Kissing Numbers Research Articles

Solving clustered low-rank semidefinite programs arising from polynomial optimization

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

On the maximum size packings of disks with kissing radius 3

L\'{a}szl\'{o} Fejes T\'{o}th and Alad\'{a}r Heppes proposed the following generalization of the kissing number problem. Given a ball in $\mathbb{R}^d$, consider a family of balls touching it, and another family of balls touching the first family. Find the maximal possible number of balls in this arrangement, provided that no two balls intersect by interiors, and all balls are congruent. They showed that the answer for disks on the plane is $19$. They also conjectured that if there are three families of disks instead of two, the answer is $37$. In this paper we confirm this conjecture.

A primal–dual penalty method via rounded weighted-ℓ1 Lagrangian duality

We propose a new duality scheme based on a sequence of smooth minorants of the weighted- penalty function, interpreted as a parametrized sequence of augmented Lagrangians, to solve non-convex constrained optimization problems. For the induced sequence of dual problems, we establish strong asymptotic duality properties. Namely, we show that (i) the sequence of dual problems is convex and (ii) the dual values monotonically increase to the optimal primal value. We use these properties to devise a subgradient based primal–dual method, and show that the generated primal sequence accumulates at a solution of the original problem. We illustrate the performance of the new method with three different types of test problems: A polynomial non-convex problem, large-scale instances of the celebrated kissing number problem, and the Markov–Dubins problem. Our numerical experiments demonstrate that, when compared with the traditional implementation of a well-known smooth solver, our new method (using the same well-known solver in its subproblem) can find better quality solutions, i.e. ‘deeper’ local minima, or solutions closer to the global minimum. Moreover, our method seems to be more time efficient, especially when the problem has a large number of constraints.

A Short Solution of the Kissing Number Problem in Dimension Three

In this note, we give a short solution of the kissing number problem in dimension three.

Kissing Numbers and the Centered Maximal Operator

We prove that in a metric measure space X, if for some $$p \in (1,\infty )$$ there are uniform bounds (independent of the measure) for the weak type (p, p) of the centered maximal operator, then X satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type (1, 1) of the centered operator. Thus, the following characterization is obtained: the centered maximal operator satisfies uniform weak type (1, 1) bounds if and only if the space X has the Besicovitch intersection property. In $$\mathbb {R}^d$$ with any norm, the constants coming from the Besicovitch intersection property are bounded above by the translative kissing numbers. The extensive literature on kissing numbers allows us to obtain, first, sharp estimates on the uniform bounds satisfied by the centered maximal operators defined by arbitrary norms on the plane, second, sharp estimates in every dimension when the $$\ell _\infty $$ norm is used, and third, improved estimates in all dimensions when considering euclidean balls, as well as the sharp constant in dimension 3. Additionally, we prove that the existence of uniform $$L^1$$ bounds for the averaging operators associated with arbitrary measures and radii, is equivalent to a weaker variant of the Besicovitch intersection property.

On the Lattice Hadwiger Number of Superballs and Some Other Bodies

We show that the lattice Hadwiger (= kissing) number of superballs is exponential in the dimension. The same methods can be used to show exponential growth for more general convex bodies as well.

A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

Approximating the cone of copositive kernels to estimate the stability number of infinite graphs

It has been shown that the stable set problem in an infinite compact graph, and particularly the kissing number problem, reduces to an optimization problem over the cone of copositive kernels. We propose two converging hierarchies approximating this cone. Both are extensions of existing inner hierarchies for the finite dimensional copositive cone. We implement the first two levels of the new hierarchies for the kissing number problem.

Improving the Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry

ABSTRACTThe kissing number of is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin [Mittelmann and Vallentin 10], based on the semidefinite programming bound of Bachoc and Vallentin [Bachoc and Vallentin 08], computed the best known upper bounds for the kissing number for several values of n ⩽ 23. In this article, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for n = 9, …, 23.

A copositive formulation for the stability number of infinite graphs

In the last decade, copositive formulations have been proposed for a variety of combinatorial optimization problems, for example the stability number (independence number). In this paper, we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite-dimensional copositive program. For this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures. We determine the extreme rays of the latter cone, and we illustrate this theory with the help of the kissing number problem.

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

Functionals on the spaces of convex bodies

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger’s covering conjecture and Borsuk’s partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.

