Semidefinite Programming and Harmonic Analysis

Transcription

1 1 / 74 Semidefinite Programming and Harmonic Analysis Cristóbal Guzmán CS Discrete Fourier Analysis and Applications March 7, 2012

2 2 / 74 Motivation SDP gives best relaxations known for some combinatorial optimization problems. Medium/large scale SDP are beyond the grasp of computational tractability (although polynomial). Many combinatorial applications are invariant under a group action. These symmetries lead to nice block decompositions of the SDP, making them computationally tractable.

3 3 / 74 Objectives Show the modeling capabilities of SDP. Show through examples how to exploit symmetries for simplifying SDP formulations. Generalize SDP models to infinite dimensions, through combinatorial applications.

4 4 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

5 5 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

6 6 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

7 Semidefinite Programming SDP: optimization of a linear function over the intersection of the cone of p.s.d. matrices with an affine subspace Figure: Spectrahedron given by 1 x 1 x 2 x 1 1 x 3 x 2 x / 74

8 8 / 74 Semidefinite Programming SDP: optimization of a linear function over the intersection of the cone of p.s.d. matrices with an affine subspace Figure: Spectrahedron given by 1 x 1 x 2 x 1 1 x 3 x 2 x 3 1 Primal problem (in standard form) m (P) min{c T x : A i x i B 0}. i=1 0.

9 9 / 74 Semidefinite Programming SDP: optimization of a linear function over the intersection of the cone of p.s.d. matrices with an affine subspace Figure: Spectrahedron given by 1 x 1 x 2 x 1 1 x 3 x 2 x 3 1 Primal problem (in standard form) m (P) min{c T x : A i x i B 0}. Dual problem i=1 0. (D) max{tr(bλ) : Tr(A i Λ) = c i, Λ 0}.

10 10 / 74 Remarks SDP is a matrix analogue of Linear Programming (A i, B diagonal) Expresive power: stability of linear systems, eigenvalue optimization, polynomial optimization, combinatorial relaxations, etc. Computationally Tractable : polynomial time interior point methods, and fast first-order methods available. Under strict primal (or dual) feasibility, we have strong duality OPT (P) = OPT (D) Note that these values are not necessarily achieved!

11 11 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

12 12 / 74 Independent sets in graphs G = (V, E) graph, V vertex set, E V 2 edge set. C V is independent if all pairs in C are not adjacent, ie x, y C : {x, y} / E. Figure: Blue vertices form an independent set.

13 13 / 74 Independent sets in graphs G = (V, E) graph, V vertex set, E V 2 edge set. C V is independent if all pairs in C are not adjacent, ie x, y C : {x, y} / E. Figure: Blue vertices form an independent set. For a finite graph G define the independence number α(g) = max{ C : C independent set}. Finding (or even approximating) α is NP-hard

14 14 / 74 Lovász Theta Function (1979) Theorem α(g) ϑ 1 (G). max x V y V K (x, y) s.t. Tr(K ) = 1, ϑ 1 (G) = K (x, y) = 0 if xy E K 0 Proof. Let C V be an independent set, and 1 C {0, 1} V its characteristic vector. Then K = 1 C 1 C1 T C is feasible for the SDP and x,y V K (x, y) = C. Thus, C ϑ 1 (G).

15 15 / 74 Theta Function: Dual Form min λ s.t. K (x, x) = λ 1 x V ϑ 1 (G) = K (x, y) = 1 xy / E K 0

16 16 / 74 Remarks: Theta Function: Dual Form min λ s.t. K (x, x) = λ 1 x V ϑ 1 (G) = K (x, y) = 1 xy / E K 0 Lovász relaxation is stronger than LP relaxation. Lovász upper bound is weak, there exists an infinite family of graphs with ϑ α > n 2 c logn This makes sense: the stability number is hard to approximate up to a factor n 1 ɛ For symmetric graphs, Lovász relaxation might give tighter bounds

17 17 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

18 18 / 74 Distance Graphs Let V be a (separable) compact metric space, with distance d : V V R +. µ is a finite (regular Borel) measure. Adjacency in the distance graph is determined by the distance d Definition Let I R +. The distance graph G(V, I) = (V, E) is a graph with vertex set V and edge set E = {xy : d(x, y) I}.

