  We study sampling problems associated with potentials that lack smoothness.
The potentials can be either convex or non-convex. Departing from the standard
smooth setting, the potentials are only assumed to be weakly smooth or
non-smooth, or the summation of multiple such functions. We develop a sampling
algorithm that resembles proximal algorithms in optimization for this
challenging sampling task. Our algorithm is based on a special case of Gibbs
sampling known as the alternating sampling framework (ASF). The key
contribution of this work is a practical realization of the ASF based on
rejection sampling for both non-convex and convex potentials that are not
necessarily smooth. In almost all the cases of sampling considered in this
work, our proximal sampling algorithm achieves better complexity than all
existing methods.
