Hopfield Neural Networks: A updated Approach for Using Associative Memory to Improve Matrices

Abstract

Modern Hopfield Neural Networks (MHNNs) are a class of neural networks renowned for their associative memory capabilities, which have broad applications in pattern recognition, optimization, and error correction. This dissertation explores the mathematical foundations, architectural nuances, and practical applications of MHNNs, focusing on the development of a novel energy function using the Concave-Convex Procedure (CCCP). The updated energy function enhances the network's convergence properties and robustness, addressing limitations of classical models such as low storage capacity and susceptibility to local minima.We demonstrate the efficacy of the update energy function through rigorous theoretical analysis and extensive simulation studies. Synthetic datasets with various distributional properties are generated to evaluate the network's performance in classification tasks. Our results indicate that the proposed MHNN model outperforms Original Hopfield networks and competes effectively with contemporary machine learning algorithms, including Support Vector Machines, Decision Tree, Random Forests, and Convolutional Neural Networks.In classifc Hopfield nerual network practical applications, we focus on image restoration tasks, successfully reconstructing highly corrupted images, and achieving high accuracy and low restoration error. This showcases the network's potential for real-world applications in fields such as bioinformatics, natural language processing, and healthcare diagnostics.This work not only underscores the capabilities of MHNNs in associative memory and optimization tasks but also paves the way for future innovations. By integrating advanced mathematical techniques and exploring hybrid approaches, this dissertation contributes significantly to the field of neural networks and machine learning, providing a robust framework for future research and application development.