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Clifford Sheaf Neural Networks

Kotaro Kamiya, Joel Nicholls

arXiv:2610.01322v1cs.LGstat.ML

TL;DR

Equivariant geometric-graph learning needs multivector transport, but versor conjugation preserves grades and imposes a restrictive constraint. The paper introduces CSNN with sandwich restriction maps and a positive-semidefinite sheaf Laplacian without that constraint. It characterizes the family’s grade coupling, reachability, and conditioning, including its half-dimensional reach in Cl(3, 0, 0).

  • Problem

    Conjugation-based restriction maps preserve grades and require versors for a well-behaved sheaf Laplacian, limiting cross-grade transport and complicating learning.

    Source: §1 · p. 1

  • Method

    CSNN places Clifford algebras on sheaf stalks and uses normalized reversion-sandwich restrictions to transport multivectors with a positive-semidefinite Laplacian and no versor constraint.

    Source: §1 · p. 2, §4.3 · p. 8

  • Results

    In Cl(3, 0, 0), the sandwich family spans exactly the pseudoscalar-commuting half of EndR; CSNN’s reversion member confines grades to {0, 1} and has single-ratio conditioning.

    Source: §3.2 · pp. 4–5, §7 · p. 9

  • Takeaways & Limitations

    The characterization makes grade coupling, reachable maps, and conditioning explicit axes for understanding the restriction-map design space.

    Source: §1 · p. 2, §7 · p. 9

    The sheaf-Laplacian kernel removes relational signal when transported configurations agree, constraining the linear core rather than the full model.

    Source: §7 · p. 9

Abstract

from arXiv · show

We introduce the Clifford Sheaf Neural Network (CSNN), an equivariant sheaf neural network for geometric graphs that places a Clifford algebra on each stalk of a cellular sheaf and transports multivector features along edges. The canonical choice of restriction map for sheaves with algebra-valued stalks is algebra homomorphism. Adding the constraint of equivariance, the naive choice becomes versor conjugation. However, versor conjugation is expressively weak, so we drop algebra homomorphism and arrive at the K-term sandwich. The resulting sheaf Laplacian is positive semidefinite by construction, needs no versor constraint, and still mixes grades. Our main contribution characterizes the resulting family of restriction maps along three axes: which grades a map couples, how much of the endomorphism space it reaches, and how well it is conditioned. The K-term sandwich spans half of the endomorphism space, and in Cl(3, 0, 0) it corresponds to the maps that commute with the central pseudoscalar. The number of terms controls expressivity. CSNN is the reversion member, a first-order model by construction and the grade-mixing corner of this family, developed as a sheaf construction for graph-level equivariant regression.

1 Introduction

Geometric-graph learning benefits from equivariant multivector representations and sheaf transport, but algebra-automorphism restrictions make the obvious combination awkward to learn and unable to mix grades. The paper introduces CSNN and characterizes its restriction-map family by grade coupling, reachability, and conditioning.

1 Introduction

  • Clifford algebra and cellular sheaves offer complementary inductive biases for geometric graphs: equivariant multivectors represent multiple grades, while learned transport lets neighbors agree beyond homophily.The sheaf Laplacian generalizes the graph Laplacian and supports message passing beyond homophily.Source: §1 · p. 1
  • Algebra-automorphism restrictions based on conjugation require versors for a well-behaved sheaf Laplacian and preserve grades, limiting learning and cross-grade transport.The versor constraint is awkward to maintain during learning.Source: §1 · p. 1
  • CSNN uses a K-term sandwich family, including a normalized reversion sandwich, to obtain a positive-semidefinite sheaf Laplacian without a versor constraint while transporting information across grades.The restriction maps are applied on both sides of each edge.Source: §1 · p. 2
  • The paper characterizes restriction maps by grade coupling, endomorphism-space reach, and conditioning; CSNN uses the first-order reversion member, which mixes scalar and vector grades but remains confined to them.The paper presents this as a design-space characterization and a sheaf construction for graph-level equivariant regression.Source: §1 · p. 2

2 Background

The background combines Clifford multivectors and cellular sheaves: the former represent geometric features across grades, while the latter define learned transport between incident cells. Their sheaf Laplacian measures disagreement after transport and is positive semidefinite under the stated stalk-form conditions.

