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An Implicitly Restarted Joint Bidiagonalization Algorithm for Large GSVD Computations

Kaixiao Fang, Zhongxiao Jia

TL;DR

This work tackles the computation of extreme GSVD components for large regular matrix pairs $ig\\{A,L\\}$ by developing an implicitly restarted joint bidiagonalization (IRJBD) framework. IRJBD extends implicit restarting to the JBD process, introduces compact residual bounds and a residual-based stopping criterion that avoids expensive right-GSVD computations, and employs exact and adaptive shift strategies to accelerate convergence. Theoretical results include insights into residual bounds in exact and finite precision and a proof that the JBD method cannot produce zero or infinite GSVD components, complemented by extensive numerical experiments showing IRJBD often outperforms thick-restart JBD. The approach offers a scalable, reliable tool for GSVD computations in large-scale applications such as ill-posed problems and high-dimensional data analysis, with potential for further refinements in handling zero/infinite components and in refined extraction strategies.

Abstract

The joint bidiagonalization (JBD) process of a regular matrix pair $\{A,L\}$ is mathematically equivalent to two simultaneous Lanczos bidiagonalization processes of the upper and lower parts of the Q-factor of QR factorization of the stacked matrix $(A^{\mathrm T},\,L^{\mathrm T})^{\mathrm T}$ when their starting vectors are closely related in a specific way. The resulting JBD method for computing extreme generalized singular values and corresponding left and right singular vectors of $\{A,L\}$ realizes the standard Rayleigh--Ritz projection of the generalized singular value decomposition (GSVD) problem of $\{A,L\}$ onto the two left and one right subspaces generated by the JBD process. In this paper, the implicit restarting technique is nontrivially and skillfully extended to the JBD process, and an implicitly restarted JBD (IRJBD) algorithm is developed with proper selection of crucial shifts proposed and a few key implementation details addressed in finite precision arithmetic. Compact upper bounds are established for the residual norm of an approximate GSVD component in both exact and finite precision arithmetic, which are used to design efficient and reliable stopping criteria and avoid the expensive computation of approximate right generalized singular vectors. Numerical experiments illustrate that IRJBD performs well and is more efficient than the thick-restart JBD algorithm.

An Implicitly Restarted Joint Bidiagonalization Algorithm for Large GSVD Computations

TL;DR

This work tackles the computation of extreme GSVD components for large regular matrix pairs by developing an implicitly restarted joint bidiagonalization (IRJBD) framework. IRJBD extends implicit restarting to the JBD process, introduces compact residual bounds and a residual-based stopping criterion that avoids expensive right-GSVD computations, and employs exact and adaptive shift strategies to accelerate convergence. Theoretical results include insights into residual bounds in exact and finite precision and a proof that the JBD method cannot produce zero or infinite GSVD components, complemented by extensive numerical experiments showing IRJBD often outperforms thick-restart JBD. The approach offers a scalable, reliable tool for GSVD computations in large-scale applications such as ill-posed problems and high-dimensional data analysis, with potential for further refinements in handling zero/infinite components and in refined extraction strategies.

Abstract

The joint bidiagonalization (JBD) process of a regular matrix pair is mathematically equivalent to two simultaneous Lanczos bidiagonalization processes of the upper and lower parts of the Q-factor of QR factorization of the stacked matrix when their starting vectors are closely related in a specific way. The resulting JBD method for computing extreme generalized singular values and corresponding left and right singular vectors of realizes the standard Rayleigh--Ritz projection of the generalized singular value decomposition (GSVD) problem of onto the two left and one right subspaces generated by the JBD process. In this paper, the implicit restarting technique is nontrivially and skillfully extended to the JBD process, and an implicitly restarted JBD (IRJBD) algorithm is developed with proper selection of crucial shifts proposed and a few key implementation details addressed in finite precision arithmetic. Compact upper bounds are established for the residual norm of an approximate GSVD component in both exact and finite precision arithmetic, which are used to design efficient and reliable stopping criteria and avoid the expensive computation of approximate right generalized singular vectors. Numerical experiments illustrate that IRJBD performs well and is more efficient than the thick-restart JBD algorithm.

Paper Structure

This paper contains 17 sections, 9 theorems, 80 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lanczos lower and upper bidiagonalizations
  3. The JBD process and the JBD method
  4. The JBD process
  5. The JBD method
  6. New residual, compact bounds for its norm, and a reliable and efficient stopping criterion
  7. Implicit restart of the JBD process
  8. One-step implicit restart of $Q_A$
  9. One-step implicit restart of $Q_L$
  10. Multi-step implicit restart of the JBD process
  11. A comparison with the thick restart JBD algorithm
  12. Selection of shifts
  13. Exact shifts
  14. Adaptive shifting strategy
  15. The IRJBD algorithm and some details
  16. ...and 2 more sections

Key Result

Theorem 3.1

\newlabelTh:left singular vector corresponding to zero and infinity0 Suppose alg:JBD does not break down before step $k$, and let $P_{A,l_1}$ and $P_{L,l_2}$ be defined as in partition.

Figures (1)

  • Figure 1: Results of computing the ten largest and smallest GSVD components of two matrix pairs

Theorems & Definitions (21)

  • Theorem 3.1
  • Proof 1
  • Definition 3.2
  • Theorem 3.3
  • Proof 2
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Proof 3
  • Definition 3.7
  • ...and 11 more