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 ◮ S n − 1 : unit sphere in R n ◮ Spherical Code: A finite set C ⊂ S n − 1 with cardinality | C | ◮ r 2 = | x − y | 2 = 2 − 2 � x , y � = 2 − 2 t . ◮ Interaction potential h : [ − 1 , 1) → R ◮ Riesz s -potential: h ( t ) = (2 − 2 t ) − s / 2 = | x − y | − s ◮ The h -energy of a spherical code C : � E ( n , h ; C ) := h ( � x , y � ) , x , y ∈ C , y � = x where t = � x , y � denotes Euclidean inner product of x and y . ◮ E ( n , h ; N ) = min { E h ( C ) | C ⊂ S n − 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 of degree k restricted to S n − 1 with � k + n − 3 � � 2 k + n − 2 � r k := dim Harm ( k ) = . n − 2 k ◮ Spherical harmonics (degree k ): { Y kj ( x ) : j = 1 , 2 , . . . , r k } orthonormal basis of Harm ( k ) with respect to surface measure on S n − 1 .

Gegenbauer polynomials ◮ The Gegenbauer polynomials and spherical harmonics can be defined through the Addition Formula : r k k ( t ) := 1 P ( n ) � t = � x , y � , x , y ∈ S n − 1 . Y kj ( x ) Y kj ( y ) , r k j =1 ◮ { P ( n ) k ( t ) } ∞ k =0 is orthogonal with respect to the weight (1 − t 2 ) ( 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 C S n − 1 is r k k ( � x , y � ) = 1 P ( n ) � � � � M k ( C ) := Y kj ( x ) Y kj ( y ) r k x , y ∈ C j =1 x ∈ C y ∈ C � 2 r k �� = 1 � Y kj ( x ) ≥ 0 . r k j =1 x ∈ C ◮ M k ( C ) = 0 if and only if � x ∈ C Y ( x ) = 0 for all Y ∈ Harm ( k ). ◮ If M k ( C ) = 0 for 1 ≤ k ≤ τ , then C is called a spherical τ -design and � S n − 1 p ( y ) d σ n ( y ) = 1 � p ( x ) , ∀ polys p of deg at most τ . N x ∈ C

‘Good’ potentials for lower bounds Suppose f : [ − 1 , 1] → R is of the form ∞ f k P ( n ) � f ( t ) = k ( t ) , f k ≥ 0 for all k ≥ 1 . (1) k =0 f (1) = � ∞ k =0 f k < ∞ = ⇒ convergence is absolute and uniform. Then: � E ( n , C ; f ) = f ( � x , y � ) − f (1) N x , y ∈ C ∞ P ( n ) � � = f k k ( � x , y � ) − f (1) N k =0 x , y ∈ C � f 0 − f (1) � ≥ f 0 N 2 − f (1) N = N 2 . N

‘Good’ potentials for lower bounds Suppose f : [ − 1 , 1] → R is of the form ∞ f k P ( n ) � f ( t ) = k ( t ) , f k ≥ 0 for all k ≥ 1 . (1) k =0 f (1) = � ∞ k =0 f k < ∞ = ⇒ convergence is absolute and uniform. Then: � E ( n , C ; f ) = f ( � x , y � ) − f (1) N x , y ∈ C ∞ � = f k M k ( C ) − f (1) N k =0 � f 0 − f (1) � ≥ f 0 N 2 − f (1) N = N 2 . N

Let A n , h := { f : f ( t ) ≤ h ( t ) , t ∈ [ − 1 , 1) , f k ≥ 0 , k = 1 , 2 , . . . } . Thm (Delsarte-Yudin LP Bound) For any C ⊂ S n − 1 with | C | = N E ( n , h ; C ) ≥ N 2 ( f 0 − f (1) N ) . (2) C satisfies E ( n , h ; C ) = E ( n , f ; C ) = N 2 ( f 0 − f (1) N ) ⇐ ⇒ (a) f ( t ) = h ( t ) for t ∈ {� x , y � : x � = y , x , y ∈ C } , and (b) for all k ≥ 1, either f k = 0 or M k ( C ) = 0 .

Example: n -Simplex on S n − 1 Let C be N = n + 1 points on S n − 1 forming a regular simplex. Then there is only one inner product α 0 = � x , y � for x � = y ∈ C . ◮ The first degree Gegenbauer polynomial P ( n ) 1 ( t ) = t . x ∈ C x | 2 = 0 . ◮ M 1 ( C ) = � x , y ∈ C � x , y � = | � If h is convex and increasing then linear interpolant f ( t ) = h ( α 0 ) + h ′ ( α 0 )( t + 1 / n ) has (a) f 1 = h ′ ( α 0 ) ≥ 0 and (b) f ( t ) ≤ h ( t ) = ⇒ E ( n , h ; N = n + 1) = E ( n , h ; C ) .

