SLOPE-ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION.

Area of Science:

• Statistics
• Machine Learning
• Econometrics

Background:

• Linear models with high-dimensional data (p > n) present estimation challenges.
• Existing methods like Lasso may struggle with controlling false discoveries.
• Rank-based penalization offers a novel approach to coefficient selection.

Purpose of the Study:

• Introduce Sorted L-One Penalized Estimation (SLOPE) as a new estimator for linear models.
• Develop an efficient algorithm for solving the SLOPE optimization problem.
• Evaluate SLOPE's performance in controlling the false discovery rate (FDR) and its inferential properties.

Main Methods:

• SLOPE is formulated as a convex optimization problem using a sorted L1 norm.
• An efficient algorithm with complexity comparable to Lasso is demonstrated.
• Theoretical guarantees for FDR control are derived, particularly for orthogonal designs.
• Empirical evaluation on simulated and real datasets is conducted.

Main Results:

• SLOPE provably controls the false discovery rate (FDR) at level q under orthogonal designs when using Benjamini-Hochberg (BH) critical values.
• The proposed algorithm is computationally efficient, similar to Lasso.
• Experiments show SLOPE possesses strong inferential properties and substantial power across various designs.
• SLOPE demonstrates superior performance compared to existing methods in simulations and real-world applications.

Conclusions:

• SLOPE provides a powerful and statistically sound method for high-dimensional linear regression.
• It offers guaranteed control over the false discovery rate, a critical metric in modern statistics.
• The method shows promise for broader applications in statistical inference and machine learning.