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Minimax Optimal Estimation of Mean and Covariance Functions with Spectral Regularization

Naveen Gupta, Bharath K Sriperumbudur

Abstract

Estimation of the mean and covariance functions is a fundamental problem in functional data analysis, particularly for discretely observed functional data. In this work, we study a regularization-based framework for estimating the mean and the covariance functions within a reproducing kernel Hilbert space (RKHS) setting. Our approach utilizes a spectral regularization technique under Hölder-type source conditions, allowing for a broad class of regularization schemes and accommodating a wide range of smoothness assumptions on the target functions. Unlike previous works in the literature, the proposed work does not require the target functions to belong to the underlying RKHS. Convergence rates for the proposed estimators are derived, and optimality is established by obtaining matching minimax lower bounds.

Minimax Optimal Estimation of Mean and Covariance Functions with Spectral Regularization

Abstract

Estimation of the mean and covariance functions is a fundamental problem in functional data analysis, particularly for discretely observed functional data. In this work, we study a regularization-based framework for estimating the mean and the covariance functions within a reproducing kernel Hilbert space (RKHS) setting. Our approach utilizes a spectral regularization technique under Hölder-type source conditions, allowing for a broad class of regularization schemes and accommodating a wide range of smoothness assumptions on the target functions. Unlike previous works in the literature, the proposed work does not require the target functions to belong to the underlying RKHS. Convergence rates for the proposed estimators are derived, and optimality is established by obtaining matching minimax lower bounds.
Paper Structure (16 sections, 19 theorems, 213 equations)

This paper contains 16 sections, 19 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Definitions and Notation
  5. Mean Estimation
  6. Covariance Estimation
  7. Proofs
  8. Proof of Proposition \ref{['mean_alternate_estimator']}
  9. Proof of Theorem \ref{['thm:mean']}
  10. Proof of Theorem \ref{['thm:mean-lower']}
  11. Proof of Theorem \ref{['thm:covariance']}
  12. Proof of Theorem \ref{['thm:cov-lower-1']}
  13. Proof of Theorem \ref{['thm:lower_covariance']}
  14. Supplementary Results: Mean Function Estimation
  15. Supplementary Results: Covariance Function Estimation
  16. ...and 1 more sections

Key Result

Proposition 2.1

For a bounded and continuous $k:T\times T\rightarrow \mathbb{R}$, we have where $\hat{A}_{n}: L^2(T) \to L^2(T)$ is given as and $V_{1} = \frac{1}{n}\sum_{i=1}^{n}\frac{1}{m_{i}}\sum_{j=1}^{m_{i}} Y_{ij}k^{\frac{1}{2}}(t_{ij}, \cdot)$. Here $k^\frac{1}{2}(x,\cdot):=\sum_l \sqrt{\lambda_l}\psi_l(x)\psi_l(\cdot),\,x\in T$, with $(\lambda_l,\psi_l)_{l\in\mathbb{N}}$ being the eigenvalue-eigenvector

Theorems & Definitions (38)

  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 28 more