Extension of CHMP to general hypergraphs

Develop an extension of the Cycle-Hyperedge Message Passing (CHMP) algorithm that handles nonuniform hypergraphs and arbitrary cycle orders under the sufficient cycle-coverage conditions of Theorem \ref{thm:generalsynch}.

Background

The implemented CHMP framework assumes an n-uniform hypergraph and uses a selected family of (n−1)-cycles. Theorem \ref{thm:generalsynch} establishes that, for more general hypergraphs and cycle families, suitable cycle consistency measurements contain enough information to recover hyperedge corruption levels exactly. The paper identifies the algorithmic extension required to exploit this result but does not provide its construction.

References

This suggests that Algorithm \ref{alg:CHMP} could be extended to a general hypergraph setting since the set {d_C}{h \in N_C}, for C belonging to a set of cycles satisfying the assumptions of the proposition, contains enough information to determine {s{h}*}_{h \in H}. But we leave the details of this extension for future work.

— Higher-Order Group Synchronization  (2505.21932 - Duncan et al., 28 May 2025) in Section 3.3, final paragraph of Section \ref{subsec:general}