  This paper studies iterative schemes for measure transfer and approximation
problems, which are defined through a slicing-and-matching procedure. Similar
to the sliced Wasserstein distance, these schemes benefit from the availability
of closed-form solutions for the one-dimensional optimal transport problem and
the associated computational advantages. While such schemes have already been
successfully utilized in data science applications, not too many results on
their convergence are available. The main contribution of this paper is an
almost sure convergence proof for stochastic slicing-and-matching schemes. The
proof builds on an interpretation as a stochastic gradient descent scheme on
the Wasserstein space. Numerical examples on step-wise image morphing are
demonstrated as well.
