Energy Optimization with Orthogonal Potentials on the Sphere Ryan - PowerPoint PPT Presentation

Energy Optimization with Orthogonal Potentials on the Sphere Ryan W. Matzke University of Minnesota - Twin Cities November 28, 2018 In collaboration with Dmitriy Bilyk, Alexey Glazyrin, Josiah Park, and Alex Vlasiuk Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Energy Given a potential function F : [ − 1 , 1 ] → R Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Energy Given a potential function F : [ − 1 , 1 ] → R The (discrete) energy of Z ⊂ S d , | Z | = N , with respect to F , is E F ( Z ) = 1 � F ( � x , y � ) . N 2 x , y ∈ Z Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Energy Given a potential function F : [ − 1 , 1 ] → R The (discrete) energy of Z ⊂ S d , | Z | = N , with respect to F , is E F ( Z ) = 1 � F ( � x , y � ) . N 2 x , y ∈ Z The (continuous) energy of a measure µ ∈ B ( S d ) , with respect to F , is � � I F ( µ ) = S d F ( � x , y � ) d µ ( x ) d µ ( y ) . S d Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Energy Given a potential function F : [ − 1 , 1 ] → R The (discrete) energy of Z ⊂ S d , | Z | = N , with respect to F , is E F ( Z ) = 1 � F ( � x , y � ) . N 2 x , y ∈ Z The (continuous) energy of a measure µ ∈ B ( S d ) , with respect to F , is � � I F ( µ ) = S d F ( � x , y � ) d µ ( x ) d µ ( y ) . S d 1 If µ = � x ∈ Z δ x , then | Z | I F ( µ ) = 1 � F ( � x , y � ) = E F ( Z ) . N 2 x , y ∈ Z Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Appearances of Monotonic Potentials Electrostatics (Riesz energy): F ( � x , y � ) = || x − y || − s . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Appearances of Monotonic Potentials Electrostatics (Riesz energy): F ( � x , y � ) = || x − y || − s . Stolarsky Invariance Principle: F ( � x , y � ) = || x − y || . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Appearances of Monotonic Potentials Electrostatics (Riesz energy): F ( � x , y � ) = || x − y || − s . Stolarsky Invariance Principle: F ( � x , y � ) = || x − y || . Packing Problem: � ∞ || x − y || < δ F ( � x , y � ) = || x − y || ≥ δ . 0 Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Orthogonal Potentials Frame Potential: F ( � x , y � ) = |� x , y �| 2 . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Orthogonal Potentials Frame Potential: F ( � x , y � ) = |� x , y �| 2 . Fejes Tóth Conjecture (sum of acute angles): F ( � x , y � ) = 1 π arccos( |� x , y �| ) . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Orthogonal Potentials Frame Potential: F ( � x , y � ) = |� x , y �| 2 . Fejes Tóth Conjecture (sum of acute angles): F ( � x , y � ) = 1 π arccos( |� x , y �| ) . p -Frame Potential: F ( � x , y � ) = |� x , y �| p . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Orthogonal Potentials Frame Potential: F ( � x , y � ) = |� x , y �| 2 . Fejes Tóth Conjecture (sum of acute angles): F ( � x , y � ) = 1 π arccos( |� x , y �| ) . p -Frame Potential: F ( � x , y � ) = |� x , y �| p . These orthogonal potentials are monotonic in real projective space. Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Positive Definiteness Definition A function f : [ − 1 , 1 ] → R is positive definite on S d if, for all n ∈ N , x 1 , ..., x n ∈ S d , c 1 , ..., c n ∈ R , n n � � c i c j f ( � x i , x j � ) ≥ 0 . i = 1 j = 1 Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Positive Definiteness Definition A function f : [ − 1 , 1 ] → R is positive definite on S d if, for all n ∈ N , x 1 , ..., x n ∈ S d , c 1 , ..., c n ∈ R , n n � � c i c j f ( � x i , x j � ) ≥ 0 . i = 1 j = 1 A function f is positive definite iff for all n ∈ N � 1 f ( n ; d − 1 d − 1 d − 2 ˆ ( t )( 1 − t 2 ) ) = a d , n f ( t ) C 2 dt ≥ 0 . 2 n 2 − 1 Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Positive Definiteness Definition A function f : [ − 1 , 1 ] → R is positive definite on S d if, for all n ∈ N , x 1 , ..., x n ∈ S d , c 1 , ..., c n ∈ R , n n � � c i c j f ( � x i , x j � ) ≥ 0 . i = 1 j = 1 A function f is positive definite iff for all n ∈ N � 1 f ( n ; d − 1 d − 1 d − 2 ˆ ( t )( 1 − t 2 ) ) = a d , n f ( t ) C 2 dt ≥ 0 . 2 n 2 − 1 Theorem σ is a minimizer of I F ( µ ) if and only if F is positive definite. Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Negative Definiteness Definition A function f : [ − 1 , 1 ] → R is negative definite on S d if, for all n ∈ N , x 1 , ..., x n ∈ S d , c 1 , ..., c n ∈ R , n n � � c i c j f ( � x i , x j � ) ≤ 0 . i = 1 j = 1 A function f is negative definite iff for all n ∈ N � 1 f ( n ; d − 1 d − 1 d − 2 ˆ ( t )( 1 − t 2 ) 2 dt ≤ 0 . ) = a d , n f ( t ) C 2 n 2 − 1 Theorem σ is a maximizer of I F ( µ ) if and only if F is negative definite. Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Frame Potential Definition We call a finite set of unit vectors { x 1 , ..., x N } ⊂ S d a finite unit norm tight frame (FUNTF) if there exists some constant A > 0 such that for all y ∈ R d + 1 N |� y , x j �| 2 = A || y || 2 . � j = 1 Definition The frame potential of { x i } N i = 1 is N i = 1 ) = 1 FP ( { x i } N � |� x i , x j �| 2 , N 2 i , j = 1 Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Frame Potential Definition The frame potential of { x i } N i = 1 is N i = 1 ) = 1 � FP ( { x i } N |� x i , x j �| 2 . N 2 i , j = 1 The frame potential of µ ∈ B ( S d ) is � � S d |� x , y �| 2 d µ ( x ) d µ ( y ) . FP ( µ ) = S d Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Frame Potential Definition The frame potential of { x i } N i = 1 is N i = 1 ) = 1 � FP ( { x i } N |� x i , x j �| 2 . N 2 i , j = 1 The frame potential of µ ∈ B ( S d ) is � � S d |� x , y �| 2 d µ ( x ) d µ ( y ) . FP ( µ ) = S d Theorem (Benedetto, Fickus (2003)) 1 If N ≥ d + 1 , the minimum value of the frame potential is d + 1 , and the (local/global) minimizers are precisely the FUNTF’s in R d + 1 . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Fejes Tóth Conjecture Conjecture (Fejes Tóth (1959)) Let d ≥ 1 , N = m ( d + 1 ) + k with m ∈ N 0 and 0 ≤ k ≤ d, and F ( � x , y � ) = 1 π arccos( |� x , y �| ) . Then E F ( Z ) is maximized by the point set Z = { z 1 , . . . , z N } ⊂ S d with z p ( d + 1 )+ i = e i . In this case, the energy is k ( k − 1 )( m + 1 ) 2 + 2 km ( d + 1 − k )( m + 1 ) + ( d − k )( d + 1 − k ) m 2 . 2 N 2 In particular, if N = m ( d + 1 ) , the sum is maximized by m copies of the orthonormal basis: E F ( Z ) = 1 d max 2 · d + 1 . Z ⊂ S d # Z = N Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Bounding the Energy G ( t ) = 1 2 − 69 50 π t 2 ≥ 1 π arccos( | t | ) = F ( t ) . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Bounding the Energy G ( t ) = 1 2 − 69 50 π t 2 ≥ 1 π arccos( | t | ) = F ( t ) . Figure: The graph of the function G ( t ) − F ( t ) for 0 ≤ t ≤ 1. Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

