Clifford Sheaf Neural Networks

Kotaro Kamiya; Joel Nicholls

cs.LGOct 1, 2026

Clifford Sheaf Neural Networks

Authors: Kotaro Kamiya, Joel Nicholls

Organizations: SyntheticGestalt KK

Abstract

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.

Figures & tables

Figure 1: The cell poset for a single edge, viewed as the base space. The two vertices lie below the edge they bound, and the arrows are the incidence relations along which restriction maps transport features.

Figure 2: Grade coupling of the sandwich family in

\mathrm{Cl}(3,0,0)

. Nodes are the grades

0

–

3

; an arc joins two grades that a map couples. Conjugation is grade-preserving; the reversion sandwich

s_{c}

couples

\{0,1\}

and

\{2,3\}

; the

\gamma

-sandwich couples

\{0,3\}

and

\{1,2\}

; a general or

K

-term sandwich couples all six pairs. Reachable dimensions are listed in Figure 3 .

Figure 3: Inclusion of restriction-map families in

\mathrm{Cl}(3,0,0)

(rotations,

SO(3)

) and

\mathrm{Cl}(3,0,1)

(rigid motions,

SE(3)

). Each edge denotes inclusion (

\subseteq

, lower in higher), and each number is the dimension of that family in

\mathrm{End}_{\mathbb{R}}

. Unnormalized sandwich dimensions are shown, normalizing fixes the overall scale and removes one dimension (reversion

7\!\to\!6

\mathrm{Cl}(3,0,0)

16\!\to\!15

\mathrm{Cl}(3,0,1)

). The braced families are equivariant,

s_{Rc\widetilde{R}}(Rx\widetilde{R})=R\,s_{c}(x)\,\widetilde{R}

. Sandwich sums fill exactly half of the linear maps; in

\mathrm{Cl}(3,0,0)

these are the ones commuting with the central pseudoscalar.

Figure 4: A single CSNN layer. Node multivector features are transported across each edge by normalized reversion sandwiches, combined through the sheaf Laplacian, and used to update node features and positions, gated by scalar invariants.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Symbol	Term	Definition
Clifford algebra
$\mathrm{Cl}(p,q,r)$	Clifford algebra	Signature $(p,q,r)$ on $\mathbb{R}^{n}$ , $n=p+q+r$ : the quotient of the tensor algebra by $v^{2}=Q(v)$ ; real dimension $2^{n}$ .
$\mathrm{Cl}(3,0,0)$	Euclidean case	$n=3$ , dimension $8$ ; symmetry group $SO(3)$ .
$\mathrm{Cl}(3,0,1)$	projective case, PGA	$n=4$ , dimension $16$ ; symmetry group $SE(3)$ ; degenerate scalar product with an $8$ -dimensional radical.
$g_{ij}$	metric	Defined by $e_{i}e_{j}+e_{j}e_{i}=2g_{ij}$ .
$\mathrm{Cl}^{(k)}$	grade- $k$ subspace	Spanned by the $k$ -blades, the products of $k$ orthogonal basis vectors; dimension $\binom{n}{k}$ .

Appendix

Table 5

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O(n)

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