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Cross-Validated Loss-Based Covariance Matrix Estimator Selection in High Dimensions.

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Journal of Computational and Graphical Statistics : a Joint Publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America
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Selecting the best covariance matrix estimator in high dimensions is challenging. This study introduces a cross-validation method to optimally choose estimators, demonstrating its effectiveness in simulations and real-world data analysis.

Keywords:
covariance matrix estimationcross-validationdimension reductionhigh-dimensional statisticsloss-based estimation

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • The sample covariance matrix is optimal in low-dimensional settings but inadequate for high-dimensional data.
  • Numerous alternative covariance matrix estimators exist for high-dimensional scenarios.
  • Selecting the best estimator among available options remains a significant challenge.

Purpose of the Study:

  • To develop a principled method for selecting optimal covariance matrix estimators in high-dimensional settings.
  • To establish the theoretical foundation for cross-validated loss-based estimation in this context.
  • To provide a practical solution for choosing among diverse covariance matrix estimators.

Main Methods:

  • Utilizing the framework of cross-validated loss-based estimation.
  • Proposing a general class of loss functions for covariance matrix estimation.
  • Establishing finite-sample risk bounds and conditions for asymptotic optimality of the cross-validation selector.

Main Results:

  • Demonstrated the optimality of the proposed cross-validation selector in numerical experiments across various data-generating processes.
  • Validated the procedure's effectiveness in moderate sample sizes.
  • Showcased practical benefits in a dimension reduction application using single-cell transcriptome sequencing data.

Conclusions:

  • The proposed cross-validation procedure provides a robust and theoretically sound method for selecting optimal covariance matrix estimators.
  • This approach addresses a critical challenge in high-dimensional statistics, improving the reliability of downstream analyses.
  • The method offers significant practical advantages, particularly in complex biological data applications like single-cell RNA sequencing.