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Fixed-strength spherical designs

Travis Dillon

TL;DR

This work analyzes fixed-strength spherical designs in high dimension, revealing that minimal-size signed designs grow only as $O_t(d^t)$, far smaller than naive degrees-of-freedom expectations. It establishes a tight connection between spherical and Gaussian designs, providing explicit transfer and projection principles that allow constructing smaller designs in one space from the other. The authors also develop unweighted Gaussian designs via $t$-wise independence, obtain optimal signed-design constructions, and present two robust notions of approximate designs with sharp lower and upper bounds, including tensor-approximate designs with near-optimal $\varepsilon^{-2}$ scaling. The results unify algebraic, geometric, probabilistic, and optimization techniques to advance understanding of design sizes in the fixed-strength regime and offer tools for practical construction and analysis of approximate and signed designs.

Abstract

A spherical $t$-design is a finite subset $X$ of the unit sphere such that every polynomial of degree at most $t$ has the same average over $X$ as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as $t \to \infty$, has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where $t$ is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed $t$-designs, which is far less than predicted by a typical degrees-of-freedom heuristic. We also establish a method to ``project'' spherical designs between dimensions, prove a variety of results on approximate designs, and construct new $t$-wise independent subsets of $\{1,2,\dots,q\}^d$ which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.

Fixed-strength spherical designs

TL;DR

This work analyzes fixed-strength spherical designs in high dimension, revealing that minimal-size signed designs grow only as , far smaller than naive degrees-of-freedom expectations. It establishes a tight connection between spherical and Gaussian designs, providing explicit transfer and projection principles that allow constructing smaller designs in one space from the other. The authors also develop unweighted Gaussian designs via -wise independence, obtain optimal signed-design constructions, and present two robust notions of approximate designs with sharp lower and upper bounds, including tensor-approximate designs with near-optimal scaling. The results unify algebraic, geometric, probabilistic, and optimization techniques to advance understanding of design sizes in the fixed-strength regime and offer tools for practical construction and analysis of approximate and signed designs.

Abstract

A spherical -design is a finite subset of the unit sphere such that every polynomial of degree at most has the same average over as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as , has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed -designs, which is far less than predicted by a typical degrees-of-freedom heuristic. We also establish a method to ``project'' spherical designs between dimensions, prove a variety of results on approximate designs, and construct new -wise independent subsets of which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.

Paper Structure

This paper contains 16 sections, 38 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Prior bounds for designs
  3. Upper bounds
  4. Optimal constructions for small t
  5. Converting spherical and gaussian designs
  6. Producing Gaussian designs from spherical, and vice versa
  7. Projecting spherical designs
  8. Smaller unweighted Gaussian designs
  9. Optimal signed designs
  10. Approximate designs
  11. L2-approximate designs
  12. Approximate designs via tensors
  13. Open questions
  14. Appendix
  15. Spherical moments of monomials
  16. ...and 1 more sections

Key Result

Theorem 1.2

For all positive integers $d$ and $t$, there is a number $N(d,t)$ such that for all $n \geq N(d,t)$, there is an unweighted spherical $t$-design in $\mathbb{R}^d$ with exactly $n$ points.

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.2: Seymour, Zaslavsky Seymour-Zaslavsky
  • Theorem 1.3: Delsarte, Goethals, Seidel delsarte-spherical-codes-designs
  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem D
  • Theorem E
  • Proposition 2.1
  • Proposition 2.2: Kane kane-small-designs
  • ...and 34 more