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Dynamics of Structured Complex-Valued Hopfield Neural Networks

Rama Murthy Garimella, Marcos Eduardo Valle, Guilherme Vieira, Anil Rayala, Dileep Munugoti

TL;DR

The paper investigates the dynamics of structured complex-valued Hopfield neural networks, introducing Hermitian, skew-Hermitian, and two braided weight-matrix forms. It offers theoretical results showing $L\le 2$ cycles for Hermitian weights, $L=4$ for skew-Hermitian weights, and $L=8$ for braided Hermitian/skew-Hermitian weights under parallel updates, complemented by extensive computational experiments across nine matrix-structure configurations. The authors also study CvHNNs in polar form, revealing that phase structure drives diverse cycle lengths and can realize Hermitian or skew-Hermitian behavior under specific relations between magnitude and phase. Overall, the work highlights that cycle dynamics in CvHNNs are highly sensitive to weight-matrix structure and sign constraints, with potential implications for designing robust associative memories and memory-augmented computation in complex or hypercomplex domains. These insights pave the way for learning rules and integration with modern architectures, including hypercomplex and attention-based systems, to exploit structured CvHNN memory dynamics.

Abstract

In this paper, we explore the dynamics of structured complex-valued Hopfield neural networks (CvHNNs), which arise when the synaptic weight matrix possesses specific structural properties. We begin by analyzing CvHNNs with a Hermitian synaptic weight matrix and establish the existence of four-cycle dynamics in CvHNNs with skew-Hermitian weight matrices operating synchronously. Furthermore, we introduce two new classes of complex-valued matrices: braided Hermitian and braided skew-Hermitian matrices. We demonstrate that CvHNNs utilizing these matrix types exhibit cycles of length eight when operating in full parallel update mode. Finally, we conduct extensive computational experiments on synchronous CvHNNs, exploring other synaptic weight matrix structures. The findings provide a comprehensive overview of the dynamics of structured CvHNNs, offering insights that may contribute to developing improved associative memory models when integrated with suitable learning rules.

Dynamics of Structured Complex-Valued Hopfield Neural Networks

TL;DR

The paper investigates the dynamics of structured complex-valued Hopfield neural networks, introducing Hermitian, skew-Hermitian, and two braided weight-matrix forms. It offers theoretical results showing cycles for Hermitian weights, for skew-Hermitian weights, and for braided Hermitian/skew-Hermitian weights under parallel updates, complemented by extensive computational experiments across nine matrix-structure configurations. The authors also study CvHNNs in polar form, revealing that phase structure drives diverse cycle lengths and can realize Hermitian or skew-Hermitian behavior under specific relations between magnitude and phase. Overall, the work highlights that cycle dynamics in CvHNNs are highly sensitive to weight-matrix structure and sign constraints, with potential implications for designing robust associative memories and memory-augmented computation in complex or hypercomplex domains. These insights pave the way for learning rules and integration with modern architectures, including hypercomplex and attention-based systems, to exploit structured CvHNN memory dynamics.

Abstract

In this paper, we explore the dynamics of structured complex-valued Hopfield neural networks (CvHNNs), which arise when the synaptic weight matrix possesses specific structural properties. We begin by analyzing CvHNNs with a Hermitian synaptic weight matrix and establish the existence of four-cycle dynamics in CvHNNs with skew-Hermitian weight matrices operating synchronously. Furthermore, we introduce two new classes of complex-valued matrices: braided Hermitian and braided skew-Hermitian matrices. We demonstrate that CvHNNs utilizing these matrix types exhibit cycles of length eight when operating in full parallel update mode. Finally, we conduct extensive computational experiments on synchronous CvHNNs, exploring other synaptic weight matrix structures. The findings provide a comprehensive overview of the dynamics of structured CvHNNs, offering insights that may contribute to developing improved associative memory models when integrated with suitable learning rules.

Paper Structure

This paper contains 17 sections, 3 theorems, 16 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Complex-Valued Hopfield Neural Networks
  4. CvHNNs with Hermitian weight matrix
  5. CvHNN with skew-Hermitian weight matrix
  6. CvHNNs with braided weight matrices
  7. Computational Experiments with Synchronous Complex-valued Hopfield Neural Networks
  8. A is symmetric and B is symmetric
  9. A is symmetric and B is arbitrary
  10. A is antisymmetric and B is antisymmetric
  11. A is antisymmetric and B is arbitrary
  12. A is arbitrary and B is symmetric
  13. A is arbitrary and B is antisymmetric
  14. A is arbitrary and B is arbitrary
  15. Conclusions from the computational experiments with synchronous CvHNNs
  16. ...and 2 more sections

Key Result

Theorem 1

Consider a complex-valued Hopfield neural network $R=(M,T)$ where $M \in \mathbb{C}^{N \times N}$ is Hermitian.

Figures (10)

  • Figure 1: Probability of cycle length in synchronous CvHNNs with a skew-Hermitian weight matrix and either null thresholds ($T=0$) or random arbitrary thresholds ($T$).
  • Figure 2: Probability of cycle length in synchronous CvHNNs with a braided Hermitian and braided skew-Hermitian weight matrix, considering both null thresholds ($T=0$) and random thresholds ($T$ arbitrary).
  • Figure 3: Probability of cycle length in synchronous structured CvHNNs when both matrices A and B are symmetric.
  • Figure 4: Probability of cycle length in synchronous structured CvHNNs when $A$ is symmetric and $B$ is an arbitrary matrix.
  • Figure 5: Probability of cycle length in synchronous structured CvHNNs when both $A$ and $B$ are antisymmetric matrices.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • proof
  • Example 1
  • Definition 1: Braided Hermitian Matrix
  • Definition 2: Braided Skew-Hermitian Matrix
  • Theorem 3
  • Example 2