  Sliced optimal transport reduces optimal transport on multi-dimensional
domains to transport on the line. More precisely, sliced optimal transport is
the concatenation of the well-known Radon transform and the cumulative density
transform, which analytically yields the solutions of the reduced transport
problems. Inspired by this concept, we propose two adaptions for optimal
transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we
introduce the vertical slice transform, which integrates along all circles
orthogonal to a given direction. Secondly, we introduce a semicircle transform,
which integrates along all half great circles with an appropriate weight
function. Both transforms are generalized to arbitrary measures on the sphere.
While the vertical slice transform can be combined with optimal transport on
the interval and leads to a sliced Wasserstein distance restricted to even
probability measures, the semicircle transform is related to optimal transport
on the circle and results in a different sliced Wasserstein distance for
arbitrary probability measures. The applicability of both novel sliced optimal
transport concepts on the sphere is demonstrated by proof-of-concept examples
dealing with the interpolation and classification of spherical probability
measures. The numerical implementation relies on the singular value
decompositions of both transforms and fast Fourier techniques. For the
inversion with respect to probability measures, we propose the minimization of
an entropy-regularized Kullback--Leibler divergence, which can be numerically
realized using a primal-dual proximal splitting algorithm.
