Uniform Memory Retrieval with Larger Capacity for Modern Hopfield Models

We propose a two-stage optimization formulation for the memory retrieval dynamics of modern Hopfield models, termed $\mathtt{U\text{-}Hop}$. Our key contribution is a learnable feature map $\Phi$ which transforms the Hopfield energy function into a 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 uniformed memory (local minimum) distribution, followed by **(Stage II:)** standard Hopfield energy minimization for memory retrieval. This results in significant reduction of possible meta-stable states in the Hopfield energy function, thus preventing memory confusion. Empirically, with real-world datasets, we demonstrate that $\mathtt{U\text{-}Hop}$ outperforms all existing modern Hopfield models and SOTA similarity measures, achieving a substantial margin in both associative memory retrieval and deep learning tasks. Code is available at [GitHub](https://github.com/MAGICS-LAB/UHop); future updates are on [arXiv](https://arxiv.org/abs/2404.03827).