Computing upper bounds for densest packings with congruent copies of a convex body (Part 2) David de Laat (TU Delft) Fernando Oliveira (FU Berlin → Universidade de S˜ ao Paulo) Frank Vallentin (Universit¨ at zu K¨ oln) Second ERC SDModels Workshop, October 7–9, 2013, FU Berlin
Berlin, two years ago. . .
Numerical results image credit: WWW
Conclusions We are developing the first step of an algorithmic solution ⋆ for a large class of packing problems Complexity of body K is reflected in the complexity ⋆ of the computation Numerical calculations are challenging ⋆ but seem to be in reach (in dimensions 2 , 3 )
Theory
Codes and anticodes in Cayley graphs ⇒ xy − 1 ∈ Σ Cayley ( G, Σ ) x ∼ y ⇐ Σ ⊆ G, Σ = Σ − 1 group undirected graph on G α = 2 / 5 may contain loops Cayley ( Z / 5 Z , { 1 , 4 } ) I ✓ G independent: 8 x, y 2 I, x 6 = y, x 6⇠ y ⇢ e 62 Σ � (anti-) coding problem: if e 2 Σ find indep. sets in Cayley ( G, Σ ) which are as “large” as possible max. packing density: α ( Cayley ( G, Σ ))
Examples a) k -intersecting permutations ↓ G = S n , Σ = { σ : σ has < k fixed points } a.c. b) k -intersecting transformations difficulty G = GL ( n, F q ) , Σ = { A : rank ( A − I ) > n − k } c) distance- 1 -avoiding sets G = R n , Σ = S n − 1 d) sphere packings G = R n , Σ = B � n c. e) packing of congruent convex bodies G = R n o SO ( n ) , Σ = { ( x, A ) : K � \ x + A K � 6 = ; }
Known results a), b) optima realized by ”sunflowers” I = { σ : σ (1) = 1 , . . . , σ ( k ) = k } proved (for n large wrt. k ) by Ellis, Friedgut, Pilpel (2011) I = { A : Ae 1 = e 1 , . . . , Ae k = e k } conjectured by DeCorte, de Laat, V. (2013)
c)—e) wide open c) see Christine’s talk d) only known for n = 2 , 3 α ∈ [0 . 85 , 1 − 10 − 26 ] e) K = regular tetradedron Chen, Engel, Glotzer (2010) Gravel, Elser, Kallus (2011) α ∈ [0 . 92 , ?] K = regular pentagon Kuperberg 2 (1992)
Bounds a)–e) upper bound come from spectral techniques (convex optimization & harmonic analysis) anticodes: n R G f ( x ) dµ ( x ) α ≤ sup : f : G → R pos. type f ( e ) o f ( x ) = 0 if x ∈ Σ f positive type: � f ( x i x − 1 j ) � ∀ x 1 , . . . , x N ∈ G : 1 ≤ i,j ≤ N is pos. semidefinite f (0) f (1) f (2) f (3) f (4) f (4) f (0) f (1) f (2) f (3) G = Z / 5 Z f (3) f (4) f (0) f (1) f (2) f (2) f (3) f (4) f (0) f (0) f (1) f (2) f (3) f (4) f (0) if G finite, then optimal solution is Lov´ asz’ ϑ ( G )
anticodes: n R G f ( x ) dµ ( x ) α ≤ sup : f : G → R pos. type f ( e ) o f ( x ) = 0 if x ∈ Σ Z If I ⊆ G indep., then 1 I ∗ ˜ 1 I ( y )1 I ( yx − 1 ) dµ ( y ) 1 I ( x ) = G ˜ f ( x ) = f ( x − 1 ) is feasible codes: f ( e ) n G f ( x ) dµ ( x ) : f : G → R pos. type α ≤ sup R o f ( x ) 0 if x 62 Σ if G = F n q , then optimal solution is Delsarte’s LP bound
Computing the bounds ? parametrize cone of positive type functions & use conic optimization construction of positive type functions π : G → U ( H π ) unitary representation, h ∈ H π then f ( x ) = ( π ( x ) h, h ) is positive type Gelfand-Raikov 1942: ? all positive type functions are of this form ? extreme rays of cone of pos. type functions ? come from irreducible rep.
Segal-Mautner 1950: If G is nice and if f is rapidly decreasing: f is pos. type ⇐ ⇒ optimization variable Z trace( π ( x ) ˆ f ( x ) = f ( π )) d ν ( π ) b G for positive, trace-class operators b f ( π ) : H π → H π b G = { irred. unitary rep. of G } / ∼ ν = Plancherel measure on b G Z ˆ Fourier transform f ( x ) π ( x − 1 ) dµ ( x ) f ( π ) = G
Σ closed under conjugation a)—d) ⇒ can restrict to central pos. type functions = f central: f ( xy ) = f ( yx ) Z χ π ( x ) ¯ χ π irreducible character f ( x ) = f ( π ) d ν ( π ) b G ¯ f ( π ) ≥ 0 ∀ π ∈ b G G = Z / 5 Z 4 e 2 π ikx/ 5 ¯ X f ( x ) = f ( k ) SDP collapses to LP k =0 ? can be analyzed by hand for a), c) ? b) not yet d) Cohn-Elkies (2003) LP bound ?
