Dense Associative Memory with Epanechnikov Energy

We propose a novel energy function for Dense Associative Memory (DenseAM) networks, the log-sum-ReLU (LSR), inspired by optimal kernel density estimation. Unlike the common log-sum-exponential (LSE) function, LSR is based on the Epanechnikov kernel and enables exact memory retrieval with exponential capacity without requiring exponential separation functions. Moreover, it introduces abundant additional \emph{emergent} local minima while preserving perfect pattern recovery -- a characteristic previously unseen in DenseAM literature. Empirical results show that LSR energy has significantly more local minima (memories) that have comparable log-likelihood to LSE-based models. Analysis of LSR's emergent memories on image datasets reveals a degree of creativity and novelty, hinting at this method's potential for both large-scale memory storage and generative tasks.

@article{hoover2025_2506.10801,
  title={ Dense Associative Memory with Epanechnikov Energy },
  author={ Benjamin Hoover and Zhaoyang Shi and Krishnakumar Balasubramanian and Dmitry Krotov and Parikshit Ram },
  journal={arXiv preprint arXiv:2506.10801},
  year={ 2025 }
}