On the universal optimality of the 600 -cell: the Levenshtein framework lifted Peter Dragnev Purdue University Fort Wayne Joint work with: P. Boyvalenkov (Bulgarian Academy of Sciences), D. Hardin, E. Saff (Vanderbilt), M. Stoyanova (Sofia University, Bulgaria) Optimal and Random Point Configurations, February 26 – March 2, 2018 ICERM, Providence, RI PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 1 / 44
Outline Minimal Energy, Spherical Harmonics, Gegenbauer Polynomials Delsarte-Yudin Linear Programming Dual Programming Heuristics 1 / N -Quadrature and ULB space Levenshtein Framework - ULB Theorem Test Functions - Levenshtein Framework Lifted The Universality of the 600-cell Revisited PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 2 / 44
Minimal Energy Problem Spherical Code: A finite set C ⊂ S n − 1 with cardinality | C | = N . r 2 = | x − y | 2 = 2 − 2 � x , y � = 2 − 2 t . Interaction potential h : [ − 1 , 1 ) → R The h -energy of a spherical code C ⊂ S n − 1 : � E ( n , h ; C ) := h ( � x , y � ) , x , y ∈ C , y � = x where t = � x , y � denotes Euclidean inner product of x and y . Minimal Energy Problem: Find E ( n , h ; N ) := min { E ( n , h ; C ) | C ⊂ S n − 1 , | C | = N } . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 3 / 44
Absolutely Monotone Potentials Interaction potential h : [ − 1 , 1 ) → R Absolutely monotone potentials: C ∞ + := { h | h ( k ) ( t ) ≥ 0 , t ∈ [ − 1 , 1 ) , k ≥ 0 } . Examples: Newton potential: h ( t ) = ( 2 − 2 t ) − ( n − 2 ) / 2 = | x − y | − ( n − 2 ) ; Riesz s -potential: h ( t ) = ( 2 − 2 t ) − s / 2 = | x − y | − s ; Log potential: h ( t ) = − log( 2 − 2 t ) = − log | x − y | ; Gaussian potential: h ( t ) = exp( 2 t − 2 ) = exp( −| x − y | 2 ) ; PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 4 / 44
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 , n := dim Harm ( k ) = . n − 2 k Spherical harmonics (degree k ): { Y kj ( x ) : j = 1 , 2 , . . . , r k , n } orthonormal basis of Harm ( k ) with respect to normalized ( n − 1 ) -dimensional surface area measure on S n − 1 . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 5 / 44
Spherical Harmonics and Gegenbauer Polynomials The Gegenbauer polynomials and spherical harmonics can be defined through the Addition Formula ( t = � x , y � ): r k k ( � x , y � ) = 1 P ( n ) k ( t ) := P ( n ) � x , y ∈ S n − 1 . Y kj ( x ) Y kj ( y ) , r k j = 1 { P ( n ) k = 0 orthogonal w/weight ( 1 − t 2 ) ( n − 3 ) / 2 and P ( n ) k ( t ) } ∞ k ( 1 ) = 1. Gegenbauer polynomials P ( n ) k ( t ) are special types of Jacobi ( t ) orthogonal w.r.t. weight ( 1 − t ) α ( 1 + t ) β on polynomials P ( α,β ) k [ − 1 , 1 ] , where α = β = ( n − 3 ) / 2 . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 6 / 44
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 , 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 ∈ Π τ ( R n ) . p ( x ) , N x ∈ C PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 7 / 44
‘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: PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 8 / 44
‘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 . (2) 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 PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 9 / 44
‘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 . (2) 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 PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 9 / 44
