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Beyond Sub-Gaussian Measurements: High-Dimensional Structured Estimation with Sub-Exponential Designs.

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This study extends high-dimensional structured estimation to sub-exponential distributions, introducing exponential width for analysis. Results show sample complexity and error depend on exponential width, with implications for Lasso and Group Lasso estimators.

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Area of Science:

  • Statistics
  • Machine Learning
  • High-Dimensional Data Analysis

Background:

  • Existing high-dimensional structured estimation results are limited to sub-Gaussian distributions.
  • Norm-regularized estimators like Lasso are sensitive to design matrix and noise distributions.
  • Current analyses rely on Gaussian width, which is insufficient for sub-exponential settings.

Purpose of the Study:

  • To analyze high-dimensional structured estimation under sub-exponential distributions.
  • To develop new theoretical tools, specifically exponential width, for this setting.
  • To provide non-asymptotic estimation error bounds and sample complexity for norm-regularized estimators.

Main Methods:

  • Introduction of exponential width as a key metric for sub-exponential distributions.
  • Application of generic chaining to bound exponential width by Gaussian width.
  • Utilizing VC-dimension analysis for specific estimators like Lasso and Group Lasso.

Main Results:

  • Sample complexity and estimation error in sub-exponential settings depend on exponential width for any norm.
  • Exponential width is bounded by a constant factor times Gaussian width, enabling Gaussian-based results.
  • Lasso and Group Lasso exhibit sample complexity of the same order as in Gaussian designs.

Conclusions:

  • This work provides the first comprehensive analysis for high-dimensional structured estimation in sub-exponential settings.
  • The findings are applicable to various sub-exponential families, including log-concave and extreme-value distributions.
  • The developed framework offers a more robust understanding of estimator performance beyond sub-Gaussian assumptions.