  We propose a two-stage memory retrieval dynamics for modern Hopfield models,
termed $\mathtt{U\text{-}Hop}$, with enhanced memory capacity. Our key
contribution is a learnable feature map $\Phi$ which transforms the Hopfield
energy function into kernel space. This transformation ensures convergence
between the local minima of energy and the fixed points of retrieval dynamics
within the kernel space. Consequently, the kernel norm induced by $\Phi$ serves
as a novel similarity measure. It utilizes the stored memory patterns as
learning data to enhance memory capacity across all modern Hopfield models.
Specifically, we accomplish this by constructing a separation loss
$\mathcal{L}_\Phi$ that separates the local minima of kernelized energy by
separating stored memory patterns in kernel space. Methodologically,
$\mathtt{U\text{-}Hop}$ memory retrieval process consists of: (Stage I)
minimizing separation loss for a more uniform memory (local minimum)
distribution, followed by (Stage II) standard Hopfield energy minimization for
memory retrieval. This results in a significant reduction of possible
metastable states in the Hopfield energy function, thus enhancing memory
capacity by preventing memory confusion. Empirically, with real-world datasets,
we demonstrate that $\mathtt{U\text{-}Hop}$ outperforms all existing modern
Hopfield models and state-of-the-art similarity measures, achieving substantial
improvements in both associative memory retrieval and deep learning tasks. Code
is available at https://github.com/MAGICS-LAB/UHop ; future updates are on
arXiv:2404.03827
