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

1. Introduction.

• A filter subspace maps through convolution to an operator subspace, whose Plücker coordinates define the projective parametrization.The paper studies this subspace-valued parameter rather than an ordered family of filter vectors.Source: §1 · p. 1
• The proof establishes injectivity of the convolution map, injectivity of the parametrization’s differential, and a closed embedding of the operator sub-Grassmannian.The finiteness, birationality, singleton fibers, and smoothness follow from the closed-embedding result.Source: §1 · p. 2
• For k = 4 and q = 2, Singular recovers the projective image ideal and checks it against the main theorem.The computation is presented as a concrete check of the general proof.Source: §1 · p. 2
• The main theorem establishes that the Grassmann–Plücker parametrization is a closed embedding.The parameter space is isomorphic to its image; the map is finite and birational, every fiber is a singleton, and the image is smooth.Source: §1 · p. 2
• Unlike prior work on polynomial convolutional networks, this paper uses a higher-dimensional filter subspace as its single-layer parameter and records complete operator-subspace Plücker coordinates.The cited prior work studies a projective parametrization factoring through a Segre–Veronese embedding.Source: §1 · p. 2
• The results concern a single-layer linear subspace of convolution operators, not nonlinear activations, multilayer compositions, loss functions, or training performance.The paper explicitly makes no claims about those settings.Source: §1 · p. 2

2.1. Finite-stride convolution.

• Finite-stride convolution induces an injective linear map from the filter space to the operator space.Injectivity is proved for the concrete convolution model, rather than imposed as an extra assumption.Source: §2.1 · pp. 3–4

2.2. Filter-subspace parameters.

• The parameter space is the Grassmannian of q-dimensional filter subspaces, and convolution sends each such subspace to a q-dimensional operator subspace.The image remains q-dimensional because the convolution map is injective.Source: §2.1 · p. 3, §2.2 · p. 4

2.3. The Plücker parametrization.

• The Grassmann–Plücker parametrization applies the Plücker embedding to the convolution-induced operator subspace and defines the model’s projective neural variety.Equivalently, the induced exterior-power map takes the filter subspace’s Plücker coordinates to those of its operator subspace.Source: §2.3 · p. 5

3.1. Standard affine coordinates on a Grassmannian.

• Standard graph charts describe Grassmannians by affine coordinates, giving smoothness and dimension q(n − q); the transitive general linear group action also yields irreducibility.Each chart is an affine space with q(n − q) coordinates.Source: §3.1 · p. 7

3.2. The tangent space.

• At each q-plane U, the Grassmannian tangent space is naturally identified with Hom(U, V/U).This quotient-space description represents first-order changes of the subspace independently of the chosen complement.Source: §3.2 · pp. 7–8

3.3. Plücker coordinates and the classical embedding.

• Plücker coordinates are projective coordinates obtained from the maximal minors of a basis matrix, independent of the chosen basis.Changing basis multiplies all coordinates by the same nonzero determinant.Source: §3.3 · p. 8
• The classical Plücker map is a closed embedding, with its image defined by the quadratic Plücker relations.The paper cites this theorem rather than reproving it.Source: §3.3 · p. 8

4.1. Meaning of regularity.

• Here, “regular parametrization” means the differential has maximal rank everywhere, unlike the broader convention that any algebraic-variety morphism is regular.The authors use this terminology following work on polynomial convolutional networks [11, Theorem 4.5].Source: §4.1 · p. 9

4.2. The convolution-induced Grassmann map is a morphism.

• The convolution-induced map between Grassmannians is a morphism, as its graph-coordinate expression is linear and regular.Composing it with the Plücker morphism gives the projective parametrization.Source: §4.2 · p. 9

4.3. Differential of the convolution-induced Grassmann map.

• The differential sends a tangent map A to CU ◦ A ◦ (C|U)−1, where CU maps quotient classes [v] to [C(v)].Injectivity of CU makes the convolution-induced Grassmann map’s differential injective.Source: §4.3 · pp. 10–11

4.4. Differential of the Plücker map.

• The differential of the Plücker map is injective at each point because the classical Plücker map is a closed immersion.The cited argument uses that a closed immersion is unramified and induces an injective map on tangent spaces.Source: §4.4 · p. 11

4.5. The regularity theorem.

• The differential of the full parametrization Φ is injective at every point.The proof combines injectivity of the Plücker differential with injectivity of the convolution-induced Grassmann map’s differential.Source: §4.5 · p. 12

5.1. The Grassmannian isomorphism induced by a linear isomorphism.

• The linear isomorphism onto C(K) induces an isomorphism of the corresponding Grassmannians, reducing the problem to the inclusion Gr(q, C(K)) into Gr(q, H).The induced map and its inverse are regular.Source: §5.1 · pp. 12–13

5.2. A Plücker-coordinate proof for a sub-Grassmannian.

• The sub-Grassmannian is characterized by vanishing Plücker coordinates whose indices include a basis vector outside W.The resulting locus is closed, and the inclusion identifies Gr(q, W) isomorphically with it.Source: §5.2 · pp. 13–15
• Standard affine Grassmannian charts verify that the inclusion and its inverse on the image are regular, completing the closed-embedding proof.On each chart, the inclusion inserts zero coordinates.Source: §5.2 · p. 15

