神经网络的拓扑结构受架构限制，与权重无关。
On the Topology of Neural Network Superlevel Sets
- 基于微分方程条件的激活函数使输出具可预测拓扑
- 超水平集的贝蒂数受网络架构唯一控制，与权重无关
- 适用于研究模型拓扑稳定性与泛化能力的理论分析
我们证明，满足Riccati型常微分方程条件的激活函数的神经网络，在解析域上产生Pfaffian输出，其形式仅由网络架构决定。因此，超水平集以及神经网络参数化向量场的李括号秩下降点集，其拓扑复杂度（特别是总贝蒂数）仅依赖于架构，且在所有权重下保持统一有界。
原文摘要 · Abstract (English)
We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, an assumption arising in recent universal approximation results in the uniform topology, produce Pfaffian outputs on analytic domains with format controlled only by the architecture. Consequently, superlevel sets, as well as Lie bracket rank drop loci for neural network parameterized vector fields, admit architecture-only bounds on topological complexity, in particular on total Betti numbers, uniformly over all weights.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。