Bounding the Energy G ( t ) = 1 2 − 69 50 π t 2 ≥ 1 π arccos( | t | ) = F ( t ) . Figure: The graph of the function G ( t ) − F ( t ) for 0 ≤ t ≤ 1. From results of Benedetto and Fickus on frame potential, we have d µ ∈ B ( S d ) I G ( µ ) = 1 69 2 ( d + 1 ) ≤ µ ∈ B ( S d ) I F ( µ ) ≤ max max 2 − 50 π ( d + 1 ) . Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

✶ On the Circle Proofs: Geometric (Fodor, Vígh, Zarnocz), Fourier expansion, Chebyshev expansion. Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

On the Circle Proofs: Geometric (Fodor, Vígh, Zarnocz), Fourier expansion, Chebyshev expansion. For x ∈ S 1 , define the antipodal quadrants in the direction of x as √ 2 Q ( x ) = { y : |� x , y �| > 2 } . We have a Quadrant Stolarsky Principle: Proposition (Bilyk, Matzke (2018)) For an N-point set Z ⊂ S 1 , 2 � N � � � 2 = 1 � � � � � � D L 2 , quad ( Z ) ✶ Q ( x ) ( z i ) − σ Q ( x ) d σ ( x ) � � � N � S 1 � � i = 1 N = 1 4 − 1 π · 1 � arccos |� z i , z j �| . N 2 i , j = 1 Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere

p-Frame Potential Definition For p ∈ ( 0 , ∞ ) , µ ∈ B ( S d ) , and Z ⊆ S d , we define the p-frame potential of µ as � � S d |� x , y �| p d µ ( x ) d µ ( y ) FP ( µ, p ) = S d and the p-frame potential of Z as 1 � |� x , y �| p . FP ( Z , p ) = | Z | 2 x , y ∈ Z Ryan W. Matzke Energy Optimization with Orthogonal Potentials on the Sphere