标题： $α$-Cap 过程：二进制神经元随机几何网络的连续模型 显示英文标题

标题： The $α$-Cap Process: A Continuous Model for Random Geometric Networks of Binary Neurons

摘要： 递归的二进制神经元网络是人工智能中的一个基础概念。 虽然这些网络传统上被假设为完全连接的，但当图结构不同时，复杂的动态行为可能会出现。 特别感兴趣的一种图结构是几何随机图，它模拟了生物神经网络中存在的空间依赖性。 在这样的图类中，全局状态依赖性往往会使得分析变得复杂，从而促使研究它们在连续极限下的动态行为。 在本工作中，我们提出并分析了一个连续模型，用于描述二进制神经元状态在$\mathbb{R}^d$中的演化，该模型通过一个函数$\psi:\mathbb{R}^d\to[0,1]$来编码某一点的神经活动。 我们的分析涵盖了由卷积和锐化定义的一类过程；我们证明，当通过这种过程演化时，$\psi$的等高集渐近收敛到$\mathbb{R}^d$中的球体。 值得注意的是，这一过程的一个特例是通过平均曲率运动界面的 Merriman-Bence-Osher (MBO) 方案[MBO92]，我们提供了对其行为的新分析。 我们的结果建立了一种几何随机图与经典的界面运动模型之间的意外联系，为空间结构与神经动力学之间的相互作用提供了新的见解。 显示英文摘要

摘要： Recurrent networks of binary neurons are a foundational concept in artificial intelligence. While these networks are traditionally assumed to be fully connected, complex dynamics can emerge when the graph structure is varied. One graph structure of particular interest is the geometric random graph, which models the spatial dependencies present in biological neural networks. In such classes of graphs, global state dependencies tend to complicate analysis, motivating the study of their dynamics in the continuum limit. In this work, we propose and analyze a continuous model for the evolution of binary neuron states in $\mathbb{R}^d$ via a function $\psi:\mathbb{R}^d\to[0,1]$ encoding the neural activity at a point. Our analysis encompasses a class of processes defined by convolution and sharpening; we demonstrate that, when evolved this process, the level sets of $\psi$ asymptotically converge to balls in $\mathbb{R}^d$. Notably, a special case of this process is the Merriman-Bence-Osher (MBO) scheme for the motion of interfaces by mean curvature[MBO92], and we provide a novel analysis of its behavior. Our results establish a surprising connection between geometric random graphs and classical models of interface motion, offering new insights into the interplay between spatial structure and neural dynamics.