Can geometric projection onto π-reversible generators recover the swapping algorithm?

Determine whether the geometric projection approach that projects a proposal Markov generator onto the π-reversible space (as in Billera–Holmes–Vogtmann, Dalalyan–Montanari, and Choi–Wolfer) can recover more complicated Markov Chain Monte Carlo algorithms such as the swapping algorithm; specifically, establish whether an appropriate projection or geometric construction within the π-reversible framework yields the transition kernel of the swapping algorithm.

Background

Prior geometric approaches construct π-reversible samplers by projecting proposal generators onto the π-reversible space, recovering algorithms such as Metropolis–Hastings and Barker proposals. These methods, however, inherently produce reversible kernels and have not been shown to reproduce more complex, potentially nontrivial, algorithms.

The paper’s rate-distortion framework succeeds in recovering the swapping algorithm (and other non-reversible schemes), highlighting a gap in the projection-based approach. Clarifying whether the π-reversible geometric projection paradigm can reproduce the swapping algorithm would connect two unifying viewpoints and potentially broaden the applicability of projection-based MCMC design.