Metropolis algorithms are classical tools for sampling from target distributions, with broad applications in statistics and scientific computing. Their convergence speed is governed by the spectral gap of the associated Markov operator. Recently, Andrieu et al. (2024) derived the first explicit bounds for the spectral gap of Random-Walk Metropolis when the target distribution is smooth and strongly log-concave. However, existing literature rarely discuss non-smooth targets. In this work, we derive explicit spectral gap bounds for the Random-Walk Metropolis and Metropolis-adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincare or log-Sobolev inequality, beyond the strongly log-concave setting. Our theoretical results are further supported by numerical experiments.

翻译：Metropolis算法是从目标分布中采样的经典工具，在统计学和科学计算领域具有广泛应用。其收敛速度由相关马尔可夫算子的谱隙决定。最近，Andrieu等人（2024）在目标分布光滑且强对数凹的条件下，首次推导出随机游走Metropolis算法谱隙的显式界。然而，现有文献很少讨论非光滑目标分布。本工作中，我们针对一大类非光滑分布，推导出随机游走Metropolis算法和Metropolis调整Langevin算法的显式谱隙界。此外，通过将我们的分析与Goyal等人（2025）的最新结果相结合，我们将这些界推广到满足Poincaré不等式或对数Sobolev不等式的目标分布，突破了强对数凹条件的限制。数值实验进一步验证了我们的理论结果。