发表机构
Zhili College, Tsinghua University; Department of Mathematical Sciences, Tsinghua University（清华大学志道书院; 清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出卷积滤波器子空间的格拉斯曼-普吕克参数化，证明其为闭嵌入，参数化有限双有理且纤维单元素，投影神经簇光滑，还通过Singular验证特例并探讨与滤波器冗余等的联系。
AI 中文摘要
我们提出了一种对单个卷积层中滤波器的几何参数化方式：参数不再是有序的滤波器向量族，而是滤波器空间中固定维度的子空间。对于一维有限步长卷积，滤波器到卷积算子的对应关系给出了一个单射线性映射$\u2102:\u2124\to H$，该映射将$\u2124$中的滤波器子空间$\text{Gr}(q,\u2124)$映射到$H$中的算子子空间$\text{Gr}(q,H)$；将其与普吕克嵌入复合，得到投影参数化$\u03a6:\text{Gr}(q,\u2124)\to\mathbb{P}(\bigwedge^q H)$。利用$T_U\text{Gr}(q,\u2124)\cong\text{Hom}(U,\u2124/U)$，我们计算了诱导格拉斯曼映射的微分，并证明$\u03a6$的微分在每一点都是单射的。随后，我们利用普吕克坐标的消失方程和格拉斯曼上的标准仿射坐标，证明$\text{Gr}(q,\u2102(\u2124))\hookrightarrow\text{Gr}(q,H)$是一个闭嵌入，进而$\u03a6$也是闭嵌入。因此，参数空间与其投影像同构，该参数化是有限的且双有理等价于其像，每个纤维都是单元素集，所得的投影神经簇是光滑的。对于$k=4$和$q=2$，我们还使用Singular软件恢复像理想，并检查其维数、次数、图表秩和光滑性，该计算用于说明而非替代一般证明。最后，我们讨论了其与滤波器冗余和低秩卷积的可能联系，同时区分已证明的几何结果与需要数值验证的应用方案。
英文摘要
We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subspace of the filter space. For one-dimensional finite-stride convolution, the filter-to-convolution-operator correspondence gives an injective linear map $\mathcal{C}:\mathcal{K}\to H$. This map sends filter subspaces in $\mathrm{Gr}(q,\mathcal{K})$ to operator subspaces in $\mathrm{Gr}(q,H)$; composing it with the Plücker embedding yields a projective parametrization $Φ:\mathrm{Gr}(q,\mathcal{K})\to\mathbb{P}(\bigwedge^q H)$. Using $T_U\mathrm{Gr}(q,\mathcal{K})\cong\mathrm{Hom}(U,\mathcal{K}/U)$, we compute the differential of the induced Grassmannian map and show that the differential of $Φ$ is injective at every point. We then use the vanishing equations for Plücker coordinates and standard affine coordinates on a Grassmannian to prove that $\mathrm{Gr}(q,\mathcal{C}(\mathcal{K}))\hookrightarrow\mathrm{Gr}(q,H)$ is a closed embedding, and hence that $Φ$ is a closed embedding. Consequently, the parameter space is isomorphic to its projective image, the parametrization is finite and birational onto its image, every fiber is a singleton, and the resulting projective neural variety is smooth. For $k=4$ and $q=2$, we also use Singular to recover the image ideal and check its dimension, degree, chart rank, and smoothness. This computation illustrates, rather than replaces, the general proof. Finally, we discuss possible connections with filter redundancy and low-rank convolution, while distinguishing the proved geometric results from application proposals requiring numerical validation.
Comments24 pages; includes a symbolic computational example using Singular
