Generalized singular value decomposition
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition. The two versions differ because one version decomposes two (or more) matrices (much like higher order PCA) 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 (GSVD) is a matrix decomposition more general than the singular value decomposition. It is used to study the conditioning and regularization of linear systems with respect to quadratic semi-norms.
Let , or . Given matrices and , their GSVD is given by
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.
The weighted version of the generalized singular value decomposition (GSVD) 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 I is the Identity Matrix and where and are orthonormal given their constraints ( and ). Additionally, and are positive definite matrices (often diagonal matrices of weights). This form of the GSVD is the core of certain techniques, such as generalized principal component analysis and Correspondence analysis.
The GSVD has been successfully applied to signal processing and big data, e.g., in genomic signal processing. These applications also inspired a higher-order GSVD (HO GSVD) and a tensor GSVD.
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