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Folded Transport MCMC: Eliminating Label Switching by Sampling on a Fundamental Domain
[Submitted on 3 Jun 2026 (v1), last revised 18 Jun 2026 (this ve · 2026-06-04 · via stat updates on arXiv.org

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Abstract:In Bayesian mixture models and other exchangeable-component models, the posterior is invariant under permutation of component labels, creating m! equivalent modes-the label-switching problem. Standard MCMC methods either mix poorly across these modes or rely on post-hoc relabelling that cannot guarantee the sampler has converged. We propose Folded Transport MCMC (FolT-MCMC), which eliminates label switching before sampling by restricting the Markov chain to a fundamental domain-a sorted or reflected subspace containing exactly one representative from each symmetric mode. The proposal is a learned normalising flow whose density is symmetrised over the group orbits, ensuring correct targeting on the reduced space. We show that this construction preserves a computable convergence diagnostic based on the oscillation of the log-density ratio, and that the diagnostic becomes sharper on the fundamental domain whenever the original-space flow under-covers one or more symmetric modes. Experiments on Gaussian mixtures (d=2-20), label-switching targets (up to 24 equivalent modes), a standard Bayesian three-component mixture posterior, and real accelerometer data from a supertall building show improvement ratios of 2x to 145x, with the folded diagnostic stable across dimensions while the unfolded diagnostic collapses.

Submission history

From: Jun Hu [view email]
[v1] Wed, 3 Jun 2026 00:26:46 UTC (505 KB)
[v2] Thu, 18 Jun 2026 08:57:45 UTC (973 KB)