  Wasserstein distance (WD) and the associated optimal transport plan have been
proven useful in many applications where probability measures are at stake. In
this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is
based on the transport map induced by an optimal one-dimensional projection of
the two input distributions. We draw connections between min-SWGG and
Wasserstein generalized geodesics in which the pivot measure is supported on a
line. We notably provide a new closed form for the exact Wasserstein distance
in the particular case of one of the distributions supported on a line allowing
us to derive a fast computational scheme that is amenable to gradient descent
optimization. We show that min-SWGG is an upper bound of WD and that it has a
complexity similar to as Sliced-Wasserstein, with the additional feature of
providing an associated transport plan. We also investigate some theoretical
properties such as metricity, weak convergence, computational and topological
properties. Empirical evidences support the benefits of min-SWGG in various
contexts, from gradient flows, shape matching and image colorization, among
others.
