# Folded Transport MCMC: Eliminating Label Switching by Sampling on a Fundamental Domain

arXiv: https://arxiv.org/abs/2606.04307 · HTML: https://mlsanity.com/paper/2606.04307/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 22 references that are in the map.
- First posted on arXiv 3 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Self-Certifying Transport MCMC via Dual Spectral-Gap Certificates](https://mlsanity.com/paper/2605.30722/) (2026)
- [Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows](https://mlsanity.com/paper/2606.01078/) (2026)

## Abstract (arXiv)

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.