Modern Hopfield networks (MHNs) have recently gained significant attention in the field of artificial intelligence because they can store and retrieve a large set of patterns with an exponentially large memory capacity. A MHN is generally a dynamical system defined with Lagrangians of memory and feature neurons, where memories associated with in-distribution (ID) samples are represented by attractors in the feature space. One major problem in existing MHNs lies in managing out-of-distribution (OOD) samples because it was originally assumed that all samples are ID samples. To address this, we propose the rectified Lagrangian (RegLag), a new Lagrangian for memory neurons that explicitly incorporates an attractor for OOD samples in the dynamical system of MHNs. RecLag creates a trivial point attractor for any interaction matrix, enabling OOD detection by identifying samples that fall into this attractor as OOD. The interaction matrix is optimized so that the probability densities can be estimated to identify ID/OOD. We demonstrate the effectiveness of RecLag-based MHNs compared to energy-based OOD detection methods, including those using state-of-the-art Hopfield energies, across nine image datasets.

本研究针对现代霍普菲尔德网络在处理外部分布样本时的不足，提出了一种名为修正拉格朗日（RegLag）的方法，通过在动态系统中显式引入外部分布样本的吸引子来优化记忆神经元的拉格朗日。研究表明，该方法在识别外部分布样本方面具有明显的效果，相较于基于能量的外部分布检测方法，能够显著提高准确性。

现代霍普菲尔德网络中的外部分布检测的修正拉格朗日方法