High-order rotor Hopfield neural networks for associative memory

Multistate associative memory models have shown a remarkable ability to remember non-binary data in recent years, including the complex-valued Hopfield neural networks (CHNNs) and their advanced counterpart of rotor Hopfield neural networks (RHNNs). However, the noise robustness of these models deteriorates significantly as the number of stored patterns and the resolution increase. To address this issue, inspired by the complex connections observed in biological neural systems, high-order connections are incorporated into CHNNs and RHNNs, resulting in the high-order complex-valued Hopfield neural networks (HCHNNs) and the high-order rotor Hopfield neural networks (HRHNNs). By abstracting virtual neurons, high-order connection-based update equations and projection rules are simultaneously modified as complex versions. The maximum storage capacity of the network is increased from $N$ to nearly $(N+M)$, where $N$ and $M$ represent the number of neurons and the number of high-order connections. The associative memory capabilities of HRHNNs were validated on the CIFAR-10, MNIST, and CelebA datasets, demonstrating superior robustness to noise compared to RHNNs as the number of memory patterns increased.