论文标题
大都会马尔可夫链的显式收敛范围：等级，光谱差距和轮廓
Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles
论文作者
论文摘要
对于任何值差异的任何值，我们在$ r^d $上获得了随机步行大都市算法的光谱差距的第一个显式界限，当该差异时，该算法适当地缩小了正确的$ d^{ - 1} $依赖性的正确$ d^{ - 1} $依赖性，以适合正常不变的分布。我们还可以在$ {\ rm l}^2 $中获得明确的界限，以供广泛的模型进行混合时间。在获得这些结果时，我们完善了使用等值光谱不平等的使用来获得电导轮廓界限，这也能够在更广泛的模型类别中推导显式界限。我们还为预处理的曲柄 - 尼古尔森·马尔可夫链获得了相似的结果，在合适的假设下获得了尺寸独立的界限。
We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.
