Discrete time Hopfield neural network: convergence theorem: perturbation analysis

In this research paper, 𝜺−𝒑𝒆𝒓𝒕𝒖𝒓𝒃𝒂𝒕𝒊𝒐𝒏 of diagonal elements of symmetric synaptic weight matrix, 𝑾̅̅̅ ( with 𝜺>𝟎 ) of Hopfield Associative Memory (HAM) ( resulting in updated synaptic weight matrix 𝑾̂=𝑾̅̅̅+𝜺 𝑰 ) is assumed to ensure that the sufficient condition of convergence theorem is satisfied. It is proved that under such perturbation, stable states of HAMs based on synaptic weight matrices 𝑾̂,𝑾̅̅̅ are same. This result is generalized to prove that if 𝑾̂=𝑾̅̅̅+𝑹̅, ( where 𝑾̅̅̅,𝑹̅ have the same eigenvectors ), the stable states of HAMs based on 𝑾̂,𝑾̅̅̅ are same. It is proved that ( in a well defined sense ), if 𝑾̅̅̅ is positive definite, from the view point of dynamics of HAM the threshold vector can be assumed to be a zero vector. These results are interesting from the viewpoint of dynamics of HAM under practical perturbation models.