Designing various component analysis at will

This paper provides a generic framework of component analysis (CA) methods introducing a new expression for scatter matrices and Gram matrices, called Generalized Pairwise Expression (GPE). This expression is quite compact but highly powerful: The framework includes not only (1) the standard CA methods but also (2) several regularization techniques, (3) weighted extensions, (4) some clustering methods, and (5) their semi-supervised extensions. This paper also presents quite a simple methodology for designing a desired CA method from the proposed framework: Adopting the known GPEs as templates, and generating a new method by combining these templates appropriately.

💡 Research Summary

The paper introduces a unifying mathematical framework for a broad class of component analysis (CA) techniques by defining a Generalized Pairwise Expression (GPE) for scatter and Gram matrices. Traditional CA methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Canonical Correlation Analysis (CCA) are shown to be special cases of GPE when specific choices of pairwise weights and transformation matrices are made.

GPE is expressed as a weighted sum over all data pairs:
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