Coding Problem: Separation Consider ∆( C ) := min x � = y ∈ C | x − y | Suppose f ∈ C [ − 1 , 1] has nonnegative Gegenbauer coefficients f k ≥ 0 and that f ( t ) ≤ 0 for t ∈ [ − 1 , t 0 ) for some t 0 ∈ ( − 1 , 1). Let M = max t f ( t ) and define � 0 − 1 ≤ t ≤ t 0 h ( t ) = . M t 0 < t ≤ 1 Then: ◮ f ∈ A ( n , h ). ◮ E ( n , C ; h ) > 0 ⇐ ⇒ ∆( C ) < cos( t 0 ). ◮ f 0 − f (1) > 0 = ⇒ N ≤ f (1) / f 0 if there is any C with N ∆( C ) ≥ cos( t 0 ) 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 ( f 0 − f (1) N ) , subject to f ∈ A n , h . For a subspace Λ ⊂ C ([ − 1 , 1]), we consider N 2 ( f 0 − f (1) / N ) . W ( n , N , Λ; h ) := sup (3) f ∈ Λ ∩ A n , h

1 / N -Quadrature Rules and Hermite Interpolation ◮ For a subspace Λ ⊂ C ([ − 1 , 1]) and N > 1, we say { ( α i , ρ i ) } k i =1 is a 1 / N -quadrature rule exact for Λ if − 1 ≤ α i < 1, ρ i > 0 for i = 1 , 2 , . . . , k , and � 1 k f ( t )(1 − t 2 ) ( n − 3) / 2 dt = f (1) � f 0 = γ n N + ρ i f ( α i ) , ( f ∈ Λ) . − 1 i =1 ◮ For f ∈ Λ ∩ A n , h , k k f 0 − f (1) � � = ρ i f ( α i ) ≤ ρ i h ( α i ) , N i =1 i =1 and so k � W ( n , N , Λ; h ) ≤ ρ i h ( α i ) . (4) i =1 ◮ If there is some f ∈ Λ ∩ A n , h such that f ( α i ) = h ( α i ) 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 = Π m (polynomials of degree m ) with nodes {� x , y � | x � = y ∈ C } . Definition A spherical code C ⊂ S n − 1 is sharp if there are m inner products between distinct points in it and C is a spherical (2 m − 1)-design. Theorem (Cohn and Kumar, 2006) If C ⊂ S n − 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 , 2 k − 1)(the DGS bound) there exist real numbers − 1 ≤ α 1 < α 2 < · · · < α k < 1 and ρ 1 , ρ 2 , . . . , ρ k , ρ i > 0 for i = 1 , 2 , . . . , k , such that the equality k f 0 = f (1) � + ρ i f ( α i ) N i =1 holds for every real polynomial f ( t ) of degree at most 2 k − 1. ◮ The numbers α i , i = 1 , 2 , . . . , k , are the roots of the equation P k ( t ) P k − 1 ( s ) − P k ( s ) P k − 1 ( t ) = 0 , where s = α k , P i ( t ) = P ( n − 1) / 2 , ( n − 3) / 2 ( t ) is a Jacobi i 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 , 2 k − 1). Then the Levenshtein nodes { α i } provide the bounds k � E ( n , N , h ) ≥ N 2 ρ i h ( α i ) . i =1 The Hermite interpolants at these nodes are the optimal polynomials which solve the finite LP in the class P τ ∩ A n , h .

Improvement of ULB and Test Functions Test functions (Boyvalenkov, Danev, Boumova, ‘96) k Q j ( n , α k ) := 1 ρ i P ( n ) � N + ( α i ) . j i =1 Subspace ULB Improvement Theorem (BDHSS, 2016) Let { ( α i , ρ i ) } 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 k W ( n , N , Λ; h ) = N 2 � ρ i h ( α i ) . i =1 Suppose Λ ′ = Λ � span { P ( n ) : j ∈ I} for some index set I ⊂ N . j If Q ( n ) i =1 ρ i P ( n ) N + � k := 1 ( α i ) ≥ 0 for j ∈ I , then j j k � W ( n , N , Λ ′ ; h ) = W ( n , N , Λ; h ) = N 2 ρ i h ( α i ) . i =1

ULB Improvement for (4 , 24)-codes The case n = 4, N = 24 is important. C 4 consists of the minimal length vectors in D 4 lattice. | C 4 | = 24. ◮ Kissing numbers in R 4 - solved by Musin in 2003 using modification of linear programming bounds. ◮ C 4 is conjectured to be maximal code but not yet proved. ◮ C 4 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 Π 5 ) are: { α 1 , α 2 , α 3 } = {− . 817352 ..., − . 257597 ..., . 474950 ... } { ρ 1 , ρ 2 , ρ 3 } = { 0 . 138436 ..., 0 . 433999 ..., 0 . 385897 ... } , The test functions for (4 , 24)-codes are: Q 6 Q 7 Q 8 Q 9 Q 10 Q 11 Q 12 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 { ( α i , ρ i ) } 4 i =1 { α 1 , α 2 , α 3 , α 4 } = {− 0 . 86029 ..., − 0 . 48984 ..., − 0 . 19572 , 0 . 478545 ... } { ρ 1 , ρ 2 , ρ 3 , ρ 4 } = { 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 − α 1 ) 2 . . . ( t − α 4 ) 2 ) ∈ Λ ∩ A n , h s. t. , H ′ ( α i ) = h ′ ( α i ) , H ( α i ) = h ( α i ) , i = 1 , . . . , 4 exists, and hence, improved ULB holds 4 E (4 , 24; h ) ≥ N 2 � ρ i h ( α i ) . i =1 Moreover, the new test functions Q ( n ) ≥ 0 , j = 0 , 1 , . . . , and hence j H ( t ) is the optimal LP solution among all polynomials in A 4 , h .