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Revisiting time-variant complex conjugate matrix equations with their corresponding real field time-variant large-scale linear equations, neural hypercomplex numbers space compressive approximation approach

Jiakuang He, Dongqing Wu

TL;DR

Addresses solving time-variant complex conjugate matrix equations (TVSSCME) for large-scale systems by transforming to a real-field LSLE and introducing a neural space-compressive framework. The approach NHNSCAA compresses the high-dimensional error into a low-dimensional hypercomplex form $E_{C1}(\\tau) \\in \\mathbb{C}^{m\\times n}$ (and $E_{C2}(\\tau) \\in \\mathbb{C}^{m\\times n}$ for the conjugate system), enabling ZND-based dynamics and a Con-CZND1_conj construction. The work establishes the equivalence between the real-field LSLE $W_R(\\tau) X_R(\\tau) = B_R(\\tau)$ and TVSSCME and demonstrates convergence numerically under $\\gamma=10$ with a comparison to the real-field model. The results indicate a viable neural, space-compressive pathway for HD matrix equations with potential impact on deep learning, control, and scientific computing.

Abstract

Large-scale linear equations and high dimension have been hot topics in deep learning, machine learning, control,and scientific computing. Because of special conjugate operation characteristics, time-variant complex conjugate matrix equations need to be transformed into corresponding real field time-variant large-scale linear equations. In this paper, zeroing neural dynamic models based on complex field error (called Con-CZND1) and based on real field error (called Con-CZND2) are proposed for in-depth analysis. Con-CZND1 has fewer elements because of the direct processing of complex matrices. Con-CZND2 needs to be transformed into the real field, with more elements, and its performance is affected by the main diagonal dominance of coefficient matrices. A neural hypercomplex numbers space compressive approximation approach (NHNSCAA) is innovatively proposed. Then Con-CZND1 conj model is constructed. Numerical experiments verify Con-CZND1 conj model effectiveness and highlight NHNSCAA importance.

Revisiting time-variant complex conjugate matrix equations with their corresponding real field time-variant large-scale linear equations, neural hypercomplex numbers space compressive approximation approach

TL;DR

Addresses solving time-variant complex conjugate matrix equations (TVSSCME) for large-scale systems by transforming to a real-field LSLE and introducing a neural space-compressive framework. The approach NHNSCAA compresses the high-dimensional error into a low-dimensional hypercomplex form (and for the conjugate system), enabling ZND-based dynamics and a Con-CZND1_conj construction. The work establishes the equivalence between the real-field LSLE and TVSSCME and demonstrates convergence numerically under with a comparison to the real-field model. The results indicate a viable neural, space-compressive pathway for HD matrix equations with potential impact on deep learning, control, and scientific computing.

Abstract

Large-scale linear equations and high dimension have been hot topics in deep learning, machine learning, control,and scientific computing. Because of special conjugate operation characteristics, time-variant complex conjugate matrix equations need to be transformed into corresponding real field time-variant large-scale linear equations. In this paper, zeroing neural dynamic models based on complex field error (called Con-CZND1) and based on real field error (called Con-CZND2) are proposed for in-depth analysis. Con-CZND1 has fewer elements because of the direct processing of complex matrices. Con-CZND2 needs to be transformed into the real field, with more elements, and its performance is affected by the main diagonal dominance of coefficient matrices. A neural hypercomplex numbers space compressive approximation approach (NHNSCAA) is innovatively proposed. Then Con-CZND1 conj model is constructed. Numerical experiments verify Con-CZND1 conj model effectiveness and highlight NHNSCAA importance.
Paper Structure (15 sections, 4 theorems, 34 equations, 13 figures)

This paper contains 15 sections, 4 theorems, 34 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Neural HN space compressive approximation approach
  4. Numerical Experimentation, Verification and Extension
  5. Con-CZND1_conj model
  6. $\gamma$ - swallowed and Space compressive approximation
  7. Extension
  8. Main code start
  9. MatrixF
  10. MatrixA
  11. MatrixC
  12. ZNDTranLeftHandSide
  13. AFMlinear
  14. ZNDTranRightHandSide
  15. Con_CZND1_Deal

Key Result

Lemma 1

Where $\tau \ge 0$ represents the real-time, $A(\tau)\in \mathbb{C}^{m\times n}$, $B(\tau)\in \mathbb{C}^{s\times t}$, $X(\tau)\in \mathbb{C}^{n\times s}$, are time-variant matrices, the following equation can be obtained:

Figures (13)

  • Figure 1: Recent main solving methods of LSLE.
  • Figure 2: Schematic figure of \ref{['eq.define.complexmatrix']} structure.
  • Figure 3: Judge whether $X^*(\tau)$ and $X_{\mathrm{R}}^*(\tau)$ are equivalent $\left (\lambda(\cdot)(n)\text{ denotes different eigenvalue for } n\in\mathbb{N^*}\right )$.
  • Figure 4: Solution $X_{\mathrm{R}}(\tau)$ computed by Con-CZND2 \ref{['eq.solve.linearerrconcznd2']} model and exact solution $X_{\mathrm{R}}^*(\tau)$ ($\gamma$ equals 10).
  • Figure 5: Schematic of neural HN space compressive approximation approach (NHNSCAA) operation by reverse thinking.
  • ...and 8 more figures

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • proof
  • Definition 2
  • proof
  • Theorem 1
  • proof
  • ...and 1 more