SLIDE 1 A semidefinite programming hierarchy for packing problems in discrete geometry
David de Laat (TU Delft) Joint work with Frank Vallentin (Universit¨ at zu K¨
Applications of Real Algebraic Geometry Aalto University – February 28, 2014
SLIDE 2 Contents
- 1. Modeling geometric packing problems
- 2. Convergence to the optimal density
- 3. Duality theory
- 4. Harmonic analysis on subset spaces
- 5. Reduction to semidefinite programs
SLIDE 3
Packing problems in discrete geometry
SLIDE 4
Packing problems in discrete geometry
◮ These problems can be modeled as maximum independent set
problems in graphs on infinitely many vertices
SLIDE 5
Packing problems in discrete geometry
◮ These problems can be modeled as maximum independent set
problems in graphs on infinitely many vertices Spherical cap packings What is the maximum number of spherical caps of size t in Sn−1 such that no two caps intersect in their interiors? G = (V, E), V = Sn−1, E = {{x, y} : x · y ∈ (t, 1)}
SLIDE 6
Packing problems in discrete geometry
◮ These problems can be modeled as maximum independent set
problems in graphs on infinitely many vertices Spherical cap packings What is the maximum number of spherical caps of size t in Sn−1 such that no two caps intersect in their interiors? G = (V, E), V = Sn−1, E = {{x, y} : x · y ∈ (t, 1)}
◮ Independent sets correspond to valid packings
SLIDE 7
The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
SLIDE 8 The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
- ◮ The Lasserre hierarchy for this problem (Laurent, 2003):
last(G) = max
x∈V
y{x} : y ∈ RI2t
≥0, y∅ = 1, Mt(y) 0
SLIDE 9 The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
- ◮ The Lasserre hierarchy for this problem (Laurent, 2003):
last(G) = max
x∈V
y{x} : y ∈ RI2t
≥0, y∅ = 1, Mt(y) 0
- ◮ I2t is the set of independent sets of cardinality at most 2t
SLIDE 10 The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
- ◮ The Lasserre hierarchy for this problem (Laurent, 2003):
last(G) = max
x∈V
y{x} : y ∈ RI2t
≥0, y∅ = 1, Mt(y) 0
- ◮ I2t is the set of independent sets of cardinality at most 2t
◮ Mt(y) is the matrix with rows and columns indexed by It and
Mt(y)J,J′ =
if J ∪ J′ ∈ I2t,
SLIDE 11 The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
- ◮ The Lasserre hierarchy for this problem (Laurent, 2003):
last(G) = max
x∈V
y{x} : y ∈ RI2t
≥0, y∅ = 1, Mt(y) 0
- ◮ I2t is the set of independent sets of cardinality at most 2t
◮ Mt(y) is the matrix with rows and columns indexed by It and
Mt(y)J,J′ =
if J ∪ J′ ∈ I2t,
◮ ϑ′(G) = las1(G) ≥ las2(G) ≥ . . . ≥ lasα(G)(G) = α(G)
SLIDE 12 The Lasserre hierarchy for finite graphs
◮ Maximum independent set problem for a finite graph as a 0/1
polynomial optimization problem: α(G) = max
v∈V
xv : xv ∈ {0, 1} for v ∈ V, xu+xv ≤ 1 for {u, v} ∈ E
- ◮ The Lasserre hierarchy for this problem (Laurent, 2003):
last(G) = max
x∈V
y{x} : y ∈ RI2t
≥0, y∅ = 1, Mt(y) 0
- ◮ I2t is the set of independent sets of cardinality at most 2t
◮ Mt(y) is the matrix with rows and columns indexed by It and
Mt(y)J,J′ =
if J ∪ J′ ∈ I2t,
◮ ϑ′(G) = las1(G) ≥ las2(G) ≥ . . . ≥ lasα(G)(G) = α(G) ◮ ϑ(G) is the Lov´
asz ϑ-number which specializes to the Delsarte LP-bound when G is the binary code graph
SLIDE 13
Generalization to infinite graphs
◮ Linear programming bound for spherical cap packings
(Delsarte, 1977 / Kabatiansky, Levenshtein, 1978)
SLIDE 14
Generalization to infinite graphs
◮ Linear programming bound for spherical cap packings
(Delsarte, 1977 / Kabatiansky, Levenshtein, 1978)
◮ Generalization of the ϑ-number
(Bachoc, Nebe, de Oliveira, Vallentin, 2009)
SLIDE 15 Topological packing graphs
◮ We consider graphs where
◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations
SLIDE 16 Topological packing graphs
◮ We consider graphs where
◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations
Definition A topological packing graph is a graph where
- the vertex set is a Hausdorff topological space
- each finite clique is contained in an open clique
