Note on the Equivalence of Orthogonalizing EM and Proximal Gradient Descent
1Department of Statistics, University of Stanford.
Summary
Orthogonalizing EM (OEM) solves penalized regression for tall data. This study reveals OEM is an instance of proximal gradient descent, a key convex optimization method.
Area of Science:
- Statistics
- Optimization
- Machine Learning
Background:
- Penalized regression is crucial for high-dimensional data.
- Existing methods like EM algorithm have limitations for 'tall' datasets (more features than samples).
- Orthogonalizing EM (OEM) was proposed to address these challenges.
Purpose of the Study:
- To analyze the theoretical underpinnings of the orthogonalizing EM (OEM) method.
- To connect OEM to established optimization frameworks.
- To demonstrate the relationship between OEM and proximal gradient descent.
Main Methods:
- Theoretical analysis of the orthogonalizing EM (OEM) algorithm.
- Comparison of OEM steps with the proximal gradient descent algorithm.
- Mathematical derivation to show OEM as a specific case of proximal gradient descent.
Main Results:
- The study demonstrates that orthogonalizing EM (OEM) is mathematically equivalent to a specific implementation of proximal gradient descent.
- This finding re-contextualizes OEM within the broader field of convex optimization.
- The connection provides a new perspective on the convergence properties and efficiency of OEM.
Conclusions:
- Orthogonalizing EM (OEM) is not a novel algorithm but an instance of proximal gradient descent.
- This insight facilitates the application of proximal gradient descent theory to understand and improve OEM.
- The findings enhance the understanding of optimization techniques for penalized regression in high-dimensional settings.
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