5.3. The main closed-embedding theorem.

• The convolution-induced map from Gr(q, K) to Gr(q, H) is a closed embedding, by factoring through the Grassmannian isomorphism and the sub-Grassmannian inclusion.The two factors are an isomorphism onto Gr(q, C(K)) and its closed inclusion into Gr(q, H).Source: §5.3 · p. 15
• Composing that map with the Plücker embedding proves that the Grassmann–Plücker parametrization is a closed embedding.The argument composes the closed embedding of the convolution-induced Grassmannian map with the closed Plücker embedding.Source: §5.3 · p. 15

6. Finite Birationality and Geometric Consequences.

• The projective image is isomorphic to the parameter space, so the parametrization is finite and birational and every image point has exactly one preimage.These are consequences of the closed embedding.Source: §6 · p. 16
• The neural variety has the dimension of Gr(q, K) and is smooth.The isomorphism preserves dimension and regularity of local rings.Source: §6 · pp. 16–17

6.1. A concrete nontrivial symbolic computation.

• For k = 4, q = 2, s = 1, d′ = 2, and d = 5, Singular elimination recovers the homogeneous ideal of the projective image.The computation eliminates the six source Plücker coordinates; the reduced Gröbner basis has 40 elements.Source: §6.1 · pp. 17–18
• The concrete computation checks the predicted dimension, full chart rank, and smoothness for this convolutional case.The image is smooth because the affine cone is singular only at its vertex.Source: §6.1 · p. 18
• The computation illustrates the general theorem rather than serving as a computer proof of it.The paper supplies a reproducible script for this low-dimensional symbolic example.Source: §6.1 · p. 18

7.1. Status of the statements in this section.

• The geometric results do not establish higher accuracy, generalization, speed, or robustness; these depend on the network, data, optimization, and implementation and need further validation.Source: §7.1 · p. 18

7.2. Change-of-basis redundancy and intrinsic degrees of freedom.

• The Grassmann parameter removes basis-choice redundancy by identifying GL(q)-related bases, but it recovers the subspace, not the original ordered filters.Source: §7.2 · p. 19

7.3. Low-rank filter families and convolutional compression.

• When filters lie approximately in a q-dimensional subspace, q basic convolution responses can be mixed into m approximate responses, but parameter savings require sufficient rank reduction and overhead accounting.Earlier separable approximations decompose particular convolution tensors; this is a modeling motivation, not an equivalent model or algorithm.Source: §7.3 · pp. 19–20

8.1. Multilayer Grassmann parameter spaces.

• A product of Grassmann spaces can describe formal multilayer parameters, but it does not yet define a deep-network function map because later inputs depend on earlier-layer parameters.A basis-compatible composition must be defined before analyzing the total map; the single-layer theorem cannot simply be applied layer by layer.Source: §8.1 · p. 20
• Grassmann-valued deep models offer possible numerical design techniques, but they represent data or activation subspaces rather than convolution-filter subspaces.The cited approaches use full-rank mappings, reorthogonalization, projection pooling, and manifold backpropagation.Source: §8.1 · p. 20

8.2. Nonlinear activation and change-of-basis equivariance.

• Coordinatewise activation depends on the chosen basis, while a symmetric-tensor alternative makes later layers parameter-dependent and requires rebuilding the total parametrization.Equivariant-network methods offer design directions but do not prove a multilayer extension.Source: §8.2 · p. 21

• The results concern complete Plücker representations of operator subspaces, not full network function classes with arbitrary activations, readouts, and classification layers.A readout without suitable GL(q) invariance or equivariance can depend on the chosen basis.Source: §9 · p. 21
• Injectivity is proved for the full input space and valid convolution; altered boundaries, deleted outputs, restricted inputs, or redundant parameters require re-proving ker C = 0.Source: §9 · p. 21
• The theory is over complex algebraic varieties, so real-manifold topology, numerical stability, and optimization dynamics do not follow directly from the closed-embedding result.Source: §9 · p. 21
• Plücker coordinates may be less economical for large-scale numerical computation than orthonormal basis or projection matrices.Source: §9 · p. 22
• The subspace dimension q is fixed, and the paper provides neither automatic rank selection nor systematic training or data experiments.It therefore does not claim improved accuracy, generalization, computational complexity, or robustness.Source: §9 · p. 22

10. Conclusion.

• The filter-to-convolution-operator map is injective, so each q-dimensional filter subspace maps to a q-dimensional operator subspace.This injectivity is proved for the concrete convolution model, rather than assumed.Source: §10 · p. 22
• The induced Grassmannian map has an injective differential, and the Plücker-coordinate equations show that the Grassmannian inclusion and Φ are closed embeddings.In standard affine coordinates, the inclusion is A ↦ (A, 0).Source: §10 · pp. 22–23
• The closed embedding makes Φ an isomorphism onto its image, finite and birational, with singleton fibers and a smooth projective neural variety.The image has dimension q(k − q).Source: §10 · p. 23
• The results apply to the paper’s single-layer linear operator subspace and complete Plücker representation, not automatically to nonlinear, multilayer, projected, or incompletely observed models.Those settings require new checks of well-definedness, injectivity, differential structure, and fibers.Source: §10 · p. 23
• For k = 4, q = 2, d = 5, d′ = 2, s = 1, Singular recovered the image ideal and confirmed dimension 4, degree 2, full chart differential rank, and smoothness.The ideal has 21 coordinate vanishings, 18 linear identifications, and one Klein quadratic.Source: §10 · p. 23

Declaration on the Use of Generative Artificial Intelligence.

• Generative AI tools assisted with manuscript organization, mathematical exposition, literature-search support, bibliographic preparation, and LaTeX formatting; the authors say they independently verified the final content.The authors assume responsibility for all content.Source: Declaration on the Use of Generative… · p. 23