Cardinality-based sparse singular value decomposition for similarity matrices

Area of Science:

• Data Science
• Linear Algebra
• Machine Learning

Background:

• Sparse decomposition methods are crucial in principal component analysis (PCA) for identifying key variables.
• Existing PCA-based methods often rely on regularization parameters or direct cardinality choices for sparse eigenvectors.
• PCA's applicability is limited in certain scenarios, such as analyzing cross-correlation matrices.

Purpose of the Study:

• To extend cardinality-based sparse decomposition from PCA to singular value decomposition (SVD).
• To enable direct control over the sparsity (cardinality) of both left and right singular vectors.
• To enhance SVD for analyzing continuous data matrices, especially cross-correlation matrices.

Main Methods:

• Developed a cardinality-based SVD approach allowing independent specification of left and right singular vector cardinalities.
• Extended the method to support SVD approximations beyond rank-1.
• Enabled the creation of matrices containing sparse left and right singular vectors.

Main Results:

• Successfully generated sparse singular vectors that highlight the most impactful variables in a dataset.
• Demonstrated the ability to control the number of non-zero elements in singular vectors.
• Extended the technique to multi-rank SVD approximations, producing sparse left and right matrices.

Conclusions:

• Cardinality-based SVD offers a flexible and powerful alternative to PCA for sparse decomposition.
• The method effectively identifies significant variables by creating sparse singular vectors.
• This approach broadens the utility of sparse decomposition techniques to a wider range of data analysis problems.