Lattices, sphere packings, spherical codes and energy minimization Abhinav Kumar MIT November 10, 2009
Sphere packings Deﬁnition A sphere packing in Rn is a collection of spheres/balls of equal size which do not overlap (except for touching). The density of a sphere packing is the volume fraction of space occupied by the balls. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Lattices Deﬁnition A lattice Λ in Rn is a discrete subgroup of rank n, i.e. generated by n linearly independent vectors of Rn . Examples: integer lattice Zn , checkerboard lattice Dn , simplex lattice An , special root lattices E6 , E7 , E8 , Leech lattice Λ24 , and so on. Associated sphere packing: if m(Λ) is the length of a smallest non-zero vector of Λ, then we can put balls of radius m(Λ)/2 around each point of Λ so that they don’t overlap.
Spherical codes A spherical code C on S n−1 is a ﬁnite subset of the sphere. Example: The kissing conﬁguration of a lattice Λ is the set of minimal non-zero vectors of Λ, rescaled to the unit sphere. The minimal angle θ(C) of the code is the smallest radial angle between distinct elements of C. cos(θ(C)) = max x, y . x,y ∈C,x=y
Sphere packing problem Problem: Find a/the densest sphere packing(s) in R n . In dimension 1, we can achieve density 1 by laying intervals end to end. In dimension 2, the best possible is by using the hexagonal lattice. [Fejes T´th 1940] o
Sphere packing problem II In dimension 3, the best possible way is to stack layers of the solution in 2 dimensions. This is Kepler’s conjecture, now a theorem of Hales.
There are inﬁnitely (in fact, uncountably) many ways of doing this! These are the Barlow packings. In higher dimensions, we have some guesses for the densest sphere packing. But no proofs yet.
Lattices packing The packing problem for lattices asks for the densest lattice(s) in Rn for every n. This is equivalent to the determination of the Hermite constant γn , which arises in the geometry of numbers. The known answers are: n 1 2 3 4 5 6 7 8 24 Λ A1 A2 A3 D4 D5 E6 E7 E8 Leech In low dimensions, lattices provide good candidates for the densest sphere packings. But in high dimensions, it is expected that the densest packings will not be lattices.
Spherical code problem The spherical code problem asks: 1 Given an angle θ0 ∈ (0, π] and a dimension n, what is the maximum number of points N for which there is a spherical code on S n−1 with minimal angle at least θ0 ? 2 Given n and N, what is the maximum possible angle of an N-point code on S n−1 ? An optimal spherical code is one which solves the second problem.
Kissing problem The kissing problem is the spherical code problem for angle π/3. It can be rephrased as: how many non-overlapping unit spheres can be arranged around a central unit sphere in Rn ? Answers n = 1: Two intervals around a central interval. n = 2: Six circles around a central circle. n = 3 (Newton-Gregory problem): Twelve spheres, but the conﬁguration is not rigid or unique. Proof by Sch¨tte and van u der Waerden, 1953.
Kissing problem II n = 4: Twenty-four, coming from kissing arrangement of the D4 lattice. Proof by Musin, 2003. Believed to be unique, but unproven. n = 8: Kissing conﬁguration of E8 of 240 points. n = 24: Kissing conﬁguration of Λ24 of 196560 points. The last two are due to Odlyzko and Sloane and (independently) Levenshtein. No other answers known.
LP bounds How do we prove optimality/uniqueness of any of these? One idea: linear programming bounds. Invented by Delsarte to deal with association schemes, binary codes etc. Levenshtein used these to prove optimality and uniqueness of a number of codes or families of codes. For instance, the 240-point kissing conﬁguration in R 8 and the 196560-point kissing conﬁguration in R 24 are sharp for the linear programming bound, and therefore optimal. This even enables one to prove uniqueness. Application of the linear programming bound also allowed the resolution of the lattice packing problem in 24 dimensions [Cohn-K 2003].
Positive deﬁnite kernels Fix n ≥ 2. We say f : [−1, 1] → R is a positive deﬁnite kernel if for every code C ⊂ S n−1 , the |C | × |C | matrix f ( x, y ) x,y ∈C is positive semideﬁnite. In particular, x,y ∈C f ( x, y ) ≥ 0. Sch¨nberg (1930s) classiﬁed all the positive deﬁnite kernels. He o showed that the ultraspherical or Gegenbauer polynomials Ciλ (t), i = 0, 1, 2, . . . are PDKs and that any PDK is a non-negative linear combination of them. Here λ = n/2 − 1.
Gegenbauer polynomials The Gegenbauer polynomials arise from representation theory/harmonic analysis. They are given by the generating function ∞ (1 − 2tz + z 2 )−λ = Ciλ (t)z i i =0 So we have 1 C0 (t) = 1 2 C1 (t) = (n − 2)t 3 C2 (t) = (n − 2)(nt 2 − 1)/2 and so on.
