Grassmann--Pl"ucker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings

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 C:K→ H. This map sends filter subspaces in Gr(q,K) to operator subspaces in Gr(q,H); composing it with the Pl"ucker embedding yields a projective parametrization Phi:Gr(q,K)→mathbbP(bigwedgeq H). Using TUGr(q,K)congHom(U,K/U), we compute the differential of the induced Grassmannian map and show that the differential of Phi is injective at every point. We then use the vanishing equations for Pl"ucker coordinates and standard affine coordinates on a Grassmannian to prove that Gr(q,C(K))hookrightarrowGr(q,H) is a closed embedding, and hence that Phi 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.