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 discusses 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算法是目标分布抽样的经典工具，在统计学和科学计算领域具有广泛应用。其收敛速度由关联马尔可夫算子的谱间隙决定。近期，Andrieu等人(2024)首次推导了目标分布光滑且强对数凹时随机游走Metropolis算法谱间隙的显式界。然而，现有文献鲜有探讨非光滑目标分布的情形。本文针对一类广泛非光滑分布，推导了随机游走Metropolis算法和Metropolis调整Langevin算法的显式谱间隙界。进一步地，结合Goyal等人(2025)的最新成果，我们将这些界推广至满足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。理论结果通过数值实验得到验证。