‘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 . (2) 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 PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 9 / 44
Delsarte-Yudin LP Bound Let A n , h := { f : f ( t ) ≤ h ( t ) , t ∈ [ − 1 , 1 ) , f k ≥ 0 , k = 1 , 2 , . . . } . Thm (Delsarte-Yudin Lower Energy Bound) For any C ⊂ S n − 1 with | C | = N and f ∈ A n , h , E ( n , h ; C ) ≥ N 2 ( f 0 − f ( 1 ) N ) . (3) 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 . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 10 / 44
Linear program: Maximize D-Y lower bound Maximizing Delsarte-Yudin lower bound is a linear programming problem. Maximize N 2 ( f 0 − f ( 1 ) N ) subject to f ∈ A n , h . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 11 / 44
Linear program: Maximize D-Y lower bound Maximizing Delsarte-Yudin lower bound is a linear programming problem. Maximize 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 ) W ( n , N , Λ; h ) := sup N ) . (4) f ∈ Λ ∩ A n , h PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 12 / 44
Linear program: Maximize D-Y lower bound Maximizing Delsarte-Yudin lower bound is a linear programming problem. Maximize 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 ) W ( n , N , Λ; h ) := sup N ) . (5) f ∈ Λ ∩ A n , h Usually, Λ = span { P ( n ) } i ∈ I for some finite I , i and we replace f ( t ) ≤ h ( t ) with f ( t j ) ≤ h ( t j ) , j ∈ J for finite J . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 13 / 44
Dual Programming Heuristics Primal Program Dual Program Maximize c T x Minimize b T y subject to A T y ≥ b , y ≥ 0 subject to Ax ≤ b , x ≥ 0 PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 14 / 44
Dual Programming Heuristics Primal Program Dual Program Maximize c T x Minimize b T y subject to A T y ≥ b , y ≥ 0 subject to Ax ≤ b , x ≥ 0 f 0 − 1 � Maximize Primal f i N i ∈ I f i P ( n ) � subject to: ( t j ) ≤ h ( t j ) , j ∈ J , f i ≥ 0 . i i ∈ I PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 15 / 44
Dual Programming Heuristics Primal Program Dual Program Maximize c T x Minimize b T y subject to A T y ≥ b , y ≥ 0 subject to Ax ≤ b , x ≥ 0 f 0 − 1 � Maximize Primal f i N i ∈ I f i P ( n ) � subject to: ( t j ) ≤ h ( t j ) , j ∈ J , f i ≥ 0 . i i ∈ I � Dual Minimize ρ j h ( t j ) j ∈ J subject to: 1 ρ j P ( n ) � N + ( t j ) ≥ 0 , i ∈ I \ { 0 } , ρ j ≥ 0 . i j ∈ J PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 16 / 44
Dual Programming Heuristics - complementary slackness Add slack variables { u j } j ∈ J and { w i } i ∈ I . f 0 − 1 � Primal Maximize f i N i ∈ I f i P ( n ) � subject to: ( t j ) + u j = h ( t j ) , j ∈ J , f i ≥ 0 . i i ∈ I � Dual Minimize ρ j h ( t j ) j ∈ J subject to: 1 ρ j P ( n ) � N + ( t j ) − w i = 0 , i ∈ I \ { 0 } , ρ j ≥ 0 . i j ∈ J Complementary slackness condition for Primal Objective=Dual Objective: f i · w i = 0, i ∈ I , and ρ j · u j = 0, j ∈ J . PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 17 / 44
1 / N -Quadrature Rules For a subspace Λ ⊂ C [ − 1 , 1 ] we say { ( α i , ρ i ) } k i = 1 with − 1 ≤ α i < 1, ρ i > 0 for i = 1 , 2 , . . . , k is a 1 / N -quadrature rule exact for Λ if � 1 k f ( t )( 1 − t 2 ) ( n − 3 ) / 2 dt = f ( 1 ) � f 0 = γ n + ρ i f ( α i ) , ( f ∈ Λ) . N − 1 i = 1 k ⇒ f 0 − f ( 1 ) � = = ρ i f ( α i ) . N i = 1 PB, PD, DH, ES, MS Levenshtein framework lifted March 2, 2018 18 / 44
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