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 \ge 7.6\times 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 (10^{47}) to (10^4); (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 \ge 1.7\times 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.

翻译：运输马尔可夫链蒙特卡洛通过训练归一化流来预处理梅特罗波利斯-黑斯廷斯提议，在复杂后验分布上实现了高经验效率；然而，尚无前期工作为此类采样器提供数值上非平凡且严格的谱间隙界。我们首次建立了此类界。针对香蕉族上的独立梅特罗波利斯-黑斯廷斯采样器，我们在(D=2)（覆盖原始空间）下认证了(γ^\ast = 0.828)，在(D=5)（覆盖经解析解卷的高斯空间，并在所述数值Lipschitz认证下通过网格认证的梯度界）下认证了(γ^\ast \ge 7.6\times 10^{-4})，两者均在95%置信度下严格成立。该框架基于三大支柱：（i）采用缩比约束的谱归一化将流Lipschitz常数从(10^{47})降至(10^4)；（ii）基于覆盖的经验振荡界将平凡的分析界替换为数据依赖的证书；（iii）振荡正则化训练在不影响密度拟合质量的情况下将经验振荡降低60-90%，从而将实用证书扩展至(D=20)（(γ^\ast \ge 1.7\times 10^{-4})）。对四个额外目标（高斯混合、剪切建筑、尼尔漏斗、贝叶斯逻辑回归）的测试识别出三个精确障碍：边界曲率、目标刚度与尾部覆盖失配。仿射-样条对比表明，在相同负对数似然下，更简单的架构产生更紧的证书，颠覆了通常的表达能力层级。