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Decentralized Neural Networks for Robust and Scalable Eigenvalue Computation

Ronald Katende

TL;DR

A novel method for eigenvalue computation using a distributed cooperative neural network framework that enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices, surpassing traditional centralized algorithms in large-scale eigenvalue computations.

Abstract

This paper introduces a novel method for eigenvalue computation using a distributed cooperative neural network framework. Unlike traditional techniques that face scalability challenges in large systems, our decentralized algorithm enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices. Each agent employs a localized neural network, refining its estimates through communication with neighboring agents. Our empirical results confirm the algorithm's convergence towards the true eigenvalue, with estimates clustered closely around the true value. Even in the presence of communication delays or network disruptions, the method demonstrates strong robustness and scalability. Theoretical analysis further validates the accuracy and stability of the proposed approach, while empirical tests highlight its efficiency and precision, surpassing traditional centralized algorithms in large-scale eigenvalue computations.

Decentralized Neural Networks for Robust and Scalable Eigenvalue Computation

TL;DR

A novel method for eigenvalue computation using a distributed cooperative neural network framework that enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices, surpassing traditional centralized algorithms in large-scale eigenvalue computations.

Abstract

This paper introduces a novel method for eigenvalue computation using a distributed cooperative neural network framework. Unlike traditional techniques that face scalability challenges in large systems, our decentralized algorithm enables multiple autonomous agents to collaboratively estimate the smallest eigenvalue of large matrices. Each agent employs a localized neural network, refining its estimates through communication with neighboring agents. Our empirical results confirm the algorithm's convergence towards the true eigenvalue, with estimates clustered closely around the true value. Even in the presence of communication delays or network disruptions, the method demonstrates strong robustness and scalability. Theoretical analysis further validates the accuracy and stability of the proposed approach, while empirical tests highlight its efficiency and precision, surpassing traditional centralized algorithms in large-scale eigenvalue computations.
Paper Structure (25 sections, 5 theorems, 37 equations, 2 figures)

This paper contains 25 sections, 5 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Eigenvalue Problems and Matrix Partitioning
  4. Matrix Partitioning
  5. Eigenvalues of Submatrices
  6. Neural Networks for Eigenvalue Approximation
  7. Training Neural Networks for Submatrices
  8. Loss Function for Neural Network Training
  9. Parameter Fine-tuning
  10. Collaborative Eigenvalue Estimation
  11. Update Rule for Submatrix Eigenvalues
  12. Global Eigenvalue Update Rule
  13. Methodology
  14. Neural Network Design
  15. Distributed Framework
  16. ...and 10 more sections

Key Result

Theorem 1

Under assumptions (i)–(iii) in section true above, the sequence of eigenvalue estimates $\{ \lambda_j^{(i, k)} \}$ generated by the algorithm converges exponentially to a consensus value $\lambda_j^{(k)}$ as $k \to \infty$ where $0 < \delta < 1$ depends on the spectral properties of the Laplacian matrix of the communication graph $\mathcal{G}$.

Figures (2)

  • Figure 1: Communication graph
  • Figure 2: Convergence of distributed cooperative Eigenvalue computation

Theorems & Definitions (19)

  • Definition 1: Consensus Error
  • Theorem 1: Convergence to Consensus
  • proof
  • Corollary 1
  • proof
  • Definition 2: Estimation Error
  • Theorem 2: Error Bound
  • proof
  • Theorem 3: Per-Iteration Complexity
  • proof
  • ...and 9 more