

:= 1

N + k i=1 ρiP(n) j

(αi) ≥ 0 for j ∈ I, then W(n, N, Λ′; h) = W(n, N, Λ; h) = N2

k

• i=1

ρih(αi).

ULB Improvement for (4, 24)-codes

The case n = 4, N = 24 is important. C4 consists of the minimal length vectors in D4 lattice. |C4| = 24.

◮ Kissing numbers in R4 - solved by Musin in 2003 using

modification of linear programming bounds.

◮ C4 is conjectured to be maximal code but not yet proved. ◮ C4 is not universally optimal - Cohn, Conway, Elkies, Kumar -

2008.

ULB Improvement for (4, 24)-codes

For n = 4, N = 24 Levenshtein nodes and weights (exact for Π5) are: {α1, α2, α3} = {−.817352..., −.257597..., .474950...} {ρ1, ρ2, ρ3} = {0.138436..., 0.433999..., 0.385897...}, The test functions for (4, 24)-codes are: Q6 Q7 Q8 Q9 Q10 Q11 Q12 0.0857 0.1600 −0.0239 −0.0204 0.0642 0.0368 0.0598 Motivated by this we define Λ := span{P(4)

0 , . . . , P(4) 5 , P(4) 8 , P(4) 9 }.

ULB Improvement for (4, 24)-codes - Main Theorem

Theorem

The collection of nodes and weights {(αi, ρi)}4

i=1

{α1, α2, α3, α4} = {−0.86029..., −0.48984..., −0.19572, 0.478545...} {ρ1, ρ2, ρ3, ρ4} = {0.09960..., 0.14653..., 0.33372..., 0.37847...}, define a 1/N-quadrature rule that is exact for Λ. A Hermite-type interpolant H(t) = H(h; (t − α1)2 . . . (t − α4)2) ∈ Λ ∩ An,h s. t. , H(αi) = h(αi), H′(αi) = h′(αi), i = 1, . . . , 4 exists, and hence, improved ULB holds E(4, 24; h) ≥ N2

4

• i=1

ρih(αi). Moreover, the new test functions Q(n)

j

≥ 0, j = 0, 1, . . . , and hence H(t) is the optimal LP solution among all polynomials in A4,h.

LP Optimal Polynomial for (4, 24)-code

1.0 0.5 0.5 1.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure: The (4, 24)-code optimal interpolant - Coulomb potential

Sketch of the proof

The following lemma plays an important role in the proof of the positive definiteness of the Hermite-type interpolants described in Theorem 2.

Lemma

Suppose T := {t1 ≤ · · · ≤ tk} ⊂ [a, b] is a set of nodes and B := {g1, . . . , gk} is a linearly independent set of functions on [a, b] such that the matrix gB = (gi(tj))k

i,j=1 is invertible

(repetition of points in the multiset yields corresponding derivatives). Let H(t, h; span(B)) denote the Hermite-type interpolant associated with T. Then H(t, h; span(B)) =

k

• i=1

h[t1, . . . , ti]H(t, (t−t1) · · · (t−ti−1); span(B)), (5) where h[t1, . . . , ti] are the divided differences of h.

600 cell

◮ C600 = 120 points in R4. Each x ∈ C has 12 nearest neighbors

forming an icosahedron (Voronoi cells are dodecahedra).

◮ 8 inner products between distinct points in C600:

{−1, ±1/2, 0, (±1 ± √ 5)/4}.

◮ 2*7+1 or 2*8 interpolation conditions (would require 14 or 15

design)

◮ C600 is an 11 design, but almost a 19 design (only 12-th

moment is nonzero). I.e. quadrature rule from C is exact on subspace Λ = Π19 ∩ {P(4)

12 }⊥. ◮ Andreev (1999) found polynomial in Π17 to show 600-cell

Cohn and Kumar find family of 17-th degree polynomials that proves universal optimality of C600 and they require f11 = f12 = f13 = 0; Λ0

17 = Π17 ∩ {P(4) 11 , P(4) 12 , P(4) 13 }⊥ with

Lagrange condition at -1.

600 cell

◮ Levensthein: n = 4, N = 120, quadrature: 6 nodes exact for

degree = 11.

◮ Test functions: Q11, Q12 > 0, Q13, Q14 < 0. ◮ Find quadrature rule for Λ15 = Π15 ∩ {P(4) 12 , P(4) 13 }⊥. ◮ Verify Hermite interpolation works in Λ15. ◮ New test functions Q11, Q12 > 0 so this solves the linear

program in Π15.

◮ Degree 17. Try Λ1 17 = Π17 ∩ {P(4) 12 , P(4) 13 }⊥, double

interpolation at -1. It works.

◮ Degree 17. Try Λ2 17 = Π17 ∩ {P(4) 11 , P(4) 12 }⊥, double

interpolation -1. It works.

◮ Degree 17. All solutions form triangle.