Computation of point sets on the sphere with good separation, - PowerPoint PPT Presentation

Computation of point sets on the sphere with good separation, covering or polarization Rob Womersley, r.womersley@unsw.edu.au School of Mathematics and Statistics, University of New South Wales Good sets of 400 points for packing, covering S 2 (ICERM sp-18) Separation, Covering, Polarization April 2018 1 / 24

Outline Polarization Spheres and point sets 1 Parametrizations Spheres and point sets Best packing 2 Separation/Packing Best covering Covering/Mesh norm 3 Riesz s -energy Best polarization 4 (ICERM sp-18) Separation, Covering, Polarization April 2018 2 / 24

Spheres and point sets Spheres and point sets Unit sphere Unit sphere S d = � x ∈ R d +1 : | x | = 1 � Sets of distinct points X N = { x 1 , . . . , x N } ⊂ S d d +1 | x | 2 = x · x � x · y = x i y i , i =1 Distance: : x , y ∈ S d | x − y | 2 = 2(1 − x · y ) Euclidean distance Geodesic distance: dist ( x , y ) = arccos( x · y ) | x − y | = 2 sin( dist ( x , y ) / 2) Spherical cap centre z ∈ S d , radius α x ∈ S d : dist ( x , z ) ≤ α � � C ( z ; α ) = (ICERM sp-18) Separation, Covering, Polarization April 2018 3 / 24

Spheres and point sets Separation/Packing Packing/Separation Separation: δ ( X N ) := min i � = j dist ( x i , x j ) Packing radius = δ ( X N ) / 2 X N ⊂ S d δ ( X N ) ∼ c sep d N − 1 /d Best packing: δ N := max (ICERM sp-18) Separation, Covering, Polarization April 2018 4 / 24

Spheres and point sets Covering/Mesh norm Mesh norm/Covering radius/Fill radius Covering radius: h ( X N ) := max j =1 ,...,N dist ( x , x j ) min x ∈ S d ρ ( X N ) := 2 h ( X N ) Mesh ratio: δ ( X N ) ≥ 1 X N ⊂ S d h ( X N ) ∼ c cov d N − 1 /d Best covering: h N := min (ICERM sp-18) Separation, Covering, Polarization April 2018 5 / 24

Spheres and point sets Riesz s -energy Riesz energy and sums of distances  N N 1  � � if s � = 0;   | x i − x j | s    i =1 j =1   j � = i E ( s ; X N ) = N N 1  � � log | x i − x j | , if s = 0 .      i =1  j =1  j � = i  X N ⊂ S d E ( s ; X N ) min s > 0;  E s,N = X N ⊂ S d E ( s ; X N ) max s ≤ 0 .  Asymptotics ( N → ∞ ) for s = 0 (Log), 0 < s < d , s = d , s > d s > d uniformly distributed As s → ∞ get best packing (separation) Borodachov, Hardin & Saff monograph (ICERM sp-18) Separation, Covering, Polarization April 2018 6 / 24

Spheres and point sets Polarization Polarization Function N 1 � U s ( x , X N ) := sign ( s ) | x − x j | s . j =1 Polarization U ∗ s ( X N ) = min x ∈ S d U s ( x , X N ) Optimal set of N points X ∗ N satisfy X N ⊂ S d U ∗ M s,N := max s ( X N ) = max X N ⊂ S d min x ∈ S d U s ( x , X N ) . M s,N ≥ E s,N N − 1 As s → ∞ get best covering Erd´ elyi and Saff (2013), ..., Borodachov, Hardin & Saff monograph (ICERM sp-18) Separation, Covering, Polarization April 2018 7 / 24

Spheres and point sets Polarization Polarization N = 12, d = 2, s = 3 (ICERM sp-18) Separation, Covering, Polarization April 2018 8 / 24

Spheres and point sets Parametrizations Parametrizations Criteria invariant under rotation of point set permutation of points Criteria depend only on distance/angle/inner product between points, or distance/angle/inner product with another point on S d Aim: always feasible X N ⊂ S d Spherical parametrization For S 2 : polar angle θ ∈ [0 , π ] , azimuthal angle φ ∈ [0 , 2 π ) Derivative discontinuities at poles Rotation to fix first point at north pole ( θ = 0 ) second point on prime meridian ( φ = 0 ) Issues if using gradient differences to estimate second order information Minimax = ⇒ derivative discontinuities/generalized gradients eg. | x | = max( x, − x ) = min v s.t. v ≥ x, v ≥ − x (ICERM sp-18) Separation, Covering, Polarization April 2018 9 / 24

Spheres and point sets Parametrizations Inner products Matrix of distinct points X = [ x 1 · · · x N ] ∈ R d +1 × N |A ( X N ) | ≤ N ( N − 1) Set A ( X N ) = { z = x i · x j ∈ [ − 1 , 1) , j > i } , 2 Best packing: X N ⊂ S d max min i � = j x i · x j Matrix of inner products Z = X T X ∈ R N × N Z is symmetric, positive semi-definite X � 0 = ⇒ SDP diag ( Z ) = e where e = (1 , . . . , 1) T ∈ R d +1 rank ( Z ) = d + 1 = ⇒ fixed (low) rank correlation matrix PK points, N = 400, maximum inner product = 0.98296 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 1 2 3 4 5 6 7 10 4 (ICERM sp-18) Separation, Covering, Polarization April 2018 10 / 24

