Dimension-independent spectral gap of polar slice sampling
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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 aforementioned result of Roberts Rosenthal (2002) by proving 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 main tool along this path is the development of conditions, in terms of a generalized level set function, that imply an explicit lower bound on the spectral gap in a general slice sampling context. The latter result is potentially applicable beyond the polar slice sampling framework.
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