A semidefinite programming hierarchy for packing problems in discrete geometry

Laat, David de; Vallentin, Frank

doi:10.1007/s10107-014-0843-4

Cited by 27 publications

(51 citation statements)

References 38 publications

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Section: Introductionmentioning

confidence: 99%

“…In particular, there have emerged improvements to the lower bounds for N (18) [24,36] and improvements to the upper bounds for N (d) where d = 14, 16, 17, 18, 19, 20 [1, 2, 14, 15, 16]. There have also been various recent improvements to upper bounds for N (d) for d 24 using semidefinite programming, see [12,21,22,28,38].The asymptotic behaviour of N (d) is quadratic in d with a general upper bound of d(d+1)/2 [23, Theorem 3.5] and a general lower bound of (32d 2 + 328d + 29)/1089 [15, Corollary 2.8]. One can also consider the related problem of, for fixed α ∈ (0, 1), finding N α (d), the maximum number of lines in R d through the origin with pairwise angle arccos α.…”

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confidence: 99%

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Equiangular lines in Euclidean spaces

Greaves

Koolen

Munemasa

et al. 2016

Journal of Combinatorial Theory, Series A

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We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Equiangular lines in Euclidean spaces

Greaves

Koolen

Munemasa

et al. 2016

Journal of Combinatorial Theory, Series A

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“…We show how to approximate the duals E * t by semidefinite programs that are block diagonalized into sufficiently small blocks so that it becomes possible to numerically compute the 4-point bound E 2 for interesting problems. This leads to the best known bounds for these problems, and this demonstrates the computational applicability of the moment techniques developed in [30].…”

Section: Introductionmentioning

confidence: 56%

“…, where G and f are the graph and potential function as defined in Section 2. For packing problems in discrete geometry, the t-th step of the hierarchy from [30] is given by H max…”

Section: Optimization With Infinitely Many Binary Variablesmentioning

confidence: 99%

Moment methods in energy minimization: New bounds for Riesz minimal energy problems

Laat

2019

Trans. Amer. Math. Soc.

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We use moment methods to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We compute the second step of our hierarchy for Riesz s-energy problems with five particles on the 2-dimensional unit sphere, where the s = 1 case known as the Thomson problem. This yields new sharp bounds (up to high precision) and suggests the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for 5-particles on S 2 . This is the first time a 4-point bound has been computed for a continuous problem.We define the t-th step in our hierarchy by optimizing over measures λ ∈ M(I 2t ) satisfying the three properties discussed above.

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“…Such an approach can be used to prove the Bukh-Cox, Welch-Rankin, and Levenstein bounds. Three-point bounds via semidefinite programming would likely produce even better bounds [43,44].…”

Section: Proving Optimality Of a Packingmentioning

confidence: 99%

Game of Sloanes: best known packings in complex projective space

Jasper

King

Mixon

2019

Wavelets and Sparsity XVIII

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It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane's online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table.

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A semidefinite programming hierarchy for packing problems in discrete geometry

Cited by 27 publications

References 38 publications

Equiangular lines in Euclidean spaces

Equiangular lines in Euclidean spaces

Moment methods in energy minimization: New bounds for Riesz minimal energy problems

Game of Sloanes: best known packings in complex projective space

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