Modern Complex-Valued Hopfield Network for Associative Memory of Continuous and Periodic Patterns

In this study, we propose Modern Complex-Valued Hopfield Network (Modern CVHN), a novel associative memory model designed for continuous data with inherent periodic structure. The model operates on a toroidal state space—constructed as the Cartesian product of complex unit circles—and performs memory encoding and retrieval via a softmax-based energy function that intrinsically incorporates periodicity. Through numerical experiments, we demonstrate that Modern CVHN achieves superior memory capacity and robustness to noise compared to both conventional Complex-Valued Hopfield Networks and Modern Hopfield Network, across discrete phase patterns and continuous periodic data. These findings underscore the effectiveness of energy-based modeling on toroidal manifolds for associative memory involving periodic structures. This approach offers a promising foundation for future applications in complex information processing tasks characterized by periodicity.