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An NPDo Approach for Principal Joint Block Diagonalization

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

TL;DR

The paper tackles principal joint block-diagonalization (pjbd), extracting a dominant common block structure from a set of Hermitian matrices without requiring full joint diagonalization. It introduces NPDo, a nonlinear polar decomposition-based optimization framework on the Stiefel manifold, and proves that its SCF-type iterations converge to stationary points with monotone objective growth, making it scalable to large problems. By formulating $f(P)=\sum_i \omega_i \mathrm{tr}([P_i^{H}A_iP_i]^{s_i})$ and deriving the gradient $\mathscr{H}(P)$, the approach efficiently identifies dominant blocks, with LOCG acceleration further reducing cost for large-scale cases. Empirical results demonstrate that NPDo-based pjbd outperforms traditional Givens-rotation-based full-jbd methods, particularly when only the leading components are of interest, offering a practical tool for high-dimensional block-structure discovery.

Abstract

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 {\em principal joint block-diagonalization\/} (\pjbd), which aim 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 is proposed and it is built on a {\em nonlinear polar decomposition with orthogonal 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 are presented to illustrate the effectiveness of the NPDo approach and its superiority to Givens rotation-based methods.

An NPDo Approach for Principal Joint Block Diagonalization

TL;DR

The paper tackles principal joint block-diagonalization (pjbd), extracting a dominant common block structure from a set of Hermitian matrices without requiring full joint diagonalization. It introduces NPDo, a nonlinear polar decomposition-based optimization framework on the Stiefel manifold, and proves that its SCF-type iterations converge to stationary points with monotone objective growth, making it scalable to large problems. By formulating and deriving the gradient , the approach efficiently identifies dominant blocks, with LOCG acceleration further reducing cost for large-scale cases. Empirical results demonstrate that NPDo-based pjbd outperforms traditional Givens-rotation-based full-jbd methods, particularly when only the leading components are of interest, offering a practical tool for high-dimensional block-structure discovery.

Abstract

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 -by-. This paper is concerned with {\em principal joint block-diagonalization\/} (\pjbd), which aim 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 (-by- or larger). An NPDo approach is proposed and it is built on a {\em nonlinear polar decomposition with orthogonal 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 are presented to illustrate the effectiveness of the NPDo approach and its superiority to Givens rotation-based methods.
Paper Structure (6 sections, 3 theorems, 25 equations, 2 algorithms)

This paper contains 6 sections, 3 theorems, 25 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. NPDo Approach for the Optimization Problem
  3. NPDo Approach
  4. An Algorithm for NPDo
  5. Acceleration with LOCG
  6. Principal Joint Block Diagonalization

Key Result

Lemma 2.1

Let $A\in\mathbb{C}^{n\times n}$ be Hermitian positive semidefinite, $s\ge 1$ an integer, and $Z\in\mathbb{C}^{n\times k}$ a variable. It holds thatThe original result was stated for real matrix variables $Z$, but in the complex case it can be understood in the same way as upon perturbing $Z$ to $Z+ and for all $Z,\,\widehat{Z}\in\mathbb{C}^{n\times m}$, we have

Theorems & Definitions (4)

  • Lemma 2.1: li:2024
  • Theorem 2.1: li:2024
  • proof
  • Theorem 2.2: li:2024