# Non-Vacuous Certification of Transport MCMC via Oscillation-Controlled Normalizing Flows

arXiv: https://arxiv.org/abs/2606.01078 · HTML: https://mlsanity.com/paper/2606.01078/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 33 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 13 of 13)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [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)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Rank-normalization, folding, and localization: An improved widehatR for assessing convergence of MCMC](https://mlsanity.com/paper/1903.08008/) (2019)
- [NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport](https://mlsanity.com/paper/1903.03704/) (2019)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)

## Built on by (most cited first, 1 of 1)

- [Folded Transport MCMC: Eliminating Label Switching by Sampling on a Fundamental Domain](https://mlsanity.com/paper/2606.04307/) (2026)

## Abstract (arXiv)

Transport MCMC trains a normalizing flow to precondition Metropolis--Hastings proposals, achieving high empirical efficiency on challenging posteriors; yet no prior work produces a numerically non-vacuous, rigorous spectral-gap bound for such samplers. We establish the first such bounds. For independence MH on the banana family we certify (γ^ast = 0.828) at (D = 2) (covering in the original space) and (γ^ast ≥ 7.6× 10-4) at (D = 5) (covering in an analytically unwarped Gaussian space with a grid-certified gradient bound under the stated numerical Lipschitz certification), both rigorous at 95% confidence. The framework rests on three pillars: (i) spectral normalization with reduced scale clips constrains the flow Lipschitz constant from (1047) to (104); (ii) a coverage-based empirical oscillation bound replaces the vacuous analytical bound with a data-dependent certificate; and (iii) oscillation-regularised training cuts the empirical oscillation by 60--90% at no cost to density fit, extending practical certificates through (D = 20) ((γ^ast ≥ 1.7× 10-4)). Tests on four further targets (Gaussian mixture, shear-building, Neal's funnel, Bayesian logistic regression) identify three precise barriers: boundary curvature, target stiffness, and tail-coverage mismatch. An affine-vs-spline comparison shows that simpler architectures yield tighter certificates at identical NLL, inverting the usual expressiveness hierarchy.