On the Use of Minimum Penalties in Statistical Learning

Area of Science:

• Statistics
• Machine Learning
• Multivariate Analysis

Background:

• Existing multivariate methods often estimate relationships via covariance matrices, limiting generalizability.
• There is a need for unified frameworks that estimate both regression coefficients and outcome variable associations.

Purpose of the Study:

• To propose the MinPen framework for simultaneous estimation of regression coefficients and inter-response relationships.
• To develop a novel penalty function for detecting and utilizing relationships between responses.
• To provide theoretical guarantees and extensions for the proposed methodology.

Main Methods:

• The MinPen framework employs a novel minimum function-based penalty for joint estimation.
• An iterative algorithm addresses the non-convex optimization problem.
• The framework is extended to exponential family loss functions, including multiple binomial responses.

Main Results:

• Theoretical results include high-dimensional convergence rates and model selection consistency.
• A framework for post-selection inference is established.
• The method demonstrates effective finite sample properties via simulations and data examples.

Conclusions:

• The MinPen framework offers a robust and versatile approach to multivariate analysis.
• It effectively estimates both regression parameters and outcome variable associations.
• The methodology provides a valuable tool for complex statistical modeling and inference.