Generalized LASSO with under-determined regularization matrices.

Area of Science:

• Statistics
• Machine Learning
• Optimization

Background:

• The Least Absolute Shrinkage and Selection Operator (LASSO) is a widely used penalized regression method.
• Generalized LASSO extends basic LASSO by incorporating a regularization matrix for coefficients.

Purpose of the Study:

• To establish the intrinsic connection between generalized LASSO and basic LASSO.
• To demonstrate how generalized LASSO problems can be reformulated as basic LASSO problems.

Main Methods:

• Utilizing the Lagrangian framework to transform the generalized LASSO.
• Analyzing the conditions under which the transformation is valid (even/under-determined, full rank regularization matrix).

Main Results:

• Demonstrated that generalized LASSO can be converted to basic LASSO under specific conditions.
• Showcased that published LASSO results and solvers are applicable to generalized LASSO.
• Revealed that certain LASSO variants, like robust LASSO, can be expressed in the generalized LASSO form.

Conclusions:

• A fundamental link exists between generalized and basic LASSO formulations.
• This connection facilitates the application of existing LASSO methodologies to generalized LASSO.
• The findings simplify the analysis and computation for generalized LASSO models.