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<p>Matrix joint block-diagonalization (<span>jbd</span>) frequently arises from diverse applications such as independent component analysis, blind source separation, and common principal component analysis (CPCA), among others. Particularly, CPCA aims at joint diagonalization, i.e., each block size being 1-by-1. This paper is concerned with <i>principal joint block-diagonalization</i> (<span>pjbd</span>), which aims to achieve two goals: 1)&#xa0;partial joint block-diagonalization, and 2)&#xa0;identification of dominant common block-diagonal parts for all involved matrices. This is in contrast to most existing methods, especially the popular ones based on Givens rotation, which focus on full joint diagonalization and quickly become impractical for matrices of even moderate size (300-by-300 or larger). An NPDo approach, directly aiming at dominant common block-diagonal parts, is proposed and it is built on the <i>nonlinear polar decomposition with orthonormal polar factor dependency</i> that characterizes the solutions of the optimization problem designed to achieve <span>pjbd</span>, and it is shown the associated SCF iteration is globally convergent to a stationary point while the objective function increases monotonically during the iterative process. Numerical experiments, including practical applications to multi-view subspace clustering and independent subspace analysis, are presented to illustrate the effectiveness of the NPDo approach and its superiority to Givens rotation-based methods.</p>

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An NPDo approach for principal joint block diagonalization

  • 期刊论文

  • 发表日期：

    2026年4月14日

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  • Ren-Cang Li,
  • Ding Lu,
  • Li Wang,
  • Lei-Hong Zhang

摘要

Matrix joint block-diagonalization (jbd) frequently arises from diverse applications such as independent component analysis, blind source separation, and common principal component analysis (CPCA), among others. Particularly, CPCA aims at joint diagonalization, i.e., each block size being 1-by-1. This paper is concerned with principal joint block-diagonalization (pjbd), which aims to achieve two goals: 1) partial joint block-diagonalization, and 2) identification of dominant common block-diagonal parts for all involved matrices. This is in contrast to most existing methods, especially the popular ones based on Givens rotation, which focus on full joint diagonalization and quickly become impractical for matrices of even moderate size (300-by-300 or larger). An NPDo approach, directly aiming at dominant common block-diagonal parts, is proposed and it is built on the nonlinear polar decomposition with orthonormal polar factor dependency that characterizes the solutions of the optimization problem designed to achieve pjbd, and it is shown the associated SCF iteration is globally convergent to a stationary point while the objective function increases monotonically during the iterative process. Numerical experiments, including practical applications to multi-view subspace clustering and independent subspace analysis, are presented to illustrate the effectiveness of the NPDo approach and its superiority to Givens rotation-based methods.