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
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Sn−1 : unit sphere in Rn
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Spherical Code: A finite set C ⊂ Sn−1 with cardinality |C |
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r 2 = |x − y |2 = 2 − 2hx, y i = 2 − 2t.
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Interaction potential h : [−1, 1) → R
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Riesz s-potential: h(t) = (2 − 2t)−s/2 = |x − y |−s
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The h-energy of a spherical code C :
X
E (n, h; C ) :=
h(hx, y i),
x,y ∈C ,y 6=x
where t = hx, y i denotes Euclidean inner product of x and y .
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E(n, h; N) = min{Eh (C ) | C ⊂ Sn−1 , |C | = N}.
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Absolutely monotone h: h(k) (t) ≥ 0 for all t ∈ [−1, 1) and
all k ≥ 0.
Spherical Harmonics
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Harm(k): homogeneous harmonic polynomials in n variables
of degree k restricted to Sn−1 with
k +n−3
2k + n − 2
.
rk := dim Harm(k) =
k
n−2
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Spherical harmonics (degree k): {Ykj (x) : j = 1, 2, . . . , rk }
orthonormal basis of Harm(k) with respect to surface
measure on Sn−1 .
Gegenbauer polynomials
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The Gegenbauer polynomials and spherical harmonics can be
defined through the Addition Formula:
(n)
Pk (t) :=
rk
1 X
Ykj (x)Ykj (y ),
rk
t = hx, y i, x, y ∈ Sn−1 .
j=1
I
(n)
{Pk (t)}∞
k=0 is orthogonal with respect to the weight
(n)
2
(n−3)/2
(1 − t )
on [−1, 1] and normalized so that Pk (1) = 1.
Spherical Designs
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The k-th moment of a spherical code C Sn−1 is
Mk (C ) :=
X
(n)
Pk (hx, y i)
x,y ∈C
=
rk X X
1 X
Ykj (x)Ykj (y )
=
rk
j=1 x∈C y ∈C
!2
rk
X
1 X
Ykj (x)
rk
j=1
≥ 0.
x∈C
P
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Mk (C ) = 0 if and only if
Y ∈ Harm(k).
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If Mk (C ) = 0 for 1 ≤ k ≤ τ , then C is called a spherical
τ -design and
Z
1 X
p(y ) dσn (y ) =
p(x), ∀ polys p of deg at most τ .
N
Sn−1
x∈C
x∈C
Y (x) = 0 for all
‘Good’ potentials for lower bounds
Suppose f : [−1, 1] → R is of the form
f (t) =
∞
X
(n)
fk Pk (t),
fk ≥ 0 for all k ≥ 1.
(1)
k=0
f (1) =
P∞
k=0 fk
< ∞ =⇒ convergence is absolute and uniform.
Then:
E (n, C ; f ) =
=
X
x,y ∈C
∞
X
fk
f (hx, y i) − f (1)N
X
(n)
Pk (hx, y i) − f (1)N
x,y ∈C
k=0
f (1)
.
≥ f0 N − f (1)N = N f0 −
N
2
2
‘Good’ potentials for lower bounds
Suppose f : [−1, 1] → R is of the form
f (t) =
∞
X
(n)
fk Pk (t),
fk ≥ 0 for all k ≥ 1.
(1)
k=0
f (1) =
P∞
k=0 fk
< ∞ =⇒ convergence is absolute and uniform.
Then:
E (n, C ; f ) =
=
X
x,y ∈C
∞
X
f (hx, y i) − f (1)N
fk Mk (C )
− f (1)N
k=0
f (1)
≥ f0 N − f (1)N = N f0 −
.
N
2
2
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 ) ≥ N 2 (f0 −
f (1)
).
N
C satisfies E (n, h; C ) = E (n, f ; C ) = N 2 (f0 −
f (1)
N )
(2)
⇐⇒
(a) f (t) = h(t) for t ∈ {hx, y i : x 6= 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 = hx, y i for x 6= y ∈ C .
I
I
(n)
The first degree Gegenbauer polynomial P1 (t) = t.
P
P
M1 (C ) = x,y ∈C hx, y i = | x∈C x|2 = 0.
If h is convex and increasing then linear interpolant
f (t) = h(α0 ) + h0 (α0 )(t + 1/n)
has (a) f1 = h0 (α0 ) ≥ 0 and (b) f (t) ≤ h(t) =⇒
E(n, h; N = n + 1) = E (n, h; C ) .
Coding Problem: Separation
Consider ∆(C ) := minx6=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
(
0 −1 ≤ t ≤ t0
h(t) =
.
M t0 < t ≤ 1
Then:
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f ∈ A(n, h).
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E (n, C ; h) > 0 ⇐⇒ ∆(C ) < cos(t0 ).
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(1)
> 0 =⇒ N ≤ f (1)/f0 if there is any C with
f0 − f N
∆(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 ) := N 2 (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
N 2 (f0 − f (1)/N).
