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CANONICAL THRESHOLDING FOR NON-SPARSE HIGH-DIMENSIONAL LINEAR REGRESSION.

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Summary
This summary is machine-generated.

This study introduces canonical thresholding estimators for high-dimensional linear regression, relaxing sparsity assumptions. These estimators, linked to LASSO and Principal Component Regression (PCR), offer improved performance and a new measure of problem complexity.

Keywords:
62H25High-dimensional linear regressionPrimary 62J05covariance eigenvalues decayprincipal component regressionrelative errorssecondary 62H12thresholding

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • High-dimensional linear regression often assumes sparse coefficients.
  • Existing methods may struggle when coefficients are not sparse.
  • Covariance matrix properties are crucial for understanding data structure.

Purpose of the Study:

  • To develop novel estimators for high-dimensional linear regression without sparsity assumptions.
  • To introduce a new family of estimators based on eigenvalue decay.
  • To analyze the theoretical properties and performance of these estimators.

Main Methods:

  • Proposing canonical thresholding estimators.
  • Analyzing mean squared error and prediction error bounds.
  • Investigating relative errors and introducing joint effective dimension.
  • Establishing minimax lower bounds for optimality.

Main Results:

  • Canonical thresholding estimators are proposed and analyzed.
  • Sufficient conditions for convergence based on eigenvalue decay are identified.
  • A new concept, joint effective dimension, is introduced to characterize problem complexity.
  • Minimax lower bounds demonstrate the optimality of the proposed methods.

Conclusions:

  • The proposed canonical thresholding estimators perform well in high-dimensional linear regression.
  • Eigenvalue decay is a key structural assumption for effective estimation.
  • The joint effective dimension provides a comprehensive measure of regression problem complexity.