Gradient-type algorithms for partial singular value decomposition.

Area of Science:

• Numerical analysis
• Linear algebra
• Matrix computations

Background:

• Singular Value Decomposition (SVD) is a fundamental matrix factorization technique.
• Calculating the full SVD can be computationally expensive, especially for large matrices.
• Often, only a few singular values and their corresponding vectors are needed.

Purpose of the Study:

• To develop efficient algorithms for computing a limited number of SVD terms.
• To enable the calculation of singular values corresponding to the largest or smallest terms.
• To provide methods advantageous for large-scale matrix computations.

Main Methods:

• Two novel algorithms based on gradient search and conjugate gradient search are proposed.
• The algorithms compute the SVD term by term.
• Singular values are processed in either decreasing or increasing order.

Main Results:

• The proposed algorithms efficiently compute a few terms of the SVD.
• They are particularly effective for large matrices.
• The methods allow for ordered computation of singular values.

Conclusions:

• The developed gradient and conjugate gradient algorithms offer a simple and efficient way to compute partial SVD.
• These methods are especially beneficial for applications involving large matrices where only a subset of singular values is required.