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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 aforementionedresult of Roberts & Rosenthal (2002) by developing a theory which identifiesconditions, in terms of a generalized level set function, that imply anexplicit lower bound on the spectral gap even in a general slice samplingcontext. Verifying the identified conditions for polar slice sampling yields alower bound of 1/2 on the spectral gap for arbitrary dimension if the targetdensity is rotationally invariant, log-concave along rays emanating from theorigin and sufficiently smooth. The general theoretical result is potentiallyapplicable beyond the polar slice sampling framework.

Dimension-independent spectral gap of polar slice sampling