Linear programming bound for codes Theorem Let f (t) = i fi Ciλ (t) be a positive deﬁnite kernel (i.e. all fi ≥ 0), such that f0 0 and f (t) ≤ 0 for t ∈ [−1, cos θ0 ]. Then any code C with minimal angle at least θ0 has at most f (1)/f0 points. Proof. We have |C |f (1) ≥ f ( x, y ) x,y ∈C = fi Ciλ ( x, y ) x,y ∈C i = fi Ciλ ( x, y ) i x,y ∈C 2 ≥ f0 |C | .
LP bounds II Why are these called linear programming bounds? Write f = 1 + i 0 fi Ci . Variables: fi . Constraints: fi ≥ 0 Constraints: 1 + fi Ci (t) ≤ 0 for every t ∈ [−1, cos(θ0 )]. Objective function: minimize 1 + fi Ci (1). This is a convex optimization program, and we can approximate it by a linear program by discretizing the interval [−1, 1], and restrinting to ﬁnitely many nonzero fi .
Potential energy minimization One way to ﬁnd good spherical codes: potential energy. Put N points on a sphere with a repulsive force law (e.g. electrostatic repulsion), and let the system evolve. They will tend to separate themselves to minimize potential energy. For S 2 , this is called the Thomson problem after the physicist J. J. Thomson, who asked it in connection with the plum-pudding model of the atom.
Potential energy minimization II Deﬁnition Let f : (0, 4] → R be a function. We deﬁne the f -potential energy of a code C ⊂ S n−1 to be Ef (C) = f (|x − y |2 ) x,y ∈C,x=y Note: 1 Each pair of points counted twice, so the potential energy is double that of the physicists. 2 We let f be a function of squared distance, rather than distance. This makes the formulas nicer.
Potential energy minimization III The problem is: given n, N and a potential function f , ﬁnd a spherical code of N points on S n−1 that minimizes f -potential energy.
Examples of potential functions Inverse power law: Ik (r ) = 1/r k for k ∈ R0 . For k = n/2 − 1, this gives the harmonic potential. Gaussian: Gc (r ) = exp(−cr ) for some c 0 (note that this is Gaussian as a function of distance). Aℓ (r ) = (4 − r )ℓ for nonnegative integers ℓ. All these functins are completely monotonic, i.e. (−1)m f (m) (r ) ≥ 0. The functions Aℓ span the cone of completely monotonic functions on (0, 4].
Universal optimality Note: As k 0, the potential energy minimization problem for 1/r k becomes the spherical code problem (maximize the minimal angle). Similarly, the spherical code problem is a limit of the energy minimization problems for Gaussians as well. We say a spherical code is universally optimal if it minimizes f -potential energy (among codes of its size) for all completely monotonic f . There are examples of universally optimal codes, though their existence is very uncommon. The typical situation is that we have one or more families of N-point conﬁgurations, each being optimal for Aℓ in a certain range of ℓ.
Examples N points in S 1 : regular N-gon. 2 points in S 2 : antipodal points, universally optimal. 3 points in S 2 : equilateral triangle, universally optimal. 4 points in S 2 : regular tetrahedron, universally optimal. In general, k ≤ n + 1 points on S n−1 ⊂ Rn always gives a regular simplex in a k − 1 dimensional “equatorial” subspace, under any completely monotonic function.
Examples II 5 points in S 2 : two competing conﬁgurations. Consider the conﬁguration A of two antipodal points with three points on the equator forming an equilateral triangle. t t h h h h t h @ t @@@ @ t t tt
Examples III The conﬁguration Bθ consists of a pyramid with one point on the north pole, and four points in the southern hemisphere at latitude θ = α − π/2, forming a square. t ¡t e ¡ t e ¡ et t¡ q et t α t ¡ e ¡ t e t
Examples IV For inverse power laws, some Bθ wins for steep power laws 1/r k for k 7.524+, but A wins for smaller k. Note that A maximizes angular distance, as does B0 . For the function Aℓ , the conﬁguration A wins for 1 ≤ ℓ ≤ 6, whereas some Bθ wins for ℓ ≥ 7.
LP bounds for potential energy The linear programming bounds of Delsarte for the spherical code problem (maximize N for a given θ), were adapted by Yudin to give LP bounds for potential energy. Theorem (Yudin) Let f : (0, 4] → R be any function. Suppose h : [−1, 1] → R is a polynomial such that h(t) ≤ f (2 − 2t) for all t ∈ [−1, 1], and suppose there are nonnegative coeﬃcients α0 , . . . , αd such that d h(t) = αi Ciλ (t) in terms of the Gegenbauer polynomials. Then i =0 every set of N points on S n−1 has potential energy at least N 2 α0 − Nh(1).
