Theory explaining higher-order denoising advantages

Establish a rigorous theory explaining why the higher-order CHMP pipelines, particularly CHMP + GCW, can outperform pairwise synchronization methods through the denoising and redundancy benefits of higher-order information.

Background

Numerical experiments indicate that CHMP + GCW can outperform several pairwise and standard synchronization methods, even though CHMP may estimate hyperedge corruption levels less accurately than CEMP. The authors attribute this behavior to the denoising effect of refining higher-order data into pairwise measurements and to the redundancy of higher-order information, but explicitly state that no rigorous theory currently supports this explanation.

References

Although we don't have a rigorous theory yet to support this behavior, the numerical results demonstrate the hypothesized advantages of using higher-order information directly in the synchronization process to exploit the increased redundancies and information given in the higher-order estimates.

— Higher-Order Group Synchronization  (2505.21932 - Duncan et al., 28 May 2025) in Section 7, subsection \ref{subsubsec:rotation-synch}, paragraph following Figure \ref{fig:outlier}