动态聚类及其极限微分方程
Dynamical Clustering and its Limiting Differential Equations
中文总结 AI 辅助
本文提出一种基于软分配空间流的非参数动态聚类算法，通过自适应平衡反应与扩散分量，并扩展至半监督分类，在无穷观测极限下导出反应-扩散方程，支持时间序列变化检测与高维特征聚类。
中文摘要 AI 辅助
本文提出了一种新颖的非参数聚类算法，该算法基于软分配空间 $P_k^i$ 中的流，其中 $P_k^i$ 表示点 $x^i$ 属于类别 $k$ 的概率。这些流由两个驱动因素组成：一个非线性反应分量，将 $P$ 松弛为演化的贝叶斯式后验；以及一个扩散分量，将 $P$ 松弛为其局部平均值，其扩散系数 $\ u$ 动态自适应以平衡这两个分量。该方法可扩展到半监督分类，其中部分标签已知。在观测数量趋于无穷的极限情况下，该方法产生一组非标准的反应-扩散方程，这些方程在物种之间产生尖锐的边界，并自发分离成均衡的域。包含多个扩散网络为更广泛的应用开辟了道路，包括时间序列中的状态变化检测以及基于众多特征、克服维数灾难的聚类程序。在连续极限下，该扩展产生了一类新颖的非局部扩散算子。
英文摘要
A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments $P_k^i$, representing the probability that point $x^i$ belongs to class $k$. These flows have two drivers: a nonlinear reaction component that relaxes $P$ to an evolving Bayesian-like posterior, and a diffusive component that relaxes $P$ to its local average, with a diffusivity $ν$ that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.
