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Regularity Properties for Sparse Regression.

Edgar Dobriban1, Jianqing Fan2

  • 1Department of Statistics, Stanford University, dobriban@stanford.edu.

Communications in Mathematics and Statistics
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PubMed
Summary
This summary is machine-generated.

Checking conditions for high-dimensional sparse regression is NP-hard. However, the [Formula: see text] sensitivity condition is more robust and holds in many practical scenarios, offering guidance for data analysis.

Keywords:
computational complexityhigh-dimensional statisticsrestricted eigenvaluesparse regressionℓq sensitivity

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Area of Science:

  • Statistics
  • Machine Learning
  • Computational Complexity

Background:

  • High-dimensional sparse regression relies on theoretical conditions like restricted eigenvalue and compatibility for estimator performance.
  • The practical verifiability of these core theoretical conditions has remained an open and critical question.

Purpose of the Study:

  • To rigorously investigate the computational complexity of checking central conditions in high-dimensional sparse regression theory.
  • To explore alternative conditions that may be more computationally tractable and broadly applicable.

Main Methods:

  • Proving the NP-hardness of checking restricted eigenvalue, compatibility, and [Formula: see text] sensitivity conditions.
  • Analyzing the [Formula: see text] sensitivity condition from an average-case complexity perspective.
  • Demonstrating probabilistic guarantees for the [Formula: see text] sensitivity condition under specific model assumptions.

Main Results:

  • Established that checking key conditions for Lasso and Dantzig selector performance is NP-hard.
  • Showed that the [Formula: see text] sensitivity condition is computationally weaker and more general.
  • Proved that [Formula: see text] sensitivity holds with high probability in well-behaved populations and is robust to data processing.

Conclusions:

  • The computational intractability of verifying core sparse regression conditions raises concerns about their direct application.
  • The [Formula: see text] sensitivity condition offers a more practical and robust alternative, providing valuable insights for analyzing high-dimensional correlated data.