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Characterizing Generalized Singular Value Decomposition (GSVD) Using Singular Value Expansion of Linear Operators and Its Computation


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."

Deeper Inquiries

How can the gGKB GSVD algorithm be extended to handle matrix pairs with different numbers of columns

To extend the gGKB GSVD algorithm to handle matrix pairs with different numbers of columns, we can modify the algorithm to accommodate the varying dimensions. When dealing with matrix pairs {A, L} where A ∈Rm×n and L ∈Rp×n, we can adjust the initialization step to account for the different column sizes. Additionally, in the iterative process of the gGKB algorithm, we can introduce checks and conditions to ensure that the operations are compatible with the dimensions of the matrices involved. By dynamically adjusting the computations based on the specific dimensions of the matrix pairs, we can effectively extend the gGKB GSVD algorithm to handle cases where the number of columns in the matrices differ.

What are the potential limitations or drawbacks of the proposed gGKB GSVD approach compared to other GSVD computation methods

While the gGKB GSVD approach offers a novel perspective on computing the GSVD of matrix pairs, there are potential limitations and drawbacks to consider compared to other GSVD computation methods. One limitation is the reliance on the iterative process, which may require more computational resources and time compared to direct computation methods for smaller matrices. Additionally, the gGKB GSVD approach may be more complex to implement and may require a deeper understanding of the underlying mathematical principles, making it potentially less accessible to users without a strong mathematical background. Furthermore, the gGKB GSVD approach may have limitations in handling extremely large matrices due to the iterative nature of the algorithm, which can lead to increased computational complexity and resource requirements.

Can the characterization of GSVD using singular value expansion of linear operators provide insights for developing new GSVD algorithms beyond the gGKB approach

The characterization of GSVD using singular value expansion of linear operators provides valuable insights for developing new GSVD algorithms beyond the gGKB approach. By understanding the structure of the GSVD in terms of the SVE of linear operators, researchers can explore alternative algorithmic approaches that leverage this characterization. This insight can lead to the development of more efficient and accurate GSVD computation methods that exploit the relationships between the matrices in a matrix pair. Additionally, the use of singular value expansion in characterizing GSVD opens up possibilities for exploring novel numerical techniques and optimization strategies that can enhance the computation of GSVD components. By building upon the foundation of SVE for linear operators, researchers can innovate and create advanced GSVD algorithms that offer improved performance and versatility in various applications.
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