arXiv:2608.30894 2026 Architecture 1 ideas extracted · analyzed Sep 1, 2026
The paper turns approximation of an arbitrary graph by an abelian Cayley graph into a concrete labeling-and-generator problem. The transferable asset is the inverse-pair decomposition: after labeling vertices by an abelian group, edges are partitioned into generator classes, each represented by a shared translation operator, yielding a structured sparse message-passing basis with strong parameter sharing. The degree theorem also supplies a cheap lower bound on the number of edges any fixed-degree Cayley backbone must add, which can be used to select the backbone degree and diagnose when a Cayley approximation is intrinsically poor. The most promising neural use is a hybrid graph layer consisting of a learned Cayley convolution plus a residual on the few edges not explained by that convolution.
Approximate a graph's adjacency by a learned abelian Cayley host and use one shared message-passing operator for every edge in the same inverse-pair generator class. Keep only the unexplained original edges as a residual branch, so the layer interpolates between a parameter-efficient group convolution and ordinary graph message passing.