19 19 / 74 A Familiar Example: Finite Connected Graphs Let G = (V, E) be a finite graph Let d : V V N {+ } given by { length of shortest xy-path, xy-path d(x, y) = +, xy-path If G is connected, then d is a metric on V Let µ(a) = A, so µ is the uniform measure In this case, E can be defined only in terms of the metric, namely xy E iff d(x, y) = 1

20 20 / 74 Example: Kissing Numbers τ n = maximum number of unit spheres which can touch a central unit sphere without pairwise overlapping Figure: Optimal kissing configurations for dimensions 2 and 3

21 21 / 74 Example: Kissing Numbers τ n = maximum number of unit spheres which can touch a central unit sphere without pairwise overlapping Figure: Optimal kissing configurations for dimensions 2 and : τ 3 =? 12 (Isaac Newton) or 13 (David Gregory)?

22 22 / 74 Kissing Numbers: Some History τ 1 = 2, τ 2 = 6 are trivial τ 3 = 12, Shütte, van der Waerden (1953), by elementary geometry (difficult) τ 8 = 240, τ 24 = , Odlyzko, Sloane and Levenshtein (1979), by LP (relatively easy, and elegant). τ 4 = 24, Musin (2003), by LP and elementary geometry (difficult) Unified proof (and more) with SDP!

23 23 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

24 24 / 74 Kissing Numbers and Distance Graphs V = S n 1 := {x R n : x 2 2 = 1}, the unit sphere. d(x, y) = arccos(x y) is a metric on V. e.g. d(x, x) = π. ω, the surface area induced by the Lebesgue measure is a measure on V. ω(s n 1 ) = 2π n/2 /Γ(n/2).

25 25 / 74 Kissing Numbers and Distance Graphs V = S n 1 := {x R n : x 2 2 = 1}, the unit sphere. d(x, y) = arccos(x y) is a metric on V. e.g. d(x, x) = π. ω, the surface area induced by the Lebesgue measure is a measure on V. ω(s n 1 ) = 2π n/2 /Γ(n/2). G(V, (0, θ)) is a distance graph. Stable sets in this graph are called spherical codes with minimal angular distance θ. A(n, θ) = α( G(S n 1, (0, θ)) ) Note: n-th kissing number is equivalent to A(n, π/3).

26 26 / 74 Positive Hilbert-Schmidt Kernels For Lovász SDP we need p.s.d infinite matrices C(S n 1 ) = {f : S n 1 R f continuous} Hilbert-Schmidt kernel: elements of C(S n 1 S n 1 ) Symmetric (or Hermitian) HS kernel: K (x, y) = K (y, x), for all x, y S n 1 Definition A symmetric HS kernel K is called positive if for all f C(S n 1 S n 1 ): K (x, y)f (x)f (y) dω(x) dω(y) 0 S n 1 S n 1 As in the matrix case, we will use the shorthand notation K 0 for positive kernels K.

27 27 / 74 Lovász Relaxation For a distance graph G(V, I), we get an infinite-dimensional SDP relaxation min λ s.t. K (x, x) = λ 1, x V ϑ(g(v, I)) = K (x, y) = 1, xy / E K 0

28 28 / 74 Lovász Relaxation For Kissing numbers min λ ϑ(s n 1 s.t. K (x, x) = λ 1, x S, (0, π/3)) = n 1 K (x, y) = 1, x y π/3 K 0

29 29 / 74 Theorem α(g(v, I)) ϑ(g(v, I)). Proof. Let C be a stable set in G(V, I) and let K be any feasible kernel for the relaxation. We will prove C λ. Consider the matrix (K (c, c )) c,c C2. It is p.s.d., thus 1 T K 1 = c,c K (c, c ) 0. On the other hand c,c K (c, c ) = K (c, c) + K (c, c ) c c c C K (c, c) C ( C 1) Therefore, C 1 K (c, c) = λ 1.

