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On constrained and regularized high-dimensional regression.

Xiaotong Shen1, Wei Pan2, Yunzhang Zhu1

  • 1School of Statistics, University of Minnesota, Minneapolis, MN 55455.

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Summary
This summary is machine-generated.

This study introduces a degree-of-separation condition for consistent feature selection in high-dimensional data. Optimal selection methods are presented, enabling precise parameter estimation and prediction.

Keywords:
(p, n) versus fixed p-asymptoticsConstrained regressiondifference convex programmingnonconvex regularizationparameter and nonparametric models

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • High-dimensional feature selection is vital for creating simplified estimation models.
  • Ensuring selection consistency is a key challenge in statistical modeling.

Purpose of the Study:

  • To establish a necessary and sufficient condition for selection consistency.
  • To introduce optimal feature selection methods for high-dimensional data.
  • To demonstrate improved parameter estimation and prediction accuracy.

Main Methods:

  • Derivation of a degree-of-separation condition for selection consistency.
  • Development of constrained L0 and truncated L1 methods for optimal feature selection.
  • Analysis of L0-regularization and truncated L1-regularization methods under stronger assumptions.

Main Results:

  • The minimal degree of separation is essential for selection consistency.
  • Constrained L0 and truncated L1 methods achieve selection consistency with exponentially many features.
  • These methods offer optimality in feature selection compared to existing approaches.
  • Regularization counterparts achieve similar results under slightly stricter conditions.

Conclusions:

  • The proposed methods provide optimal feature selection in high-dimensional settings.
  • Achieving selection consistency leads to enhanced parameter estimation and prediction accuracy.
  • These advancements are critical for effective high-dimensional data analysis.