Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings
Abstract
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 Plücker embedding yields a projective parametrization Φ:Gr(q,𝒦)→ℙ(bigwedgeq H). Using TUGr(q,𝒦)≅Hom(U,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 Gr(q,𝒞(𝒦))↪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.
Related
BibTeX
@misc{yuan2026grassmannplücker,
title = {{Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings}},
author = {Hongyu Yuan and Huaiqing Zuo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03361},
eprint = {2609.03361},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03361},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Grassmann--Plücker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings
AU - Hongyu Yuan
AU - Huaiqing Zuo
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03361
UR - https://arxiv.org/abs/2609.03361
AB - 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 Plücker embedding yields a projective parametrization Φ:Gr(q,𝒦)→ℙ(bigwedge^q H). Using T_UGr(q,𝒦)≅Hom(U,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 Gr(q,𝒞(𝒦))↪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.
ER -