# Quantitative spectral gap estimate and Wasserstein contraction of simple slice sampling

## TL;DR

This paper establishes a quantitative spectral gap estimate and Wasserstein contraction for simple slice sampling, enhancing understanding of its convergence properties for certain classes of probability distributions.

## Contribution

It provides the first explicit lower bound on the spectral gap of simple slice sampling for log-concave, rotationally invariant distributions, extending to broader target distributions.

## Key findings

- Wasserstein contraction is proven for simple slice sampling.
- An explicit lower bound on the spectral gap is derived.
- Results apply to distributions depending on level set volumes.

## Abstract

We prove Wasserstein contraction of simple slice sampling for approximate sampling w.r.t. distributions with log-concave and rotational invariant Lebesgue densities. This yields, in particular, an explicit quantitative lower bound of the spectral gap of simple slice sampling. Moreover, this lower bound carries over to more general target distributions depending only on the volume of the (super-)level sets of their unnormalized density.