Clifford 层状神经网络

中文摘要 AI 辅助

我们引入了 Clifford 层状神经网络（CSNN），这是一种用于几何图的等变层状神经网络，它在细胞层状的每个茎上放置一个 Clifford 代数，并沿边传输多向量特征。对于具有代数值茎的层状，限制映射的规范选择是代数同态。加上等变性的约束，朴素的选择变为 versor 共轭。然而，versor 共轭的表达能力较弱，因此我们放弃代数同态，得到 K 项三明治结构。由此产生的层状拉普拉斯算子通过构造保证半正定，无需 versor 约束，并且仍然混合各阶次。我们的主要贡献是沿着三个轴刻画由此产生的限制映射族：一个映射耦合哪些阶次，它达到自同态空间的多少，以及它的条件数如何。K 项三明治结构覆盖了自同态空间的一半，在 Cl(3, 0, 0) 中，它对应于与中心伪标量交换的映射。项数控制表达能力。CSNN 是反演成员，通过构造是一阶模型，并且是该族中混合阶次的角落，作为图级等变回归的层状构造而开发。

英文摘要

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.

发表机构

• SyntheticGestalt KK（SyntheticGestalt 株式会社）

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