30 30 / 74 Error-Correcting (Binary) Codes Model for robust communication accross noisy channels, and store/retrieve information on media Idea: send redundant messages, so receiver can detect and correct errors, under some assumptions.

31 31 / 74 Error-Correcting (Binary) Codes Model for robust communication accross noisy channels, and store/retrieve information on media Idea: send redundant messages, so receiver can detect and correct errors, under some assumptions. V = {0, 1} n, the Hamming cube. δ(x, y) = {i [n] : x i y i } is a metric on V

32 32 / 74 Error-Correcting (Binary) Codes Model for robust communication accross noisy channels, and store/retrieve information on media Idea: send redundant messages, so receiver can detect and correct errors, under some assumptions. V = {0, 1} n, the Hamming cube. δ(x, y) = {i [n] : x i y i } is a metric on V A(n, d)= maximal cardinality of C V such that every two points in C have Hamming distance at least d. Thus, A(n, d) = α({0, 1} n, {1,..., d 1}) A stable set in this graph can correct (d 1)/2 errors A(n, d) is unknown to large extend

33 33 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

34 34 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

35 35 / 74 Automorphism Group Definition Given a distance graph G(V, I), we define its automorphism group as Aut(G) = {u : V V : u bijective, d(x, y) = d(ux, uy) x, y V }

36 36 / 74 Automorphism Group Definition Given a distance graph G(V, I), we define its automorphism group as Aut(G) = {u : V V : u bijective, d(x, y) = d(ux, uy) x, y V } Examples: For the distance graph defined by shortest paths in G = (V, E), the automorphism group is Aut(G) = {u : V V : u bijective, (x, y) E iff (ux, uy) E} Deciding whether this group is trivial is as hard as the Graph Isomorphism Problem

37 37 / 74 Automorphism Group Aut(G) = {u : V V : u bij., d(x, y) = d(ux, uy) x, y V } Examples: For the Hamming cube, Aut(G) has order 2 n n!. It is generated by all n! permutations and all 2 n switches 0 1. For S n 1, the automorphism group is the orthogonal group Aut(S n 1 ) = O(R n ) := {u M n (R) : u T u = I} This group not only preserves distance, but also is invariant for, x, y = ux, uy u O(R n )

38 38 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

39 39 / 74 Symmetry and Optimization Previous examples are invariant by Aut(G) (e.g. flip-permute in {0, 1} n, rotate kissing configurations). Lovász SDP is reduced by exploting these symmetries. Definition A positive kernel K is Aut(V )-invariant if for all u Aut(V ) K (ux, uy) = K (x, y)

40 Symmetry and Optimization Previous examples are invariant by Aut(G) (e.g. flip-permute in {0, 1} n, rotate kissing configurations). Lovász SDP is reduced by exploting these symmetries. Definition A positive kernel K is Aut(V )-invariant if for all u Aut(V ) K (ux, uy) = K (x, y) Remark: If K is feasible for the Lovász SDP, then so it is its Aut(V )-invariant group average: K (x, y) = 1 Aut(V ) u Aut(V ) K (ux, uy) Change of variables by u preserves feasibility and value. 40 / 74

41 41 / 74 Lovász SDP for A(n, d): Example: Binary codes SDP min λ s.t. K (x, x) = λ 1 x {0, 1} n ϑ = K (x, y) 1 δ(x, y) {d,..., n} K 0 K is Aut({0, 1} n )-inv.

42 42 / 74 Lovász SDP for A(n, d): Example: Binary codes SDP min λ s.t. K (x, x) = λ 1 x {0, 1} n ϑ = K (x, y) 1 δ(x, y) {d,..., n} K 0 K is Aut({0, 1} n )-inv. We will show that the two last constraints can be re-written as K (x, y) = n f k Kk n (δ(x, y)) k=1 where f k 0 are constants, and Kk n(t) = n are the Krawtchouk polynomials. t=0 ( t i )( n t k i) ( 1) i,

43 43 / 74 Example: Binary codes LP By this last theorem, we have the following LP in f 1,..., f n min 1 + n ϑ k=0 f kkk n(0) = s.t. n k=0 f kkk n (t) 1 t = d,..., n f k 0 t = 1,..., n The proof of the representation theorem requires tools from Harmonic Analysis.