(3)
1/N-Quadrature Rules and Hermite Interpolation
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For a subspace Λ ⊂ C ([−1, 1]) and N > 1, we say
{(αi , ρi )}ki=1 is a 1/N-quadrature rule exact for Λ if
−1 ≤ αi < 1, ρi > 0 for i = 1, 2, . . . , k, and
Z
1
f0 = γn
2 (n−3)/2
f (t)(1−t )
−1
I
k
f (1) X
+
ρi f (αi ),
dt =
N
(f ∈ Λ).
i=1
For f ∈ Λ ∩ An,h ,
k
k
i=1
i=1
X
f (1) X
f0 −
=
ρi f (αi ) ≤
ρi h(αi ),
N
and so
W(n, N, Λ; h) ≤
k
X
ρi h(αi ).
i=1
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If there is some f ∈ Λ ∩ An,h such that f (αi ) = h(αi ) for
i = 1, . . . , k, then equality holds in (4).
(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
{hx, y i | x 6= 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
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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
k
f0 =
f (1) X
+
ρi f (αi )
N
i=1
holds for every real polynomial f (t) of degree at most 2k − 1.
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The numbers αi , i = 1, 2, . . . , k, are the roots of the equation
Pk (t)Pk−1 (s) − Pk (s)Pk−1 (t) = 0,
(n−1)/2,(n−3)/2
where s = αk , Pi (t) = Pi
polynomial.
(t) is a Jacobi
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
k
X
E(n, N, h) ≥ N 2
ρi h(αi ).
i=1
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)
k
1 X
(n)
Qj (n, αk ) :=
+
ρi Pj (αi ).
N
i=1
Subspace ULB Improvement Theorem (BDHSS, 2016)
Let {(αi , ρi )}ki=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
k
X
W(n, N, Λ; h) = N 2
ρi h(αi ).
i=1
(n)
Suppose Λ0 = Λ span {Pj : j ∈ I} for some index
P
(n)
(n)
If Qj := N1 + ki=1 ρi Pj (αi ) ≥ 0 for j ∈ I, then
L
0
W(n, N, Λ ; h) = W(n, N, Λ; h) = N
2
k
X
set I ⊂ N.
ρi h(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.
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Kissing numbers in R4 - solved by Musin in 2003 using
modification of linear programming bounds.
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C4 is conjectured to be maximal code but not yet proved.
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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
0.0857
Q7
0.1600
Q8
−0.0239
Q9
−0.0204
Q10
0.0642
Q11
0.0368
Motivated by this we define
(4)
(4)
(4)
(4)
Λ := span{P0 , . . . , P5 , P8 , P9 }.
Q12
0.0598
ULB Improvement for (4, 24)-codes - Main Theorem
Theorem
The collection of nodes and weights {(αi , ρi )}4i=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 0 (αi ) = h0 (αi ),
i = 1, . . . , 4
exists, and hence, improved ULB holds
E(4, 24; h) ≥ N 2
4
X
ρi h(αi ).
i=1
(n)
Moreover, the new test functions Qj ≥ 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
3.0
2.5
2.0
1.5
1.0
0.5
-1.0
-0.5
0.5
1.0
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 ))ki,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
X
h[t1 , . . . , ti ]H(t, (t−t1 ) · · · (t−ti−1 ); span(B)),
i=1
(5)
where h[t1 , . . . , ti ] are the divided differences of h.
600 cell
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C600 = 120 points in R4 . Each x ∈ C has 12 nearest neighbors
forming an icosahedron (Voronoi cells are dodecahedra).
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8 inner products between
√ distinct points in C600 :
{−1, ±1/2, 0, (±1 ± 5)/4}.
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2*7+1 or 2*8 interpolation conditions (would require 14 or 15
design)
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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
(4)
subspace Λ = Π19 ∩ {P12 }⊥ .
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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
(4)
(4)
(4)
f11 = f12 = f13 = 0; Λ017 = Π17 ∩ {P11 , P12 , P13 }⊥ with
Lagrange condition at -1.
600 cell
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Levensthein: n = 4, N = 120, quadrature: 6 nodes exact for
degree = 11.
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Test functions: Q11 , Q12 > 0, Q13 , Q14 < 0.
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Find quadrature rule for Λ15 = Π15 ∩ {P12 , P13 }⊥ .
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Verify Hermite interpolation works in Λ15 .
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New test functions Q11 , Q12 > 0 so this solves the linear
program in Π15 .
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Degree 17. Try Λ117 = Π17 ∩ {P12 , P13 }⊥ , double
interpolation at -1. It works.
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Degree 17. Try Λ217 = Π17 ∩ {P11 , P12 }⊥ , double
interpolation -1. It works.
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Degree 17. All solutions form triangle.
(4)
(4)
(4)
(4)
(4)
(4)