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

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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

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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 S2

(ICERM sp-18) Separation, Covering, Polarization April 2018 1 / 24

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Outline

1

Spheres and point sets Spheres and point sets Separation/Packing Covering/Mesh norm Riesz s-energy Polarization Parametrizations

2

Best packing

3

Best covering

4

Best polarization

(ICERM sp-18) Separation, Covering, Polarization April 2018 2 / 24

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Spheres and point sets Spheres and point sets

Unit sphere

Unit sphere Sd =

  • x ∈ Rd+1 : |x| = 1
  • Sets of distinct points XN = {x1, . . . , xN} ⊂ Sd

x · y =

d+1

  • i=1

xiyi, |x|2 = x · x

Distance: : x, y ∈ Sd

Euclidean distance |x − y|2 = 2(1 − x · y) Geodesic distance: dist (x, y) = arccos(x · y) |x − y| = 2 sin(dist (x, y) /2)

Spherical cap centre z ∈ Sd, radius α C (z; α) =

  • x ∈ Sd : dist (x, z) ≤ α
  • (ICERM sp-18)

Separation, Covering, Polarization April 2018 3 / 24

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Spheres and point sets Separation/Packing

Packing/Separation

Separation: δ(XN) := min

i=j dist (xi, xj)

Packing radius = δ(XN)/2 Best packing: δN := max

XN⊂Sd δ(XN) ∼ csep d N −1/d

(ICERM sp-18) Separation, Covering, Polarization April 2018 4 / 24

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Spheres and point sets Covering/Mesh norm

Mesh norm/Covering radius/Fill radius

Covering radius: h(XN) := max

x∈Sd

min

j=1,...,N dist (x, xj)

Mesh ratio: ρ(XN) := 2h(XN )

δ(XN ) ≥ 1

Best covering: hN := min

XN⊂Sd h(XN) ∼ ccov d N −1/d

(ICERM sp-18) Separation, Covering, Polarization April 2018 5 / 24

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Spheres and point sets Riesz s-energy

Riesz energy and sums of distances

E(s; XN) =                 

N

  • i=1

N

  • j=1

j=i

1 |xi − xj|s if s = 0;

N

  • i=1

N

  • j=1

j=i

log 1 |xi − xj|, if s = 0. Es,N =    min

XN⊂Sd E(s; XN)

s > 0; max

XN⊂Sd E(s; XN)

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

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Spheres and point sets Polarization

Polarization

Function Us(x, XN) := sign(s)

N

  • j=1

1 |x − xj|s . Polarization U ∗

s (XN) = min x∈Sd Us(x, XN)

Optimal set of N points X ∗

N satisfy

Ms,N := max

XN⊂Sd U ∗ s (XN) = max XN⊂Sd min x∈Sd Us(x, XN).

Ms,N ≥ Es,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

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Spheres and point sets Polarization

Polarization N = 12, d = 2, s = 3

(ICERM sp-18) Separation, Covering, Polarization April 2018 8 / 24

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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 Sd

Aim: always feasible XN ⊂ Sd Spherical parametrization

For S2: polar angle θ ∈ [0, π], azimuthal angle φ ∈ [0, 2π) 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

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Spheres and point sets Parametrizations

Inner products

Matrix of distinct points X = [x1 · · · xN] ∈ Rd+1×N Set A(XN) = {z = xi · xj ∈ [−1, 1), j > i}, |A(XN)| ≤ N(N−1)

2

Best packing: min

XN⊂Sd max i=j xi · xj

Matrix of inner products Z = XT X ∈ RN×N

Z is symmetric, positive semi-definite X 0 = ⇒ SDP diag(Z) = e where e = (1, . . . , 1)T ∈ Rd+1 rank(Z) = d + 1 = ⇒ fixed (low) rank correlation matrix

1 2 3 4 5 6 7 104

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 PK points, N = 400, maximum inner product = 0.98296

(ICERM sp-18) Separation, Covering, Polarization April 2018 10 / 24

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Best packing

Best packing

Best packing min

XN⊂Sd max i=j

xi · xj

Finite minimax problem: convert to Minimize v XN ⊂ Sd Subject to v ≥ xi · xj, 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

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Best packing

Largest inner products

PK points on S2, N = 400, Number of variables n = 797 Number active inner products: xi · xj > v − ǫ ǫ = 10−15 = ⇒ 792, ǫ = 10−6 = ⇒ 798, ǫ = 10−5 = ⇒ 801

200 400 600 800 1000 1200 1400 1600 1800 2000 10-20 10-15 10-10 10-5 100

400 PK points, v - xi xj (ICERM sp-18) Separation, Covering, Polarization April 2018 12 / 24

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Best packing

Nearest neighbour distances

PK points on S2, N = 400

(ICERM sp-18) Separation, Covering, Polarization April 2018 13 / 24

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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

