Constrained ERM Learning of Canonical Correlation Analysis: A Least Squares Perspective

Area of Science:

• Statistics
• Machine Learning
• Data Analysis

Background:

• Canonical Correlation Analysis (CCA) is vital for identifying latent relationships within multivariate data.
• Theoretical analysis of CCA often involves regularization techniques to ensure analytical consistency.
• Existing methods may not fully address the consistency properties of CCA from a least squares perspective.

Purpose of the Study:

• To investigate the theoretical consistency of Canonical Correlation Analysis (CCA) using a least squares framework.
• To develop and analyze a novel algorithm for solving CCA problems efficiently.
• To extend the proposed method to kernelized versions of CCA for broader applicability.

Main Methods:

• Constructed a constrained empirical risk minimization framework for CCA.
• Applied a two-stage randomized Kaczmarz method: noise removal followed by canonical weight vector computation.
• Provided rigorous theoretical analysis to establish statistical consistency.

Main Results:

• Demonstrated the theoretical consistency of the proposed CCA approach.
• Extended the statistical consistency results to the kernelized CCA.
• Empirical validation on synthetic and real-world datasets confirmed the algorithm's effectiveness and efficiency.

Conclusions:

• The proposed least squares-based framework and two-stage randomized Kaczmarz method offer a consistent approach to CCA.
• The method is effective and efficient for both standard and kernelized CCA.
• This work advances the theoretical understanding and practical application of CCA in multivariate data analysis.