44 44 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

45 45 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

46 46 / 74 Aut(V ) as a Topological Group Γ =Aut(V ) is a compact topological group (product and inversion are continuous) There always exists the Haar (a.k.a left-invariant) measure f (vu) = f (u) v Γ, f C(Γ) Γ Γ Examples: uniform distribution, Lebesgue measure. Given a positive kernel K, we define its group average as K (x, y) = K (ux, uy)du Γ

47 47 / 74 Γ =Aut(V ) acts on C(V ) by uf (x) = f (u 1 x) This action is linear u(αf + βg) = αuf + βug (f, g) := fg is an invariant inner product for C(V ) Γ (f, g) = (uf, ug) f, g C(V ), u Γ

48 Γ =Aut(V ) acts on C(V ) by uf (x) = f (u 1 x) This action is linear u(αf + βg) = αuf + βug (f, g) := fg is an invariant inner product for C(V ) Γ (f, g) = (uf, ug) f, g C(V ), u Γ A subspace S C(V ) is Γ-invariant if us = S, u Γ. S is Γ-irreducible if {0} and S are the only Γ-invariant subspaces of S If V is Γ-invariant, then V is Γ-invariant too Irreducible spaces are either equivalent ( linear map between orthonormal bases) or orthogonal! 48 / 74

49 49 / 74 Peter-Weyl Theorem Theorem (Peter & Weyl (1927)) All irreducible subspaces of C(V ) are of finite dimension The space C(V ) decomposes orthogonally as C(V ) = H k, k N Each space H k decomposes orthogonally in irreducible spaces m k H k = H k,i, where H k,i = Hk,i iff k = k The dimension h k of H k,i is finite, but the multiplicity m k might be infinite. i=1

50 50 / 74 Bochner s Representation Theorem Theorem (Bochner (1941)) Let e k,i,l be an complete orthonormal system for C(V ) (by Peter-Weyl). Every Aut(V )-invariant, positive K is s.t. K (x, y) = k N or more economically as h k f k,ij i,j {1,...,m k } l=1 K (x, y) = F k, Z (x,y) k, k N e k,i,l (x)e k,j,l (y), with (F k ) ij = f k,ij and (Z (x,y) k ) ij = h k l=1 e k,i,l(x)e k,j,l (y). Each F k is Hermitian and positive. The series converges absolutely and uniformly.

51 51 / 74 About Bochner s Theorem Recall the spectral theorem for symmetric matrices n K = PDP T i.e. K (x, y) = λ k e k (x)e k (y). k=1

52 52 / 74 About Bochner s Theorem Recall the spectral theorem for symmetric matrices n K = PDP T i.e. K (x, y) = λ k e k (x)e k (y). k=1 Peter-Weyl theorem is nothing but R n = e 1... e n. Matrix diagonalization contains one block of 1-dimensional (equivalent) subspaces (multiplicity n)

53 53 / 74 About Bochner s Theorem Recall the spectral theorem for symmetric matrices n K = PDP T i.e. K (x, y) = λ k e k (x)e k (y). k=1 Peter-Weyl theorem is nothing but R n = e 1... e n. Matrix diagonalization contains one block of 1-dimensional (equivalent) subspaces (multiplicity n) These are the irreducible subspaces for Γ = {1}. Not so interesting after all, since we only imposed K invariant for the identity.

54 54 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

55 55 / 74 Particular cases: Boolean Harmonics Theorem (Peter-Weyl for the Hamming Cube) C({0, 1} n ) = H 0 H 1... H n. H k is spanned by the characters χ y (x) = ( 1) y x of deg. k H k is irreducible, with dimension h k = ( ) n k

56 56 / 74 Particular cases: Boolean Harmonics Theorem (Peter-Weyl for the Hamming Cube) C({0, 1} n ) = H 0 H 1... H n. H k is spanned by the characters χ y (x) = ( 1) y x of deg. k H k is irreducible, with dimension h k = ( ) n k Remarks: Since the multiplicities are 1, positive kernels only involve psd matrices of size 1 1, ie, nonnegative scalars. Aut({0, 1} n ) acts distance-transitively on pairs δ(x, y) = δ(x, y ) u : (ux, uy) = (x, y ). This implies, invariant kernels only depend on the distance.