SLIDE 17 Topological packing graphs
◮ We consider graphs where
◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations
Definition A topological packing graph is a graph where
- the vertex set is a Hausdorff topological space
- each finite clique is contained in an open clique
◮ We consider compact topological packing graphs
SLIDE 18 Topological packing graphs
◮ We consider graphs where
◮ vertices which are close are adjacent ◮ adjacent vertices stay adjacent after slight pertubations
Definition A topological packing graph is a graph where
- the vertex set is a Hausdorff topological space
- each finite clique is contained in an open clique
◮ We consider compact topological packing graphs ◮ These graphs have finite independence number
SLIDE 19 Generalization for compact topological packing graphs
last(G) = sup
, ,
SLIDE 20 Generalization for compact topological packing graphs
last(G) = sup
,
SLIDE 21 Generalization for compact topological packing graphs
last(G) = sup
,
SLIDE 22 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
SLIDE 23 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
SLIDE 24 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
SLIDE 25 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ Vt is the set of subsets of V of cardinality ≤ t
SLIDE 26 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ Vt is the set of subsets of V of cardinality ≤ t
◮ Quotient map:
q: V t → Vt \ {∅}, (v1, . . . , vt) → {v1, . . . , vt}
SLIDE 27 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ Vt is the set of subsets of V of cardinality ≤ t
◮ Quotient map:
q: V t → Vt \ {∅}, (v1, . . . , vt) → {v1, . . . , vt}
◮ Vt \ {∅} is equipped with the quotient topology
SLIDE 28 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ Vt is the set of subsets of V of cardinality ≤ t
◮ Quotient map:
q: V t → Vt \ {∅}, (v1, . . . , vt) → {v1, . . . , vt}
◮ Vt \ {∅} is equipped with the quotient topology ◮ Vt is the disjoint union of Vt \ {∅} with {∅}
SLIDE 29 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ Vt is the set of subsets of V of cardinality ≤ t
◮ Quotient map:
q: V t → Vt \ {∅}, (v1, . . . , vt) → {v1, . . . , vt}
◮ Vt \ {∅} is equipped with the quotient topology ◮ Vt is the disjoint union of Vt \ {∅} with {∅} ◮ It gets its topology as a subset of Vt
SLIDE 30 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
SLIDE 31 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ A function K ∈ C(Vt × Vt)sym is a positive definite kernel if
(K(Ji, Jj))m
i,j=1 0
for all m ∈ N and J1, . . . , Jm ∈ Vt
SLIDE 32 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ A function K ∈ C(Vt × Vt)sym is a positive definite kernel if
(K(Ji, Jj))m
i,j=1 0
for all m ∈ N and J1, . . . , Jm ∈ Vt
◮ Cone of positive definite kernels: C(Vt × Vt)0
SLIDE 33 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ A function K ∈ C(Vt × Vt)sym is a positive definite kernel if
(K(Ji, Jj))m
i,j=1 0
for all m ∈ N and J1, . . . , Jm ∈ Vt
◮ Cone of positive definite kernels: C(Vt × Vt)0 ◮ Cone of positive definite measures:
M(Vt×Vt)0 = {µ ∈ M(Vt×Vt)sym : µ(K) ≥ 0 for all K ∈ C(Vt×Vt)0}, where µ(K) =
SLIDE 34 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
SLIDE 35 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ There is an operator At such that Mt(y), X = y, AtX for
all vectors y and matrices X
SLIDE 36 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ There is an operator At such that Mt(y), X = y, AtX for
all vectors y and matrices X
◮ Define the operator At : C(Vt × Vt)sym → C(I2t) by
Atf(S) =
f(J, J′)
SLIDE 37 Generalization for compact topological packing graphs
last(G) = sup
- λ(I=1) : λ ∈ M(I2t)≥0, λ({∅}) = 1,
A∗
t λ ∈ M(Vt × Vt)0
- ◮ There is an operator At such that Mt(y), X = y, AtX for
all vectors y and matrices X
◮ Define the operator At : C(Vt × Vt)sym → C(I2t) by
Atf(S) =
f(J, J′)
◮ The adjoint: A∗ t : M(I2t) → M(Vt × Vt)sym
SLIDE 38
Finite convergence
Theorem Suppose G is a compact topological packing graph. Then, ϑ′(G) = las1(G) ≥ · · · ≥ lasα(G)(G) = α(G).
