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Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach

Miles E. Lopes

Abstract

Let $\hatΣ=\frac{1}{n}\sum_{i=1}^n X_i\otimes X_i$ denote the sample covariance operator of centered i.i.d.~observations $X_1,\dots,X_n$ in a real separable Hilbert space, and let $Σ=\mathbb{E}(X_1\otimes X_1)$. The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error $\sqrt n\|\hatΣ-Σ\|_{\text{op}}$, in settings where the eigenvalues of $Σ$ decay as $λ_j(Σ)\asymp j^{-2β}$ for some fixed parameter $β>1/2$. Our main result shows that the bootstrap can approximate the distribution of $\sqrt n\|\hatΣ-Σ\|_{\text{op}}$ at a rate of order $n^{-\frac{β-1/2}{2β+4+ε}}$ with respect to the Kolmogorov metric, for any fixed $ε>0$. In particular, this shows that the bootstrap can achieve near $n^{-1/2}$ rates in the regime of large $β$ -- which substantially improves on previous near $n^{-1/6}$ rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a model that is compatible with both elliptical and Marčenko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.

Improved Rates of Bootstrap Approximation for the Operator Norm: A Coordinate-Free Approach

Abstract

Let denote the sample covariance operator of centered i.i.d.~observations in a real separable Hilbert space, and let . The focus of this paper is to understand how well the bootstrap can approximate the distribution of the operator norm error , in settings where the eigenvalues of decay as for some fixed parameter . Our main result shows that the bootstrap can approximate the distribution of at a rate of order with respect to the Kolmogorov metric, for any fixed . In particular, this shows that the bootstrap can achieve near rates in the regime of large -- which substantially improves on previous near rates in the same regime. In addition to obtaining faster rates, our analysis leverages a fundamentally different perspective based on coordinate-free techniques. Moreover, our result holds in greater generality, and we propose a model that is compatible with both elliptical and Marčenko-Pastur models in high-dimensional Euclidean spaces, which may be of independent interest.
Paper Structure (17 sections, 22 theorems, 151 equations)

This paper contains 17 sections, 22 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Examples of conforming models
  4. Infinite dimensions
  5. High dimensions
  6. Proof of Proposition \ref{['prop:elliptical']}
  7. Proof of Proposition \ref{['prop:MP']}
  8. Outline of the proof of Theorem \ref{['thm:main']}
  9. The term $\text{I}_n$: Dimension reduction
  10. Anti-concentration inequality for $\|G\|_{_{\scaleto{\!\triangle}{4pt}}}$
  11. Approximating $\|\cdot\|_{\textup{op}}$ with $\|\cdot\|_{_{\scaleto{\!\triangle}{4pt}}}$
  12. Upper bound for $\textup{P}(\mathcal{A}(t_2))$
  13. Upper bound for $\textup{P}(\mathcal{B}(t_1))$
  14. The terms $\text{II}_n$ and $\widehat{\text{II}}_n$: Gaussian and bootstrap approximation
  15. The term $\widehat{\text{III}}_n$: Gaussian comparison
  16. ...and 2 more sections

Key Result

Theorem 2.1

Fix any $\epsilon\in (0,1)$, and suppose that Assumption A:model holds. Then, there is a constant $c>0$ not depending on $n$ such that the event holds with probability at least $1-c/n$.

Theorems & Definitions (22)

  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 5.1
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • Proposition 5.2
  • Lemma 5.4: Lopes:EJS Theorem 2.3
  • Lemma 5.5
  • ...and 12 more