Universally optimal distribution of points on spheres

By Henry Cohn, Abhinav Kumar. 2006

440 citations recorded by the source corpus.

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m-1)-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E_8 and Leech lattices. We also prove the same result for the vertices of the 600-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/af79753494d2404b16f7b007eaeed11f64132672

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In Numerical Analysis

Citations in the corpus

Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere (2020, methodology): “To this end, we consider the Gaussian potential kernel (also known as the Radial Basis Function (RBF) kernel) Gt : S d × Sd → R+ (Cohn & Kumar, 2007; Borodachov et al., 2019): Gt(u, v) , e −t‖u−v‖2 2 = e2t·u Tv−2t, t > 0, and define the uniformity loss as the logarithm of the average pairwise…”
Minimization of the probabilistic p-frame potential (2010, background): “We refer to [10, 13, 29, 34, 36] for details and related results.”
The sphere packing problem in dimension 24 (2016, background): “ge of discrete optimization problems such as error-correcting codes [7], equal weight quadrature formulas [8], and spherical codes [13, 16]. In exceptional cases linear programming bounds are optimal [5]. However, in general linear programming bounds are not sharp and it is an open question how big ”
SPHERE PACKING BOUNDS VIA SPHERICAL CODES (2012, background): “Bn 1 ) f(0) fb(0) : The original version in [CE03] required suitable decay of f and fbat innity, and it was based on Poisson summation. These more restrictive hypotheses were removed in Section 9 of [CK07]. Here we give a more direct proof, although it has the disadvantage of not telling as much abo”
Learning with Hyperspherical Uniformity (2021, background): “Theorem 3 can be obtained from [11, 17, 38, 87] and it indicates the cross-polytope point configurations are the minimizer of MHE and MHS.”
From the Ginzburg-Landau Model to Vortex Lattice Problems (2010, result): “attice. Positive answers are found in [Rad] for packing problems, [Th] for some very short-range pairwise interaction potential, and one also ﬁnds the same conjecture and some supporting arguments in [CK] Section 9, for a certain (but diﬀerent) class of interaction potentials. But we note however ag”
The sphere packing problem in dimension 8The sphere packing problem in dimension 8 (2016, background): “In exceptional cases linear programming bounds are optimal [5].”
Upper bounds for packings of spheres of several radii (2012, background): “For better orientation we show in the plots the packings where the linear programming bound is sharp (cf. Levenshtein [28]; Cohn and Kumar [9] proved the much stronger statement that these packings provide point configurations which are universally optimal).”
Optimal simplices and codes in projective spaces (2013, background): “Since it has only two inner products between distinct points, and one of those is 0, it is tight [56] and in fact universally optimal [29].”
Jammed hard-particle packings: From Kepler to Bernal and beyond (2010, background): “Cohn and Kumar, 2007b have introduced the beautiful idea of a universally optimal configuration, a unique configuration that minimizes a class of potentials.”
The Crystallization Conjecture: A Review (2015, background): “In a famous article [159], Sarnak and Strömbergsson were able to completely solve the problem in dimensions 4, 8 and 24 (see also [50, 48, 51]).”