Classification of $N$ points becomes a simultaneous control problem when viewed through the lens of neural ordinary differential equations (neural ODEs), which represent the time-continuous limit of residual networks. For the narrow model, with one neuron per hidden layer, it has been shown that the task can be achieved using $O(N)$ neurons. In this study, we focus on estimating the number of neurons required for efficient cluster-based classification, particularly in the worst-case scenario where points are independently and uniformly distributed in $[0,1]^d$. Our analysis provides a novel method for quantifying the probability of requiring fewer than $O(N)$ neurons, emphasizing the asymptotic behavior as both $d$ and $N$ increase. Additionally, under the sole assumption that the data are in general position, we propose a new constructive algorithm that simultaneously classifies clusters of $d$ points from any initial configuration, effectively reducing the maximal complexity to $O(N/d)$ neurons.

翻译：当通过神经常微分方程（neural ODEs，即残差网络时间连续极限）的视角审视时，对$N$个点的分类问题转化为一个同步控制问题。对于单隐层每层仅含一个神经元的窄模型，已有研究表明该任务可通过$O(N)$个神经元实现。本研究聚焦于估计高效聚类分类所需神经元数量，特别是针对点集在$[0,1]^d$中独立均匀分布的最坏情形。我们的分析提出了一种量化所需神经元数量低于$O(N)$概率的新方法，重点刻画了当$d$和$N$同时增大时的渐近行为。此外，在仅假设数据处于一般位置的前提下，我们提出了一种新的构造性算法，该算法可对任意初始构型中的$d$个点进行同步聚类分类，有效将最大复杂度降低至$O(N/d)$个神经元。