Phase Kernel Lifts Capacity of Dense Associative Memory

Dense associative memory underlies modern attention mechanisms; however, conventional Modern Hopfield Network (MHN) with linear kernels yields overlapping attractor basins and spurious states under high memory load, limiting their theoretically exponential storage capacity in practice. We construct a class of nonlinear phase kernels that reshape the energy dynamics into deep, well-separated basins, and prove that any kernel in this class preserves exponential storage capacity while guaranteeing convergence to fixed points. As a representative instantiation, we introduce the Sigmoidal Phase Hopfield Network (S-Hop), which guarantees monotone energy descent and mitigates the practical loss of memory capacity. Experiments on MNIST and CIFAR-10 show that S-Hop achieves up to a $50\times$ increase in critical capacity over baseline models. We also provide a systematic capacity measurement of Hopfield networks on TinyImageNet, where S-Hop exhibits improved retrieval robustness and noise resilience. The phase-kernel design provides a principled route toward more robust dense associative memory.