SLIDE 39 Finite convergence
Theorem Suppose G is a compact topological packing graph. Then, ϑ′(G) = las1(G) ≥ · · · ≥ lasα(G)(G) = α(G).
◮ If S is an independent set, then
χS =
δR is feasible for lasα(G)(G)
SLIDE 40 Finite convergence
Theorem Suppose G is a compact topological packing graph. Then, ϑ′(G) = las1(G) ≥ · · · ≥ lasα(G)(G) = α(G).
◮ If S is an independent set, then
χS =
δR is feasible for lasα(G)(G)
◮ We show that the measures χS are precisely the extreme
points of the feasible region of lasα(G)(G)
SLIDE 41 Finite convergence
Theorem Suppose G is a compact topological packing graph. Then, ϑ′(G) = las1(G) ≥ · · · ≥ lasα(G)(G) = α(G).
◮ If S is an independent set, then
χS =
δR is feasible for lasα(G)(G)
◮ We show that the measures χS are precisely the extreme
points of the feasible region of lasα(G)(G)
◮ Using vector valued notation: λ =
probability measure σ on the set of independent sets
SLIDE 42 Finite convergence
Theorem Suppose G is a compact topological packing graph. Then, ϑ′(G) = las1(G) ≥ · · · ≥ lasα(G)(G) = α(G).
◮ If S is an independent set, then
χS =
δR is feasible for lasα(G)(G)
◮ We show that the measures χS are precisely the extreme
points of the feasible region of lasα(G)(G)
◮ Using vector valued notation: λ =
probability measure σ on the set of independent sets
◮ Then, λ(I=1) =
- χS(I=1) dσ(S) =
- |S| dσ(S) ≤ α(G)
SLIDE 43 Duality theory
◮ A duality theory is important for concrete computations:
◮ In the maximization problems last(G) we need optimal
solutions to get upper bounds
◮ In the dual minimization problems any feasible solution
provides an upper bound
SLIDE 44 Duality theory
◮ A duality theory is important for concrete computations:
◮ In the maximization problems last(G) we need optimal
solutions to get upper bounds
◮ In the dual minimization problems any feasible solution
provides an upper bound
◮ We obtain the dual by conic duality:
last(G)∗ = inf
- K(∅, ∅) : K ∈ C(Vt × Vt)0,
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅}
SLIDE 45 Duality theory
◮ A duality theory is important for concrete computations:
◮ In the maximization problems last(G) we need optimal
solutions to get upper bounds
◮ In the dual minimization problems any feasible solution
provides an upper bound
◮ We obtain the dual by conic duality:
last(G)∗ = inf
- K(∅, ∅) : K ∈ C(Vt × Vt)0,
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅}
last(G) = last(G)∗ and the optimum in last(G) is attained
SLIDE 46 Duality theory
◮ A duality theory is important for concrete computations:
◮ In the maximization problems last(G) we need optimal
solutions to get upper bounds
◮ In the dual minimization problems any feasible solution
provides an upper bound
◮ We obtain the dual by conic duality:
last(G)∗ = inf
- K(∅, ∅) : K ∈ C(Vt × Vt)0,
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅}
last(G) = last(G)∗ and the optimum in last(G) is attained
◮ These are infinite dimensional conic programs
SLIDE 47 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
