On the minimax spherical designs
نویسندگان
چکیده
Distributing points on a (possibly high-dimensional) sphere with minimal energy is long-standing problem in and outside the field of mathematics. This paper considers novel function that arises naturally from statistics combinatorial optimization, studies its theoretical properties. Our result solves both exact optimal spherical point configurations certain cases asymptotics under general assumptions. Connections between our results L1-principal component analysis quasi-Monte Carlo methods are also discussed.
منابع مشابه
On Tight Spherical Designs
Let X be a tight t-design of dimension n for one of the open cases t = 5 or t = 7. An investigation of the lattice generated by X using arithmetic theory of quadratic forms allows to exclude infinitely many values for n.
متن کاملExtremal Spherical Designs on S
A spherical t-design is a system of m points on the unit sphere S ⊂ R such that the equal weight cubature rule (|S2|/m) mj=1 f(xj) gives ∫ S2 f(x)dx for all polynomials f of degree at most t. Typically the interest is in finding spherical t-designs with the smallest number of points. Goethals and Seidel proved a lower bound m ≥ t/4 + O(t), which is not achievable for t ≥ 3. Upper bounds of m = ...
متن کاملOn the Riesz Energy of Spherical Designs
We show how polynomial techniques can be applied for obtaining upper and lower bounds on the Riesz energy of spherical designs.
متن کاملNew spherical 4-designs
Hardin, R.H. and N.J.A. Sloane, New spherical 4-designs, Discrete Mathematics 106/107 (1992) 255-264. This paper gives a number of new spherical 4-designs, and presents numerical evidence that spherical 4-designs containing n points in k-dimensional space with k G 8 exist precisely for the following values of n and k: n even and 22 for k = 1; n 2 5 for k = 2; n = 12, 14, >I6 for k=3;n~2Ofork=4;...
متن کاملOn Spherical Designs of Some Harmonic Indices
A finite subset Y on the unit sphere Sn−1 ⊆ Rn is called a spherical design of harmonic index t, if the following condition is satisfied: ∑ x∈Y f(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) of degree t. Also, for a subset T of N = {1, 2, · · · }, a finite subset Y ⊆ Sn−1 is called a spherical design of harmonic index T, if ∑ x∈Y f(x) = 0 is satisfied for all real homo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Random Structures and Algorithms
سال: 2022
ISSN: ['1042-9832', '1098-2418']
DOI: https://doi.org/10.1002/rsa.21087