Core Concepts
The nontrivial part of the GSVD of a matrix pair {A, L} is characterized by the singular value expansions of two linear operators induced by {A, L}, and a new iterative method called the generalized Golub-Kahan bidiagonalization (gGKB) is proposed to compute the extreme nontrivial GSVD components.
Abstract
The content provides a new understanding of the generalized singular value decomposition (GSVD) of a matrix pair {A, L} from the viewpoint of singular value expansion (SVE) of linear operators.
Key highlights:
By introducing two linear operators A and L induced by {A, L}, the authors show that the nontrivial part of the GSVD of {A, L} is nothing but the SVEs of A and L.
This result completely characterizes the structure of GSVD for any matrix pair with the same number of columns.
As a direct application, the authors propose a new iterative method called the generalized Golub-Kahan bidiagonalization (gGKB) to compute the extreme nontrivial GSVD components of {A, L}.
The gGKB process is a natural extension of the standard Golub-Kahan bidiagonalization (GKB) for large-scale SVD computation.
Several basic properties of the gGKB process are studied using the GSVD characterization, and preliminary results about the convergence and accuracy of the gGKB GSVD algorithm are provided.
Numerical experiments demonstrate the effectiveness of the proposed gGKB GSVD method.
Stats
The GSVD of {A, L} has the following structure:
The nontrivial generalized singular values are: ∞, ..., ∞ (q1 times), cq1+1/sq1+1, ..., cq1+q2/sq1+q2 (q2 times), 0, ..., 0 (q3 times).
The nontrivial GSVD components are linked by the vector-form relations: Axi = cipA,i, Lxi = sipL,i, siAT pA,i = ciLT pL,i for i = 1, ..., r.
Quotes
"The nontrivial part of the GVSD of {A, L} is nothing but the SVEs of the two linear operators induced by {A, L}."
"The gGKB process is a natural extension of the standard Golub-Kahan bidiagonalization (GKB) for large-scale SVD computation."