SLIDE 48 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2
SLIDE 49 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2
SLIDE 50 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2}
SLIDE 51 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2} ◮ Representation: O(3) → L(C(V2)), gf(x) = f(g−1x)
SLIDE 52 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2} ◮ Representation: O(3) → L(C(V2)), gf(x) = f(g−1x) ◮ Bochner’s theorem: A kernel K ∈ C(V2 × V2)0 is of the form
K(J, J′) =
∞
Fk, Zk(J, J′)
SLIDE 53 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2} ◮ Representation: O(3) → L(C(V2)), gf(x) = f(g−1x) ◮ Bochner’s theorem: A kernel K ∈ C(V2 × V2)0 is of the form
K(J, J′) =
∞
Fk, Zk(J, J′)
◮ ., . – trace inner product
SLIDE 54 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2} ◮ Representation: O(3) → L(C(V2)), gf(x) = f(g−1x) ◮ Bochner’s theorem: A kernel K ∈ C(V2 × V2)0 is of the form
K(J, J′) =
∞
Fk, Zk(J, J′)
◮ ., . – trace inner product ◮ Fk – positive semidefinite matrices (Fourier coefficients)
SLIDE 55 Harmonic analysis on V2
◮ We use harmonic analysis on Vt and SOS characterizations to
- btain finite dimensional semidefinite programs
◮ Assume V = S2 and t = 2 ◮ Symmetry: transitive action of O(3) on S2 ◮ Induced action on V2 by g∅ = ∅ and g{v1, v2} = {gv1, gv2} ◮ Representation: O(3) → L(C(V2)), gf(x) = f(g−1x) ◮ Bochner’s theorem: A kernel K ∈ C(V2 × V2)0 is of the form
K(J, J′) =
∞
Fk, Zk(J, J′)
◮ ., . – trace inner product ◮ Fk – positive semidefinite matrices (Fourier coefficients) ◮ Zk – zonal matrices corresponding to the above representation
SLIDE 56
Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
SLIDE 57
Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
SLIDE 58 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
SLIDE 59 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
SLIDE 60 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
SLIDE 61 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
SLIDE 62 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅})
SLIDE 63 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V )
SLIDE 64 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics
SLIDE 65 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics ◮ C(V ) ⊙ C(V ) =
- ⊕k≥0 Hk ⊙ Hk
- ⊕
- ⊕0≤k<k′ Hk ⊗ Hk′
SLIDE 66 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics ◮ C(V ) ⊙ C(V ) =
- ⊕k≥0 Hk ⊙ Hk
- ⊕
- ⊕0≤k<k′ Hk ⊗ Hk′
◮ Hk ⊙ Hk ≃ H0 ⊕ H2 ⊕ · · · ⊕ H2k
SLIDE 67 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics ◮ C(V ) ⊙ C(V ) =
- ⊕k≥0 Hk ⊙ Hk
- ⊕
- ⊕0≤k<k′ Hk ⊗ Hk′
◮ Hk ⊙ Hk ≃ H0 ⊕ H2 ⊕ · · · ⊕ H2k ◮ Hk ⊗ Hk′ ≃ H(−1)k+k′ |k−k′|
⊕ · · · ⊕ H(−1)k+k′
k+k′
SLIDE 68 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics ◮ C(V ) ⊙ C(V ) =
- ⊕k≥0 Hk ⊙ Hk
- ⊕
- ⊕0≤k<k′ Hk ⊗ Hk′
◮ Hk ⊙ Hk ≃ H0 ⊕ H2 ⊕ · · · ⊕ H2k ◮ Hk ⊗ Hk′ ≃ H(−1)k+k′ |k−k′|
⊕ · · · ⊕ H(−1)k+k′
k+k′
◮ H(−1)k
k
= Hk are the irreducible representations of SO(3)
SLIDE 69 Harmonic analysis V2
◮ How do we find the zonal matrices Zk?
