Minimal Isometric Embeddings of Graphs into Cayley Graphs of Finite Abelian Groups

What the math gives to ML

The paper gives a constructive way to turn graph edge classes into coordinates in a finite abelian group while enforcing every cycle-consistency constraint exactly. Its key transferable asset is the quotient construction: arbitrary integer edge-class increments are factored by the lattice generated by signed cycle incidences, and Smith normal form exposes the resulting cyclic factors and compact coordinates. This can provide graph neural networks with exact relative positional encodings and finite-group equivariant message passing, avoiding heuristic Laplacian eigenvectors or unbounded shortest-path embeddings. The first practical adaptation should use a selected edge partition, compute the quotient once per graph or graph family, and inject the resulting group differences into attention biases or relation-specific message functions.