Are groups essential, or can more general transform families replace them?

Determine whether the use of group structures is essential in algorithms that employ them—such as group-based blocking in Gibbs samplers, group-informed data augmentation, or guided Metropolis–Hastings—or whether more general families of transformations could achieve equal or superior performance while maintaining necessary invariance and convergence guarantees.

Background

Several existing algorithms employ group actions for reasons other than encoding target symmetries (e.g., coordinate-free blocking in Gibbs samplers, additional parametrizations in data augmentation, or choosing proposal directions in guided Metropolis–Hastings).

The authors explicitly raise the unresolved question of whether groups are intrinsically necessary in these designs, or if broader classes of transformations could provide the same or improved benefits, motivating a foundational investigation into the role and necessity of groups versus general transform families.