A least-squares framework for Component Analysis.

Area of Science:

• Machine Learning
• Data Science
• Computational Statistics

Background:

• Component Analysis (CA) methods like PCA, LDA, and CCA are vital for feature extraction in modeling and classification.
• Eigen-problem formulations of CA offer closed-form solutions but present analytical and computational challenges, including the small sample size problem.

Purpose of the Study:

• To propose a unified least-squares framework for formulating various Component Analysis methods.
• To demonstrate the benefits of this unified framework, including intuitive interpretation, efficient computation, and overcoming common drawbacks.

Main Methods:

• A novel least-squares weighted kernel reduced rank regression (LS-WKRRR) framework is introduced.
• Existing CA methods (PCA, LDA, CCA, LPP, SC) are reformulated within the LS-WKRRR framework.
• Weighted generalizations and new CA techniques are derived.

Main Results:

• PCA, LDA, CCA, LPP, and SC are shown to be specific instances of LS-WKRRR.
• The LS-WKRRR formulation provides intuitive normalization factors and efficient numerical solutions.
• The framework effectively addresses the small sample size problem and facilitates method extension.

Conclusions:

• The unified least-squares framework offers a cohesive approach to understanding and applying diverse CA methods.
• This formulation enhances computational efficiency, interpretability, and extends the applicability of CA techniques, particularly in data-scarce scenarios.