Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result of Roberts & Rosenthal (2002) by developing a theory which identifies conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap even in a general slice sampling context. Verifying the identified conditions for polar slice sampling yields a lower bound of 1/2 on the spectral gap for arbitrary dimension if the target density is rotationally invariant, log-concave along rays emanating from the origin and sufficiently smooth. The general theoretical result is potentially applicable beyond the polar slice sampling framework.

翻译：极化切片抽样是一种用于近似抽样的马尔可夫链构造方法，在满足目标分布与初始分布的适当假设条件下，其性能可被证明与状态空间维数无关。我们通过建立一种理论框架扩展了Roberts & Rosenthal (2002)的上述结论：该理论基于广义水平集函数识别出对谱隙给出显式下界的条件，即使在一般切片抽样框架下亦可适用。针对极化切片抽样验证所识别条件后，若目标密度具有旋转不变性、沿径向射线满足对数凹性且充分光滑，则对于任意维数均可获得谱隙的下界为1/2。该一般性理论结果可能适用于除极化切片抽样框架之外的更广泛场景。