densest packings with congruent copies of a convex body (Part 2) - - PowerPoint PPT Presentation

Frank Vallentin (Universit¨ at zu K¨

• ln)

David de Laat (TU Delft)

Computing upper bounds for

densest packings with

congruent copies of a convex body

(Part 2)

Fernando Oliveira (FU Berlin → Universidade de S˜ ao Paulo)

Berlin, two years ago. . .

Numerical results

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

• f the computation

Numerical calculations are challenging but seem to be in reach (in dimensions 2, 3)

Theory

Codes and anticodes in Cayley graphs

I ✓ G independent: 8x, y 2 I, x 6= y, x 6⇠ y

• max. packing density: α(Cayley(G, Σ))

Cayley(G, Σ) group Σ ⊆ G, Σ = Σ−1 x ∼ y ⇐ ⇒ xy−1 ∈ Σ undirected graph on G may contain loops

(anti-) coding problem: if ⇢ e 62 Σ e 2 Σ

• which are as “large” as possible

find indep. sets in Cayley(G, Σ)

Examples

↓

difficulty e) packing of congruent convex bodies G = Rn, Σ = B

G = Rn o SO(n), Σ = {(x, A) : K \ x + AK 6= ;} d) sphere packings

      

c. a) k-intersecting permutations G = Sn, Σ = {σ : σ has < k fixed points} b) k-intersecting transformations c) distance-1-avoiding sets G = GL(n, Fq), Σ = {A : rank(A − I) > n − k} G = Rn, Σ = Sn−1

      

a.c.

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 : Ae1 = e1, . . . , Aek = ek} conjectured by DeCorte, de Laat, V. (2013)

c)—e) wide open c) see Christine’s talk d) only known for n = 2, 3 K = regular pentagon α ∈ [0.92, ?]

e) K = regular tetradedron α ∈ [0.85, 1 − 10−26]

Bounds

anticodes: α ≤ sup n R

f(e) : f : G → R pos. type f(x) = 0 if x ∈ Σ

• if G finite, then optimal solution is Lov´

asz’ ϑ(G) f positive type: ∀x1, . . . , xN ∈ G : f(xix−1

(convex optimization & harmonic analysis) a)–e) upper bound come from spectral techniques

codes: f(e) R

α ≤ sup n f(x)  0 if x 62 Σ

• if G = Fn

anticodes: α ≤ sup n R

f(e) : f : G → R pos. type f(x) = 0 if x ∈ Σ

• If I ⊆ G indep., then 1I ∗ ˜

1I(x) = Z

1I(y)1I(yx−1)dµ(y) is feasible

Computing the bounds

? parametrize cone of positive type functions & use conic optimization construction of positive type functions π : G → U(Hπ) unitary representation, h ∈ Hπ 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. ? ? ?

b G = {irred. unitary rep. of G}/ ∼ ν = Plancherel measure on b G ˆ f(π) = Z

f(x)π(x−1) dµ(x) Fourier transform

Segal-Mautner 1950:

If G is nice and if f is rapidly decreasing:

f(x) = Z

b G

trace(π(x) ˆ f(π)) dν(π)

f is pos. type ⇐ ⇒ for positive, trace-class operators b f(π) : Hπ → Hπ

• ptimization variable

a)—d) Σ closed under conjugation = ⇒ can restrict to central pos. type functions f central: f(xy) = f(yx) χπ irreducible character f(x) = Z

χπ(x) ¯ f(π) dν(π) ¯ 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 Rn o SO(n) πa : G → U(L2(S1)) [πa(x, A)ϕ] (ξ) = e2πiax·ξϕ(A−1ξ) f(x, A) = 2π Z ∞ trace(πa(x, A) ˆ f(a))a da a > 0 in polar coordinates x = ρ(cos θ, sin θ), A = ✓cos α − sin α sin α cos α ◆ f(ρ, θ, α) = Z ∞ X

ˆ f(a)r,sis−re−i(sα+(r−s)θ)Js−r(2πaρ)a da

solve this rigorously on a computer

Explicit computations

When ˆ f(a)r,s =

X

fr,s;ka2ke−πa2 and setting the right ˆ f(a)r,s to zero

f(ρ, θ, α) = Z ∞ X

ˆ f(a)r,sis−re−i(sα+(r−s)θ)Js−r(2πaρ)a da

forces to become a polynomial times exponential If is a sum of squares, then f is pos. type eπa2

X

ˆ f(a)yrys ∈ R[a, y−N, . . . , yN]

geometric condition f(x, A)  0 if x 62 K AK Question: How to describe this set in general?

Algorithm to determine {K − AK : A ∈ SO(n)} when K is polytope?

complete SDP (with only a few minor mistakes)

continued complete SDP (with only a few minor mistakes)

A step back

We need more training. Γ = (Rn × {1, 2},

binary sphere packings

(x, r) ∼ (y, s) ⇐ ⇒ x − y ∈ (0, Rr + Rs) w(x, r) = Rn

If the matrix-valued function g(ρ) = ✓g11(ρ) g12(ρ) g12(ρ) g22(ρ) ◆ with grs(ρ) = Z ∞

X

fr,s;ka2ke−πa2J(n−2)/2(2πaρ)an−1 da satisfies ⇣Pd

⌘

a simple normalization condition involving w grs(ρ) ≤ 0 if ρ ∈ [Rr + Rs, ∞) Then αw(G) ≤ max{g11(0), g22(0)} ? ? ?

code bound

Much simpler than the pentagons f(ρ, θ, α) = Z ∞ X

ˆ f(a)r,sis−re−i(sα+(r−s)θ)Js−r(2πaρ)a da ˆ f(a)r,s =

X

fr,s;ka2ke−πa2 grs(ρ) = Z ∞

X

fr,s;ka2ke−πa2J(n−2)/2(2πaρ)an−1 da vs. 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

— our choice: |µ−1

(2πt)

— 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)

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

α ∈ [0.92, ?]

0.98 Oliveira, V. (2013) ? custom made C++ library for generating and analyzing SDPs with SOS constraints

? geometric constraint modeled by a mixture of sampling and SOS back to the pentagons ? 0.98 can probably be improved

Tetrahedra?

? still a challenge ? needs more automatization (also the harmonic analysis part) ? needs more theory for numerical

• ptimization with SOS constraints

(condition numbers, special numerical solvers)