Best packing Best packing Best packing X N ⊂ S d max min x i · x j i � = j Finite minimax problem: convert to Minimize v X N ⊂ S d Subject to v ≥ x i · x j , 1 ≤ i < j ≤ N Number of variables n = Nd − d ( d +1) 2 Vertex solution/strongly unique local minimum n + 1 active constraints/inner products achieving max Positive Lagrange multipliers for active constraints/0 in interior of generalized gradient Fewer active constraints = ⇒ curvature critical More active constraints = ⇒ degeneracy (ICERM sp-18) Separation, Covering, Polarization April 2018 11 / 24

Best packing Largest inner products PK points on S 2 , N = 400 , Number of variables n = 797 Number active inner products: x i · x j > v − ǫ ǫ = 10 − 15 = ⇒ 792 , ǫ = 10 − 6 = ⇒ 798 , ǫ = 10 − 5 = ⇒ 801 400 PK points, v - x i x j 10 0 10 -5 10 -10 10 -15 10 -20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 (ICERM sp-18) Separation, Covering, Polarization April 2018 12 / 24

Best packing Nearest neighbour distances PK points on S 2 , N = 400 (ICERM sp-18) Separation, Covering, Polarization April 2018 13 / 24

Best packing Good separation for N = 4 , ..., 1050 PK points: Good packing ME points: Low Riesz s = 1 (Coulomb) energy (Kuijlaars, Saff, Sun, 2007) CV points: Good covering Euclidean separation distance on S 2 Euclidean limit 3.8093 N -1/2 10 0 PK points 3.58 N -1/2 ME points 3.42 N -1/2 CV points 3.07 N -1/2 10 -1 100 200 300 400 500 600 700 800 900 1000 Number of points N (ICERM sp-18) Separation, Covering, Polarization April 2018 14 / 24

Best covering Best covering Best covering X N ⊂ S d min max j =1 ,...,N x · x j max x ∈ S d Continuous maximin problem: convert to finite problem Facets F ( X N ) of convex hull of X N Facet F ∈ F ( X N ) = ⇒ set of d + 1 elements of { 1 , . . . , N } 2 N − 4 Delaunay triangles for X N ⊂ S 2 Circumcentres c ( F ) of facet F ∈ F ( X N ) c ( F ) equidistant from d + 1 vertices determining facet F e = (1 , . . . , 1) T ∈ R d +1 B = [ x T Solve B u = e , i , i ∈ F ] , = ⇒ z ( F ) = 1 / � u � 2 , c ( F ) = z ( F ) u Best covering: Finite maximin problem max F ∈F ( X N ) z ( F ) min X N ⊂ S d where z ( F ) = c ( F ) · x j for each j ∈ F small changes in X N can change set of facets F ( X N ) (eg. square) (ICERM sp-18) Separation, Covering, Polarization April 2018 15 / 24

Best covering Circumcentres CV points on S 2 , N = 400 2 N − 4 = 796 facets F (Delaunay triangles) 604 facets within 10 − 6 of minimum inner product CV points, N = 400, 796 facet inner products 0.9952 0.995 0.9948 0.9946 0.9944 0.9942 0.994 0.9938 0.9936 0 100 200 300 400 500 600 700 800 (ICERM sp-18) Separation, Covering, Polarization April 2018 16 / 24

Best covering Good Covering for N = 4 , ..., 1050 PK points: Good packing ME points: Low Riesz s = 1 (Coulomb) energy (Damelin, Maymeskul, 2005) CV points: Good covering Euclidean covering radius on S 2 PK points 2.79 N -1/2 10 0 ME points 2.42 N -1/2 CV points 2.26 N -1/2 Euclidean bound 2.0 N -0.5 10 -1 100 200 300 400 500 600 700 800 900 1000 Number of points N (ICERM sp-18) Separation, Covering, Polarization April 2018 17 / 24

Best covering Consistency checks Separation/Packing If you remove one of the points achieving the minimum separation, the separation cannot get worse δ N − 1 ≥ δ N Covering/Mesh norm If you add a point at the circumcentre of one of the facets achieving the maximum distance (deep hole) then covering radius cannot get worse h N +1 ≤ h N Only (good) local optima; no guarantee of global optimality Try a variety of starting point sets Try starting from a point set obtained by deleting/adding a point Try starting from local perturbations of a point set Points sets with special structure (symmetry) hard to find ... (ICERM sp-18) Separation, Covering, Polarization April 2018 18 / 24

Best polarization Best polarization Optimal polarization, parameter s > 0 N 1 � X N ⊂ S d min max | x − x j | s x ∈ S d j =1 Continuous maximin problem: convert to finite problem Find all local minimizers x achieving (close to) global minimum s ( X N ) of U s ( x , X N ) := � N U ∗ 1 j =1 | x − x j | s Assumption: Local minimizers achieving global minimum satisfy second order sufficient conditions, so are isolated Finite set M s ( X N ) = { x ∗ ∈ S d : U s ( x ∗ , X N ) = U ∗ s ( X N ) } Finite maximin problem Maximize v X N ⊂ S d for x ∗ ∈ M s ( X N ) v ≤ U s ( x ∗ , X N ) Subject to (ICERM sp-18) Separation, Covering, Polarization April 2018 19 / 24

Best polarization PE points, local minima PE points: Good polarization, N = 400 (ICERM sp-18) Separation, Covering, Polarization April 2018 20 / 24