DNA Codeword Design: Theory and Applications

This is a survey of the origin, current progress and applications of a major roadblock to the development of analytic models for DNA computing (a massively parallel programming methodology) and DNA self-assembly (a nanofabrication methodology), namely the so-called CODEWORD DESIGN problem. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves or to their complements and has been recognized as an important problem in DNA computing, self-assembly, DNA memories and phylogenetic analyses because of their error correction and prevention properties. Major recent advances include the development of experimental techniques to search for such codes, as well as a theoretical framework to analyze this problem, despite the fact that it has been proven to be NP-complete using any single concrete metric space to model the Gibbs energy. In this framework, codeword design is reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the key to finding such sets would lie in knowledge of the geometry of these spaces. A new general technique has been recently found to embed them in Euclidean spaces in a hybridization-affinity-preserving manner, i.e., in such a way that oligos with high/low hybridization affinity are mapped to neighboring/remote points in a geometric lattice, respectively. This isometric embedding materializes long-held metaphors about codeword design in terms of sphere packing and error-correcting codes and leads to designs that are in some cases known to be provably nearly optimal for some oligo sizes. It also leads to upper and lower bounds on estimates of the size of optimal codes of size up to 32–mers, as well as to infinite families of solutions to CODEWORD DESIGN, based on estimates of the kissing (or contact) number for sphere packings in Euclidean spaces. Conversely, this reduction suggests interesting new algorithms to find dense sphere packing solutions in high dimensional spheres using results for CODEWORD DESIGN previously obtained by experimental or theoretical molecular means, as well as a proof that finding these bounds exactly is NP-complete in general. Finally, some research problems and applications arising from these results are described that might be of interest for further research.

Interior Epigraph Directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality

We propose and study a new method, called the Interior Epigraph Directions (IED) method, for solving constrained nonsmooth and nonconvex optimization. The IED method considers the dual problem induced by a generalized augmented Lagrangian duality scheme, and obtains the primal solution by generating a sequence of iterates in the interior of the dual epigraph. First, a deflected subgradient (DSG) direction is used to generate a linear approximation to the dual problem. Second, this linear approximation is solved using a Newton-like step. This Newton-like step is inspired by the Nonsmooth Feasible Directions Algorithm (NFDA), recently proposed by Freire and co-workers for solving unconstrained, nonsmooth convex problems. We have modified the NFDA so that it takes advantage of the special structure of the epigraph of the dual function. We prove that all the accumulation points of the primal sequence generated by the IED method are solutions of the original problem. We carry out numerical experiments by using test problems from the literature. In particular, we study several instances of the Kissing Number Problem, previously solved by various approaches such as an augmented penalty method, the DSG method, as well as several popular differentiable solvers. Our experiments show that the quality of the solutions obtained by the IED method is comparable with (and sometimes favourable over) those obtained by the differentiable solvers.

Geometric Approaches to Gibbs Energy Landscapes and DNA Oligonucleotide Design

DNA codeword design has been a fundamental problem since the early days of DNA computing. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves, to each other or to others' complements. Such strands represent so-called domains, particularly in the language of chemical reaction networks (CRNs). The problem has shown to be of interest in other areas as well, including DNA memories and phylogenetic analyses because of their error correction and prevention properties. In prior work, a theoretical framework to analyze this problem has been developed and natural and simple versions of Codeword Design have been shown to be NP-complete using any single reasonable metric that approximates the Gibbs energy, thus practically making it very difficult to find any general procedure for finding such maximal sets exactly and efficiently. In this framework, codeword design is partially reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the size of such sets depends on the geometry of these spaces. Here, the authors describe in detail a new general technique to embed them in Euclidean spaces in such a way that oligonucleotides with high (low, respectively) hybridization affinity are mapped to neighboring (remote, respectively) points in a geometric lattice. This embedding materializes long-held metaphors about codeword design in analogies with error-correcting code design in information theory in terms of sphere packing and leads to designs that are in some cases known to be provably nearly optimal for small oligonucleotide sizes, whenever the corresponding spherical codes in Euclidean spaces are known to be so. It also leads to upper and lower bounds on estimates of the size of optimal codes of size under 20-mers, as well as to a few infinite families of DNA strand lengths, based on estimates of the kissing (or contact) number for sphere codes in high-dimensional Euclidean spaces. Conversely, the authors show how solutions to DNA codeword design obtained by experimental or other means can also provide solutions to difficult spherical packing geometric problems via these approaches. Finally, the reduction suggests a tool to provide some insight into the approximate structure of the Gibbs energy landscapes, which play a primary role in the design and implementation of biomolecular programs.

An augmented penalty function method with penalty parameter updates for nonconvex optimization

Given an augmented Lagrangian scheme for a general optimization problem, we use an epsilon subgradient step for improving the dual function. This can be seen as an update for an augmented penalty method, which is more stable because it does not force the penalty parameter to tend to infinity. We establish for this update primal–dual convergence for our augmented penalty method. As illustration, we apply our method to the test-bed kissing number problem.

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C 1 and C 2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [ C 1 , C 2 ] . In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in R N with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R 3 . Such spherical configurations come up in connection with the kissing number problem.

Bounds for codes by semidefinite programming

Delsarte’s method and its extensions allow one to consider the upper bound problem for codes in two-point homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that using as variables power sums of distances, this problem can be considered as a finite semidefinite programming problem. This method allows one to improve some linear programming upper bounds. In particular, we obtain new bounds of one-sided kissing numbers.

The kissing number in four dimensions

The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schutte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality fc(4) = 24 is proved. The proof is based on a modification of Delsarte's method.