57 57 / 74 Boolean Harmonics Representation: Proof Idea By Fourier analysis, (χ y ) y {0,1} n is an orthonormal basis for C({0, 1} n ). dim χ y =1, and is invariant under ({0, 1} n, + mod 2 ) (switches) uχ y (x) = χ y (x u) = ( 1) y ( u) χ y (x) χ y. Under the complete group Aut({0, 1} n ) (i.e., also permutations), these one dimensional subspaces are grouped together, according to their degree H k = {χ y : deg(y) = k} are Aut({0, 1} n )-invariant

58 58 / 74 Boolean Harmonics Representation: Proof Idea Claim: H k is irreducible, for k = 0,..., n. But first Definition f : {0, 1} n C is a zonal spherical function with pole e {0, 1} n if u Aut({0, 1} n ) & ue = e uf = f. Lemma Let U C({0, 1} n ) be an invariant subspace and let e {0, 1}. If the subspace of zonal spherical functions with pole e that lie in U has dimension one, then U is irreducible.

59 59 / 74 Proof of the lemma Proof. If U is not irreducible, then U = V V, both invariant. Let e 1,..., e n and f 1,..., f m orthonormal bases for V and V, resp. Let z V (x) = n e i (e)e i (x), z V (x) = i=1 m f j (e)f j (x) j=1 These are two l.i. zonal spherical functions with pole e, a contradiction.

60 60 / 74 Boolean Harmonics: Proof Idea Apply the lemma to H k, for pole e = (0,..., 0). Let f H k be a zonal spherical function with pole 0

61 61 / 74 Boolean Harmonics: Proof Idea Apply the lemma to H k, for pole e = (0,..., 0). Let f H k be a zonal spherical function with pole 0 Group stabilizing 0 = all permutations (no switching!).

62 Boolean Harmonics: Proof Idea Apply the lemma to H k, for pole e = (0,..., 0). Let f H k be a zonal spherical function with pole 0 Group stabilizing 0 = all permutations (no switching!). Thus, for any u σ n, uf = f. Take Fourier transforms u σ n ˆf (y)χuy (x) = ˆf (y)χy (x). y {0,1} n deg(y)=k y {0,1} n deg(y)=k 62 / 74

63 Boolean Harmonics: Proof Idea Apply the lemma to H k, for pole e = (0,..., 0). Let f H k be a zonal spherical function with pole 0 Group stabilizing 0 = all permutations (no switching!). Thus, for any u σ n, uf = f. Take Fourier transforms u σ n ˆf (y)χuy (x) = ˆf (y)χy (x). y {0,1} n deg(y)=k y {0,1} n deg(y)=k Therefore, all Fourier coefficients have to coincide! i.e., subspace has dimension / 74

64 64 / 74 Proof. Bochner s Representation Theorem H k s are not equivalent, so they are in different blocks. Let us compute Bochner s representation functions Z k (x, x ) = χ y (x)χ y (x ). deg(y)=k Recall that Z k (x, x ) only depends on t = δ(x, x ). Thus, Z k (x, x ) = Z k ((0,..., 0), (1,..., 1, 0,..., 0) }{{}}{{} t n t

65 65 / 74 Proof. Bochner s Representation Theorem H k s are not equivalent, so they are in different blocks. Let us compute Bochner s representation functions Z k (x, x ) = χ y (x)χ y (x ). deg(y)=k Recall that Z k (x, x ) only depends on t = δ(x, x ). Thus, Z k (x, x ) = Z k ((0,..., 0), (1,..., 1, 0,..., 0) }{{}}{{} t n t k ( )( ) t n t = ( 1) i i k i i=1