2.1 Clifford Algebras

  • In Cl(3, 0, 0), scalar, vector, bivector, and pseudoscalar grades let a node’s scalar feature and 3D position share one multivector representation.The geometric product couples these features equivariantly.Source: §2.1 · p. 2
  • The Clifford scalar product is the scalar part of reversed x times y; it is an inner product in Cl(3, 0, 0) but only positive semidefinite in Cl(3, 0, 1).In the degenerate signature its radical is spanned by blades containing the null generator e0.Source: §2.1 · p. 2

2.2 Cellular Sheaves and the Sheaf Laplacian

  • A cellular sheaf places a stalk on each cell of the graph’s cell poset, with restriction maps on incident vertex–edge pairs.For a single edge, its two vertices lie below the edge they bound, and the arrows indicate incidence relations.Source: §2.2 · p. 2, Figure 1 · p. 3
  • The sheaf uses FinVect as its target, making restrictions arbitrary linear maps between stalks while the network uses multivector structure internally.Choosing Clifford algebras as the target would force algebra automorphisms, which the paper treats as too restrictive.Source: §2.2 · p. 3
  • The sheaf Laplacian L = δ∗δ is positive semidefinite when restriction maps admit adjoints under the stalk form, and its Dirichlet energy measures disagreement after transport.The adjoint is defined with respect to the Clifford scalar product; the energy lets neighbors agree up to learned transport rather than directly.Source: §2.2 · p. 3

3 The Sandwich Restriction-Map Family

The section characterizes sandwich restriction maps as an equivariant family, comparing their grade coupling, reachability, and conditioning. CSNN uses the normalized reversion member of this family.

3.1 Equivariance

  • Sandwich maps admit adjoints, making their sheaf Laplacians positive semidefinite, and the family is Spin-equivariant when its factors transform with the input.Source: §3 · p. 3, §3.1 · p. 4

3.2 Three Axes: Grade Coupling, Reachability, and Conditioning

  • Reversion couples grades {0, 1} and {2, 3}, while a general sandwich couples all six grade pairs in Cl(3, 0, 0).Source: §3.2 · pp. 4–5
  • In Cl(3, 0, 0), every K-term sandwich commutes with the central pseudoscalar, so the family fills exactly half of EndR; Cl(3, 0, 1) reaches 128 of 256 maps.Source: §3.2 · p. 5
  • Within the reachable half, the number of terms strictly controls reach: dimension reaches 32 at K = 4 in Cl(3, 0, 0), and 112 in Cl(3, 0, 1), below its 128 ceiling at K = 8.Source: §3.2 · p. 5
  • In Cl(3, 0, 0), reversion and γ self-terms have condition number κ = ((1 + r)/(1 − r))^2, equal to 1 at r = 0 and growing as r approaches 1.Source: §3.2 · p. 5
  • CSNN is the K = 1 normalized reversion sandwich, confined to {0, 1} and conditioned by a single ratio; from scalar and vector inputs it cannot populate grade 3, unlike general or K-term sandwiches that can express chirality.Source: §3.2 · p. 5

4 The Clifford Sheaf Neural Network Layer

The layer generates equivariant restriction elements from node and edge information, transports multivectors with normalized reversion sandwiches, and updates node features through the sheaf Laplacian.

4.1 Architecture Overview

  • A CSNN layer transports node multivectors across edges with normalized reversion sandwiches, combines them through the sheaf Laplacian, and updates node features and positions using scalar-invariant gates.Source: §4 · p. 6, Figure 4 · p. 7

4.2 Generating the Restriction Element

  • Node features and edge geometry generate restriction elements covariantly, while the layer avoids raw cross-stalk feature differences because features are comparable only after transport.Source: §4.2 · p. 7
  • The restriction-element map ϕ is built from Spin-equivariant multivector neurons, whose grade-wise linear maps and invariant-scalar gates make the map equivariant.Source: §4.2 · p. 7

4.3 The Sheaf Laplacian

  • The sheaf Laplacian is orientation-independent and positive semidefinite by its Gram form, even for general restriction elements that mix grades without a versor constraint.Source: §4.3 · p. 8
  • With identity restrictions the Laplacian reduces to the standard graph Laplacian, while versor elements recover a grade-preserving connection Laplacian of the kind used by Barbero et al. (2022).Source: §4.3 · p. 8

5 First-Order Representational Placement

CSNN’s reversion-based grade confinement makes it a first-order model, placing its representation with scalar–vector architectures rather than higher-order tensor models.

5 First-Order Representational Placement

  • Under rotations, grades 0 and 3 are spin-0 and grades 1 and 2 are spin-1, but CSNN operates only with L = 0 and L = 1 representations.Source: §5 · p. 8
  • CSNN carries no rank-two tensors, placing it with scalar–vector models such as PaiNN rather than higher-order models such as Equiformer.Source: §5 · p. 8
  • This first-order boundary is structural for Cl(3, 0, 0); in Cl(3, 0, 1), reversion preserves β-eigenspaces {0, 1, 4} and {2, 3}, with positions in the latter.Source: §5 · p. 8

6 Related Work

CSNN connects Clifford algebra, sheaf, equivariant, and topological message-passing approaches while differing in its learned, versor-free edge transport and graph-level regression target.