e) relevant irred. rep. of R n o SO ( n ) π a : G → U ( L 2 ( S 1 )) a > 0 [ π a ( x, A ) ϕ ] ( ξ ) = e 2 π iax · ξ ϕ ( A − 1 ξ ) Z ∞ trace ( π a ( x, A ) ˆ f ( x, A ) = 2 π f ( a )) a da 0 in polar coordinates Z ∞ ˆ X f ( a ) r,s i s − r e − i ( s α +( r − s ) θ ) J s − r (2 π a ρ ) a da f ( ρ , θ , α ) = 0 r,s ∈ Z ✓ cos α ◆ − sin α x = ρ (cos θ , sin θ ) , A = sin α cos α
Explicit computations the problem of finding an optimal function is an infinite-dimensional SDP goal: reformulate and relax to a finite-dimensional SDP solve this rigorously on a computer
When d f r,s ; k a 2 k e − π a 2 ˆ X f ( a ) r,s = k =0 and setting the right ˆ f ( a ) r,s to zero forces Z ∞ ˆ X f ( a ) r,s i s − r e − i ( s α +( r − s ) θ ) J s − r (2 π a ρ ) a da f ( ρ , θ , α ) = 0 r,s ∈ Z to become a polynomial times exponential If N e π a 2 ˆ X f ( a ) y r y s ∈ R [ a, y − N , . . . , y N ] r,s = − N is a sum of squares, then f is pos. type
geometric condition f ( x, A ) 0 if x 62 K � A K 2 π / 10 π / 10 α = 0 − π / 10 − 2 π / 10 Question: How to describe this set in general? Algorithm to determine {K − A K : A ∈ SO ( n ) } when K is polytope?
complete SDP (with only a few minor mistakes)
complete SDP (with only a few minor mistakes) continued
A step back We need more training. binary sphere packings Γ = ( R n × { 1 , 2 } , ( x, r ) ∼ ( y, s ) ⇐ ⇒ x − y ∈ (0 , R r + R s ) w ( x, r ) = R n r · vol B n
code bound If the matrix-valued function ✓ g 11 ( ρ ) ◆ g 12 ( ρ ) g ( ρ ) = g 12 ( ρ ) g 22 ( ρ ) with d Z ∞ f r,s ; k a 2 k e − π a 2 J ( n − 2) / 2 (2 π a ρ ) a n − 1 da X g rs ( ρ ) = 0 k =0 satisfies ⇣P d ⌘ k =0 f r,s ; k a 2 k r,s =1 , 2 is psd for all a ? a simple normalization condition involving w ? g rs ( ρ ) ≤ 0 if ρ ∈ [ R r + R s , ∞ ) ? Then α w ( G ) ≤ max { g 11 (0) , g 22 (0) }
Much simpler than the pentagons Z ∞ ˆ X f ( a ) r,s i s − r e − i ( s α +( r − s ) θ ) J s − r (2 π a ρ ) a da f ( ρ , θ , α ) = 0 r,s ∈ Z d f r,s ; k a 2 k e − π a 2 ˆ X f ( a ) r,s = k =0 vs. d Z ∞ f r,s ; k a 2 k e − π a 2 J ( n − 2) / 2 (2 π a ρ ) a n − 1 da X g rs ( ρ ) = 0 k =0 it’s univariate (the function only depends on ρ ) ? no trigonometric part ? matrix sizes: only 2 × 2 ? geometric condition: simple quadratic inequality ?
Rigorous computations right choice of polynomial basis is extremely important — using monomial basis fails badly, even for very small degrees k | L n/ 2 − 1 — our choice: | µ − 1 (2 π t ) k µ k : coefficient of L n/ 2 − 1 (2 π t ) with largest absolute value k — csdp : d ≤ 31 — SDPA-gmp with 256 bits of precision: d ≤ 51
In order to get mathematical rigorous results: — perform post processing of the floating point solution — perturb to a rational solution — analyze quality-loss of this perturbation (by estimates of eigenvalues and condition numbers)
0 . 95 0 . 90 n = 2 0 . 85 0 . 80 n = 3 0 . 75 0 . 70 n = 4 0 . 65 0 . 60 n = 5 0 . 55 0 . 50 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 de Laat, Oliveira, V. (2012)
Improving Cohn-Elkies bound 1. Adding valid inequalities (bounds on average contact numbers) 2. More flexible numerical method n lower bound Rogers Cohn-Elkies new bound 4 0 . 125 0 . 13127 0 . 13126 0 . 13081 5 0 . 08839 0 . 09987 0 . 09975 0 . 09955 6 0 . 07217 0 . 08112 0 . 08084 0 . 08070 7 0 . 0625 0 . 06981 0 . 06933 0 . 06926 density given as point density (= # centers per unit volume) density given as point density (= # centers per unit volume)
back to the pentagons Kuperberg 2 (1992) α ∈ [0 . 92 , ?] 0 . 98 Oliveira, V. (2013) ? custom made C++ library for generating and analyzing SDPs with SOS constraints 14 13 24 1 1 ? geometric constraint modeled 03 2 by a mixture of sampling and SOS 2 0 20 0 02 3 30 3 4 4 42 31 0 . 98 can probably be improved 41 ?
Tetrahedra? ? needs more automatization (also the harmonic analysis part) ? needs more theory for numerical optimization with SOS constraints (condition numbers, special numerical solvers) ? still a challenge
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