Semidefinite programming bounds for codes and anticodes in Cayley - PowerPoint PPT Presentation

Semidefinite programming bounds for codes and anticodes in Cayley graphs Frank Vallentin (Universit¨ at zu K¨ oln) Semidefinite Programming and Graph Algorithms Workshop ICERM February 12, 2014

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 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 } 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 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) closely related: chromatic number of the plane 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) distinction between coding and anticoding problems ⇢ anticoding ⇢ e 62 Σ � � problem: if coding e 2 Σ packing of point measures vs. continuous measures

Examples a) k -intersecting permutations  G = S n , Σ = { σ : σ has < k fixed points }    a.c. b) k -intersecting transformations 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 = ; }  

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 if G finite, then optimal solution is Lov´ asz’ ϑ ( G ) 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

anticodes: n R G f ( x ) dµ ( x ) α ≤ sup : f : G → R pos. type f ( e ) o f ( x ) = 0 if x ∈ Σ codes: f ( e ) n G f ( x ) dµ ( x ) : f : G → R pos. type α ≤ inf 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 SDP collapses to LP ? 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

complete SDP (with only a few minor mistakes)

complete SDP (with only a few minor mistakes) continued

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 ?

Improving Cohn-Elkies bound de Laat, Oliveira, V. (2012) 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)

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 πt ) 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)

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