6 Related Work

  • Unlike Clifford networks that fix message passing or constrain transformations to the Clifford group, CSNN learns edge-wise, versor-free restriction maps within a sheaf Laplacian.Source: §6 · p. 8
  • Prior sheaf networks learn real-vector-space restrictions for node-level tasks, whereas CSNN uses Clifford stalks and normalized reversion sandwiches for graph-level regression.Source: §6 · p. 8
  • CSNN shares the scalar–vector representational class of models such as EGNN and PaiNN, while higher-order tensor models use L ≥ 2 features.Source: §6 · pp. 8–9
  • Reading restriction maps as parallel transport connects CSNN to gauge-equivariant networks and to simplicial and topological message passing (Bodnar et al., 2021; Eijkelboom et al., 2023; Liu et al., 2024a).Source: §6 · p. 9

7 Conclusion

CSNN uses normalized reversion sandwiches for a positive-semidefinite sheaf Laplacian, and its restriction-map family has a characterized expressivity ceiling. Its linear core is bounded by configurations in the sheaf-Laplacian kernel, where transported agreement removes relational signal.

  • CSNN’s normalized reversion sandwiches yield a sheaf Laplacian that is positive semidefinite by construction, without a versor constraint.Source: §7 · p. 9
  • K-term sandwiches span exactly half of EndR, and their expressivity grows with the number of terms.Source: §7 · p. 9
  • The reversion member is first-order by construction and occupies the family’s single-ratio-conditioned corner.Source: §7 · p. 9
  • When transported configurations agree in the sheaf-Laplacian kernel, relational signal vanishes, bounding the linear core rather than the model.Source: §7 · p. 9

A Notation and Definitions

The appendix collects notation for Clifford algebras, their involutions and scalar product, and the sheaf and model components used in the paper. It also defines the transport and Laplacian quantities that describe disagreement after transport.

  • Cl(3, 0, 0) is the 8-dimensional Euclidean Clifford algebra with SO(3) symmetry, while Cl(3, 0, 1) is the 16-dimensional projective algebra with SE(3) symmetry.The pseudoscalar is e123 in Cl(3, 0, 0) and the null grade-4 blade in Cl(3, 0, 1).Source: A Notation and Definitions · p. 11
  • Grade involution α, reversion β, and Clifford conjugation γ are the paper’s three named involutions; the scalar product is ⟨x, y⟩ = [ex y]0.The scalar product is positive definite in Cl(3, 0, 0) and positive semidefinite in Cl(3, 0, 1).Source: A Notation and Definitions · p. 11
  • A versor is a geometric product of invertible non-null vectors, and conjugation by a versor preserves grades.The spin group consists of unit even versors.Source: A Notation and Definitions · p. 11
  • The normalized reversion restriction sc(x) and its adjoint s∗c(y) are defined using a restriction element c and q(c).The appendix also gives the reversion self-term’s conditioning quantities for Cl(3, 0, 0).Source: A Notation and Definitions · pp. 11–12
  • A cellular sheaf assigns stalks to cells and restriction maps to incident vertex–edge pairs; its Laplacian is L = δ∗δ.The associated Dirichlet energy measures disagreement between node features after transport.Source: A Notation and Definitions · p. 12
  • CSNN’s MVFFN is an MVN-MLP-style generator block, while its multivector linear maps use shared scalar weights per grade and no bias.The model uses spin-0 and spin-1 irreducibles.Source: A Notation and Definitions · p. 12

B Proofs

The proofs establish equivariance properties of sandwich maps, grade confinement under reversion sandwiches, and a reachability limit excluding the grade involution from finite sandwich sums.

  • Spin invariance of q(c) and reversion’s anti-automorphism property establish the equivariance step for the normalized reversion sandwich.The proof states that summing terms preserves the relevant identity.Source: B Proofs · p. 12
  • Reversion sandwiches preserve the β-eigenspaces, so compositions retain this grade confinement.Source: B Proofs · p. 12
  • In Cl(3, 0, 0), every finite sandwich sum commutes with multiplication by the central pseudoscalar, while a grade-involution sandwich anticommutes with it.Source: B Proofs · p. 12
  • The grade involution α cannot be represented by any finite sandwich sum in either signature considered.Source: B Proofs · p. 13

C Expanding the Sheaf Laplacian in the Restriction Elements

The expansion substitutes normalized reversion transports and their adjoints into the node Laplacian, then expresses edge differences and sums them over neighbors.

D The Multivector-Neuron Block

The Multivector-Neuron Block composes two grade-wise linear maps around a gated nonlinearity, widening before projecting. Its shared per-grade weights and invariant scalar gates preserve Spin equivariance.

D The Multivector-Neuron Block

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