Generalized singular value decomposition


In linear algebra, the generalized singular value decomposition is the name of two different techniques based on the singular value decomposition. The two versions differ because one version decomposes two matrices and the other version uses a set of constraints imposed on the left and right singular vectors.

Higher order version

The generalized singular value decomposition is a matrix decomposition more general than the singular value decomposition. It was introduced by Van Loan in 1976 and later developed by Paige and Saunders. The SVD and the GSVD, as well as some other possible generalizations of the SVD
, are extensively used in the study of the conditioning and regularization of linear systems with respect to quadratic semi-norms
Let, or.
Given matrices and, their GSVD is given by
and
where, and are unitary matrices, and is non-singular, where. Also,
is non-negative diagonal, and is non-negative block-diagonal, with diagonal blocks; is not always diagonal. It holds that and, and that. This implies.
The ratios are called the generalized singular values of and. If is square and invertible, then the generalized singular values are the singular values, and and are the matrices of singular vectors, of the matrix. Further, if, then the GSVD reduces to the singular value decomposition, explaining the name.

Weighted version

The weighted version of the generalized singular value decomposition is a constrained matrix decomposition with constraints imposed on the left and right singular vectors of the singular value decomposition. This form of the GSVD is an extension of the SVD as such. Given the SVD of an m×n real or complex matrix M
where
Where I is the identity matrix and where and are orthonormal given their constraints. Additionally, and are positive definite matrices. This form of the GSVD is the core of certain techniques, such as generalized principal component analysis and Correspondence analysis.
The weighted form of the GSVD is called as such because, with the correct selection of weights, it generalizes many techniques

Applications

The GSVD, formulated as a comparative spectral decomposition, has been successfully applied to signal processing and data science, e.g., in genomic signal processing.
These applications inspired several additional comparative spectral decompositions, i.e., the higher-order GSVD
and the tensor GSVD.