标题： 拓扑感知神经网络中的激活函数 显示英文标题

标题： Topology-Aware Activation Functions in Neural Networks

摘要： 这项研究探讨了新颖的激活函数，这些函数增强了神经网络在训练过程中操作数据拓扑的能力。 在传统激活函数如$\mathrm{ReLU}$的局限性基础上，我们提出了$\mathrm{SmoothSplit}$和$\mathrm{ParametricSplit}$，它们引入了拓扑“切割”能力。 这些函数使网络能够有效地转换复杂的数据流形，在低维层场景中提升性能。 通过在合成和真实数据集上的实验，我们证明了$\mathrm{ParametricSplit}$在低维设置中优于传统激活函数，同时在高维设置中保持竞争力。 我们的研究结果突显了拓扑感知激活函数在推进神经网络架构方面的潜力。 代码可通过 https://github.com/Snopoff/Topology-Aware-Activations 获取。 显示英文摘要

摘要： This study explores novel activation functions that enhance the ability of neural networks to manipulate data topology during training. Building on the limitations of traditional activation functions like $\mathrm{ReLU}$, we propose $\mathrm{SmoothSplit}$ and $\mathrm{ParametricSplit}$, which introduce topology "cutting" capabilities. These functions enable networks to transform complex data manifolds effectively, improving performance in scenarios with low-dimensional layers. Through experiments on synthetic and real-world datasets, we demonstrate that $\mathrm{ParametricSplit}$ outperforms traditional activations in low-dimensional settings while maintaining competitive performance in higher-dimensional ones. Our findings highlight the potential of topology-aware activation functions in advancing neural network architectures. The code is available via https://github.com/Snopoff/Topology-Aware-Activations.