66 66 / 74 Proof. Bochner s Representation Theorem H k s are not equivalent, so they are in different blocks. Let us compute Bochner s representation functions Z k (x, x ) = χ y (x)χ y (x ). deg(y)=k Recall that Z k (x, x ) only depends on t = δ(x, x ). Thus, Z k (x, x ) = Z k ((0,..., 0), (1,..., 1, 0,..., 0) }{{}}{{} t n t k ( )( ) t n t = ( 1) i i k i i=1 = K n k (t)

67 67 / 74 Lovász SDP for A(n, d): Finally: Binary Codes Invariant Relaxation min λ s.t. K (x, x) = λ 1 x {0, 1} n ϑ = K (x, y) 1 δ(x, y) {d,..., n} K 0 K is Aut({0, 1} n )-inv. By Bochner s theorem for Boolean harmonics, we get an LP in f 1,..., f n min 1 + n ϑ k=0 f kkk n(0) = s.t. n k=0 f kkk n (t) 1 t = d,..., n f k 0 t = 1,..., n

68 68 / 74 Outline 1 Introduction Semidefinite Programming Lovász Theta Function Distance Graphs Infinite-Dimensional Lovász Relaxation 2 Exploiting Symmetry The Automorphism Group Symmetry and Optimization 3 Harmonic Analysis Rudiments of Representation Theory Application to Binary Codes Application to Kissing Numbers

69 69 / 74 Spherical Harmonics Theorem (Peter-Weyl for Spherical Harmonics) The space of complex-valued continuous functions on the unit sphere decomposes orthogonally into pairwise O(R n )-irreducible spaces C(S n 1 ) = k N H k, where H k is the space of homogeneous polynomials f of degree k which vanish under the Laplace operator f (x) = 0, x S n 1. H k has dimension h k = ( ) ( n+k 1 n 1 n+k 3 ) n 1.

70 70 / 74 Spherical Harmonics (Bochner theorem) Theorem (Schoenberg (1940)) Every positive, O(R n )-invariant kernel K C(S n 1 S n 1 ) can be written as where K (x, y) = k=0 f k Pk n (x y), The RHS converges absolutely and uniformly f k 0 for all k, and k f k < + (P n k ) k N is an orthonormal basis for L 2 ([ 1, 1], (1 t 2 ) (n 3)/2 dt) (Jacobi polynomials).

71 71 / 74 Kissing Numbers Invariant Relaxation min λ s.t. K (x, x) = λ 1, x S n 1 τ n = K (x, y) = 1, x y π/3 K 0 K is O(R n ) invariant By Schoenberg s characterization, we get an infinite-dimensional LP min λ s.t. τ n = f k 0, k N k N f kpk n(1) = λ 1 k N f kpk n (t) 1, t [ 1, 1/2]

72 72 / 74 Kissing Numbers Invariant relaxation min τ n = λ s.t. f k 0, k N k N f kpk n(1) = λ 1 k N f kpk n (t) 1, t [ 1, 1/2] We can truncate the series, and use sums-of-squares relaxations for the last constraint (recall Hilbert s seventeenth problem). This still provides upper bounds. This gives new proofs for τ 8 = 240, and τ 24 =

73 73 / 74 Other Discrete Geometric Problems Problem Vertex set Harmonic New results analysis Kissing S n 1 Compact, Unified numbers non-abelian proofs Coloring R n Non-compact, Bounds for abelian n = 4,..., 24, new asymptotics Sphere R n Non-compact, Bounds for packing abelian n = 4,..., 36 Body R n O(R n ) Non-compact, In progress Packing non-abelian

74 74 / 74 Bibliography F. Vallentin. Lecture notes: Semidefinite programing and harmonic analysis. Tutorial for HPOPT 2008, in Tilburg Univ., 2008 C. Bachoc, D.C. Gijswijt, A. Schrijver, F. Vallentin. Invariant semidefinite programs. In Handbook on Semidefinite, Conic and Polynomial Optimization (M.F. Anjos, J.B. Lasserre (ed.)). Springer, H. Cohn, N. Elkies. New upper bounds on sphere packings, I & II. Annals of Mathematics 157, W. Rudin. Fourier analysis on groups. Wiley classics library, 1990