Definition
- Decomposition: C(V2) = ⊕∞
k=0 ⊕mk i=1 Vk,i
- Where Vk,i are irreducible subspaces with Vk,i ∼ Vk,i′
- Let ek,i,1, . . . , ek,i,hk be compatible bases of Vk,i
- Then Zk(J, J′) = Ek(J)TEk(J′) with Ek(J)j,i = ek,i,j(J)
◮ C(V2) = C({∅} ∪ V2 \ {∅}) = R ⊕ C(V2 \ {∅}) ◮ C(V2 \ {∅}) = C(V ) ⊙ C(V ) ◮ C(V ) = ⊕∞ k=0Hk where Hk are degree k spherical harmonics ◮ C(V ) ⊙ C(V ) =
- ⊕k≥0 Hk ⊙ Hk
- ⊕
- ⊕0≤k<k′ Hk ⊗ Hk′
◮ Hk ⊙ Hk ≃ H0 ⊕ H2 ⊕ · · · ⊕ H2k ◮ Hk ⊗ Hk′ ≃ H(−1)k+k′ |k−k′|
⊕ · · · ⊕ H(−1)k+k′
k+k′
◮ H(−1)k
k
= Hk are the irreducible representations of SO(3)
◮ H(−1)k+1
k
are the remaining irreducible representations of O(3)
SLIDE 70
Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2
SLIDE 71
Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2 ◮ The first fundamental theorem for the orthogonal group
implies that the Zk are polynomial matrices in the inner products of the points J ∪ J′ ⊆ S2
SLIDE 72
Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2 ◮ The first fundamental theorem for the orthogonal group
implies that the Zk are polynomial matrices in the inner products of the points J ∪ J′ ⊆ S2
◮ The constraints
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅} become polynomial inequalities
SLIDE 73 Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2 ◮ The first fundamental theorem for the orthogonal group
implies that the Zk are polynomial matrices in the inner products of the points J ∪ J′ ⊆ S2
◮ The constraints
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅} become polynomial inequalities
◮ Variables: inner products between the points in S
SLIDE 74 Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2 ◮ The first fundamental theorem for the orthogonal group
implies that the Zk are polynomial matrices in the inner products of the points J ∪ J′ ⊆ S2
◮ The constraints
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅} become polynomial inequalities
◮ Variables: inner products between the points in S ◮ Coefficients: given in terms of the entries of the Fk
SLIDE 75 Sums of squares characterizations
◮ Zk(gJ, gJ′) = Zk(J, J′) for all g ∈ O(3) and J, J′ ∈ V2 ◮ The first fundamental theorem for the orthogonal group
implies that the Zk are polynomial matrices in the inner products of the points J ∪ J′ ⊆ S2
◮ The constraints
AtK(S) ≤ −1I=1(S) for S ∈ I2t \ {∅} become polynomial inequalities
◮ Variables: inner products between the points in S ◮ Coefficients: given in terms of the entries of the Fk
◮ Modeling these constraints using sums of squares
characterizations reduces the problems to finite dimensional semidefinite programs
SLIDE 76 Thank you
- D. de Laat, F. Vallentin, A semidefinite programming hierarchy for
packing problems in discrete geometry, arXiv:1311.3789 (2013), 21 pages.
Image credit: http://www.buddenbooks.com/jb/images/150a5.gif http://en.wikipedia.org/wiki/File:Disk_pack10.svg
- W. Zhang, K.E. Thompson, A.H. Reed, L. Beenken, Relationship between packing structure and porosity in fixed
beds of equilateral cylindrical particles, Chemical Engineering Science 61 (2006), 8060–8074.