100 200 300 400 500 600 700 800 900 1000

Number of points N

10-1 100

Euclidean separation distance on S2

Euclidean limit 3.8093 N-1/2 PK points 3.58 N-1/2 ME points 3.42 N-1/2 CV points 3.07 N-1/2

(ICERM sp-18) Separation, Covering, Polarization April 2018 14 / 24

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Best covering

Best covering

Best covering max

XN⊂Sd min x∈Sd

max

j=1,...,N x · xj

Continuous maximin problem: convert to finite problem Facets F(XN) of convex hull of XN

Facet F ∈ F(XN) = ⇒ set of d + 1 elements of {1, . . . , N} 2N − 4 Delaunay triangles for XN ⊂ S2

Circumcentres c(F) of facet F ∈ F(XN)

c(F) equidistant from d + 1 vertices determining facet F Solve Bu = e, B = [xT

i , i ∈ F],

e = (1, . . . , 1)T ∈ Rd+1 = ⇒z(F) = 1/u2, c(F) = z(F)u

Best covering: Finite maximin problem max

XN⊂Sd

min

F ∈F(XN ) z(F)

where z(F) = c(F) · xj for each j ∈ F small changes in XN can change set of facets F(XN ) (eg. square)

(ICERM sp-18) Separation, Covering, Polarization April 2018 15 / 24

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Best covering

Circumcentres

CV points on S2, N = 400

2N − 4 = 796 facets F (Delaunay triangles) 604 facets within 10−6 of minimum inner product

100 200 300 400 500 600 700 800 0.9936 0.9938 0.994 0.9942 0.9944 0.9946 0.9948 0.995 0.9952 CV points, N = 400, 796 facet inner products

(ICERM sp-18) Separation, Covering, Polarization April 2018 16 / 24

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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 100 200 300 400 500 600 700 800 900 1000 Number of points N 10-1 100 Euclidean covering radius on S2

PK points 2.79 N-1/2 ME points 2.42 N-1/2 CV points 2.26 N-1/2 Euclidean bound 2.0 N-0.5

(ICERM sp-18) Separation, Covering, Polarization April 2018 17 / 24

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Best covering

Consistency checks

Separation/Packing

If you remove one of the points achieving the minimum separation, the separation cannot get worse δN−1 ≥ δN

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 hN+1 ≤ hN

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

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Best polarization

Best polarization

Optimal polarization, parameter s > 0 max

XN⊂Sd min x∈Sd N

  • j=1

1 |x − xj|s

Continuous maximin problem: convert to finite problem Find all local minimizers x achieving (close to) global minimum U ∗

s (XN) of Us(x, XN) := N j=1 1 |x−xj|s

Assumption: Local minimizers achieving global minimum satisfy second

  • rder sufficient conditions, so are isolated

Finite set Ms(XN) = {x∗ ∈ Sd : Us(x∗, XN) = U ∗

s (XN)}

Finite maximin problem Maximize v XN ⊂ Sd Subject to v ≤ Us(x∗, XN) for x∗ ∈ Ms(XN)

(ICERM sp-18) Separation, Covering, Polarization April 2018 19 / 24

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Best polarization

PE points, local minima

PE points: Good polarization, N = 400

(ICERM sp-18) Separation, Covering, Polarization April 2018 20 / 24

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Best polarization

PE points, active local minima

PE points: Good polarization, N = 400

100 200 300 400 500 600 700

Triangles

391.6 391.65 391.7 391.75 391.8 391.85 391.9 391.95 392

N = 400 points, s = 1.0, Polarization = 3.91626618e+02, Ntri = 796, Nloc = 592

(ICERM sp-18) Separation, Covering, Polarization April 2018 21 / 24

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Best polarization

Good polarization s = 1, N = 4, . . ., 500

50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 Number of points N Polarization MN

s , Energy EN s / (N−1) for max polarization points on Sd, d = 2, s = 1,

Polarization MN

s

0.898 N1.014 Energy EN

s /(N−1)

0.772 N1.034 50 100 150 200 250 300 350 400 450 500 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 Number of points N Coefficient of second term for max polarization points on Sd for d = 2, s = 1 Polarization: (MN

s −N)/ N1/2

Energy: (EN

s − N2) / N3/2

−1.10610

(ICERM sp-18) Separation, Covering, Polarization April 2018 22 / 24

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Best polarization

Good polarization for increasing s

−2 −1 1 2 3 4 5 6 7 8 0.095 0.1 0.105 0.11 0.115 0.12 0.125 Riesz index s Covering radius for N = 500 good polarization points

(ICERM sp-18) Separation, Covering, Polarization April 2018 23 / 24

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Best polarization

Acknowledgements

ICERM National Computational Infrastructure (NCI), supported by the Australian Government. Linux computational cluster Katana supported by the Faculty of Science, UNSW Sydney.

Thank You

(ICERM sp-18) Separation, Covering, Polarization April 2018 24 / 24