揭示神经网络几何结构，证明多输出模型可识别性条件。
The Alexander-Hirschowitz theorem for neurovarieties
- 基于参数化微分的几何分析，推导神经变体维度。
- 在激活次数满足 $d_i\geq 2n_i-1$ 时保证非缺陷性。
- 适用于多输出架构，对理论研究者有参考价值。
我们研究多项式神经网络对应的神经变体的维数与可识别性。给出一个独立的几何证明：当激活次数满足 $d_i\geq 2n_i-1$ 时，无论输出数量如何，均保证非缺陷性，这一结果此前由有限可识别性得出。证明基于参数化映射的微分直接分析。此外，还考察了该范围外的切线与格拉斯曼-切线障碍，并在相同次数条件下证明多输出架构的全局可识别性。
原文摘要 · Abstract (English)
We study the dimension and identifiability of neurovarieties associated to polynomial neural networks. We give an independent geometric proof that the linear bounds $d_i\geq 2n_i-1$ on the activation degrees imply non defectiveness for any number of outputs, a dimension statement previously obtained from finite identifiability. The proof is based on a direct analysis of the differential of the parameterization. We also investigate secant and Grassmann-secant obstructions outside this range and prove global identifiability for multi-output architectures under the same degree bounds.
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