Examples Known universally optimal conﬁgurations of N points on S n−1 : n N Name 2 N N-gon n n+1 simplex n 2n cross polytope 3 12 icosahedron 4 120 600-cell 8 240 E8 root system 7 56 spherical kissing 6 27 spherical kissing/Schl¨ﬂi a 5 16 spherical kissing/Clebsch 24 196560 Leech lattice minimal vectors 23 4600 spherical kissing 22 891 spherical kissing 23 552 regular 2-graph 22 275 McLaughlin 21 162 Smith 22 100 Higman-Sims 3 q q +1 q+1 (q + 1)(q 3 + 1) Cameron-Goethals-Seidel They are all sharp for LP bounds [Cohn-K, 2005].
Other conjectured universal optima Work of Ballinger, Blekherman, Cohn, Giansiracusa, Kelly, Sch¨rmann. u 40 points in R10 , inner products 1, 1/6, 0, −1/3, −1/2. 64 points in R14 , inner products 1, 1/7, −1/7, −3/7. Not sharp for linear programming bounds but perhaps still universally optimal.
Other spaces We can also deﬁne potential energy for ﬁnite subsets of other compact metric spaces, and sometimes for inﬁnite subsets of noncompact spaces. For example, for compact two-point homogeneous spaces such as RPn , CPn , HPn , we can deﬁne the notion of universal optimality (for completely monotonic functions of squared chordal distance) and ﬁnd examples of universally optimal codes (joint work with Cohn, Elkies, Khatirinejad).
Euclidean space Let P be a periodic point conﬁguration in Euclidean space. Let f : R≥0 → R a suﬃciently rapidly decaying function. Deﬁne the f -potential energy of a point x ∈ P to be f (|x − y |2 ) y ∈P,y =x and the f -potential energy of P to be the (ﬁnite) average over all points x ∈ P of their potential energies. The energy minimization problem asks: given n, f and a ﬁxed point density δ (usually 1), which periodic point conﬁguration of density δ minimizes the f -potential energy over all periodic conﬁgurations in Rn (the number of translates N is allowed to vary).
Gradient descent Recently, with Cohn and Sch¨rmann, we wrote a computer u program to carry out gradient descent on spaces of periodic conﬁgurations, to search for optima for the Gaussian potential functions exp(−cr 2 ), and obtained some interesting results for the minimum energy conﬁgurations observed experimentally. In particular, we found families of formally dual periodic conﬁgurations, i.e. conﬁgurations whose average theta functions are related by the generalized Jacobi formula, which replaces z by −1/z and multiplies by an appropriate constant. The exp(−cπr 2 )-potential energy of P should be related to the exp(−πr 2 /c)-potential energy of its dual by a factor depending on the density of P, for every c.
Formal duality Every lattice has a formal dual (namely, its dual lattice). But we do not know of any other formally dual conﬁgurations recorded previously in the literature. Example + Let Dn be the union of Dn and its translate by the all-halves + vector. Then Dn is always formally self-dual (it is a lattice exactly when 4 divides n). + In addition, if we let Dn (α) be the conﬁguration obtained by + + scaling the last coordinate of all vectors in Dn by α, then Dn (α) is formally dual to Dn + (1/α), so we in fact have a family of formally dual conﬁgurations.
Inverse problem What happens if we evolve 8 points on S 2 under a 1/r k potential? The minimum for energy is not a cube! It’s a skew cube (antiprism), where the distance between the two square faces varies as k varies. Similarly, 20 points on S 2 don’t settle down to a regular dodecahedron under the inverse power laws or Gaussians. Can we design a potential function which is minimized by the cube? Can do it with potential wells, but we want a nicer function.
Inverse problem II Theorem Let f (r ) = 1/r + r /3 − 8r 2 /11 + 2r 3 /9 − r 4 /50. The cube is the unique global minimum for f -potential energy among 8-point codes in S 2 . Proof. Linear programming bounds! We engineer f so that it’s easy to come up with an h that works and gives a sharp bound for the cube. But note that f is in fact decreasing and convex as a function of distance (even though not completely monotonic).
Some questions Are there only ﬁnitely many universal optima in a given dimension? How many local optima are there? Systematic upper/lower bounds? What are good ways to ”walk” the conﬁguration space to ﬁnd good codes (other than gradient descent for potential energy)? Is it possible to beat the D4 lattice for energy, among periodic conﬁgurations or among lattices?
Papers and References Ballinger-Blekherman-Cohn-Giansiracusa-Kelly-Sch¨rmann, u Experimental study of energy-minimizing point conﬁgurations on spheres Cohn-K, Optimality and uniqueness of the Leech lattice among lattices Cohn-K, Universally optimal distribution of points on spheres Cohn-K, Algorithmic design of self-assembling structures Cohn-K, Counterintuitive ground states in soft-core models Cohn-K-Sch¨rmann, Ground states and formal duality u relations in the Gaussian core model