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High-dimensional Sobolev tests on hyperspheres

Bruno Ebner, Eduardo García-Portugués, Thomas Verdebout

TL;DR

This work derives the asymptotic null distribution of the entire Sobolev class of tests for uniformity on the hypersphere in a regime where the dimension $d_n$ and sample size $n$ both diverge, and it characterizes the high-dimensional power against integrated von Mises–Fisher local alternatives. Under $\mathcal{H}_{0,n}$, the statistic scaled by $\sigma_n$ converges to $\mathcal{N}(0,1)$ independent of the growth rate, while under $\kappa_n=\tau_n d_n^{3/4}/\sqrt{n}$ contiguous alternatives it converges to $\mathcal{N}(\Gamma\tau^2,1)$ with a weight-dependent $\Gamma$, yielding an explicit asymptotic power $1-\Phi(z_\alpha-\Gamma\tau^2)$. The results are applied to rotational and spherical symmetry testing with supporting numerical experiments, and the paper discusses how to design Sobolev weights to achieve detection in high dimensions, as well as directions for future work on high-dimensional detection thresholds for general rotationally symmetric alternatives.

Abstract

We derive the limit null distribution of the class of Sobolev tests of uniformity on the hypersphere when the dimension and the sample size diverge to infinity at arbitrary rates. The limiting non-null behavior of these tests is obtained for a sequence of integrated von Mises-Fisher local alternatives. The asymptotic results are applied to test for high-dimensional rotational symmetry and spherical symmetry. Numerical experiments illustrate the derived behavior of the uniformity and spherically symmetry tests under the null and under local and fixed alternatives.

High-dimensional Sobolev tests on hyperspheres

TL;DR

This work derives the asymptotic null distribution of the entire Sobolev class of tests for uniformity on the hypersphere in a regime where the dimension and sample size both diverge, and it characterizes the high-dimensional power against integrated von Mises–Fisher local alternatives. Under , the statistic scaled by converges to independent of the growth rate, while under contiguous alternatives it converges to with a weight-dependent , yielding an explicit asymptotic power . The results are applied to rotational and spherical symmetry testing with supporting numerical experiments, and the paper discusses how to design Sobolev weights to achieve detection in high dimensions, as well as directions for future work on high-dimensional detection thresholds for general rotationally symmetric alternatives.

Abstract

We derive the limit null distribution of the class of Sobolev tests of uniformity on the hypersphere when the dimension and the sample size diverge to infinity at arbitrary rates. The limiting non-null behavior of these tests is obtained for a sequence of integrated von Mises-Fisher local alternatives. The asymptotic results are applied to test for high-dimensional rotational symmetry and spherical symmetry. Numerical experiments illustrate the derived behavior of the uniformity and spherically symmetry tests under the null and under local and fixed alternatives.
Paper Structure (13 sections, 5 theorems, 61 equations, 5 figures, 4 tables)

This paper contains 13 sections, 5 theorems, 61 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. General setup
  4. Null behavior
  5. Non-null behavior
  6. Testing applications
  7. Rotational symmetry
  8. Spherical symmetry
  9. Numerical experiments
  10. Discussion
  11. Main proofs
  12. Required results
  13. Numerical experiments illustrating Theorems \ref{['thm:asymp']} and \ref{['thm:LANstuff']}

Key Result

Theorem 2.1

Let $d_n$ be a sequence of positive integers such that $d_n\to\infty$ as $n\to\infty$. Assume that $\boldsymbol{X}_{n,i}$, $i=1,\ldots,n$, $n=1,2,\ldots$ form a triangular array such that, for any fixed $n$, $\boldsymbol{X}_{n,1},\ldots, \boldsymbol{X}_{n,n}$ are mutually independent and uniformly d Then $\sigma_{n}^{-1} T_n\rightsquigarrow\mathcal{N}(0,1)$, with $\rightsquigarrow$ denoting weak c

Figures (5)

  • Figure 1: Empirical evaluation of $\tilde{T}_{n,k_0}\rightsquigarrow \mathcal{N}(0,1)$ under $\mathcal{H}_{0,n}$ for $k_0=1$.
  • Figure 2: Empirical evaluation of $\tilde{T}_{n,k_0}\rightsquigarrow \mathcal{N}(0,1)$ under $\mathcal{H}_{0,n}$ for $k_0=3$.
  • Figure 3: Empirical evaluation of $\tilde{T}_{n,k_0}\rightsquigarrow \mathcal{N}(1,1)$ under $\mathcal{H}_{1,n}$ for $k_0=1$.
  • Figure 4: Empirical evaluation of $\sigma_n^{-1}T_n\rightsquigarrow \mathcal{N}(0,1)$ under $\mathcal{H}_{1,n}$ with $v_{k,d_n}=1_{\{k\leq k_0\}}$ and $k_0=3$.
  • Figure 5: Empirical evaluation of $\sigma_n^{-1}T_n\rightsquigarrow \mathcal{N}(1,1)$ under $\mathcal{H}_{1,n}$ with $v_{k,d_n}=\delta_{k,1}+[k! d_n^{-k}]^{1/4}1_{\{1<k\leq k_0\}}$ and $k_0=3$.

Theorems & Definitions (6)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Corollary 3.1
  • Theorem B.1: Theorem 35.12 in Billingsley2012
  • Lemma B.1