  Discriminating between distributions is an important problem in a number of
scientific fields. This motivated the introduction of Linear Optimal
Transportation (LOT), which embeds the space of distributions into an
$L^2$-space. The transform is defined by computing the optimal transport of
each distribution to a fixed reference distribution, and has a number of
benefits when it comes to speed of computation and to determining
classification boundaries. In this paper, we characterize a number of settings
in which LOT embeds families of distributions into a space in which they are
linearly separable. This is true in arbitrary dimension, and for families of
distributions generated through perturbations of shifts and scalings of a fixed
distribution.We also prove conditions under which the $L^2$ distance of the LOT
embedding between two distributions in arbitrary dimension is nearly isometric
to Wasserstein-2 distance between those distributions. This is of significant
computational benefit, as one must only compute $N$ optimal transport maps to
define the $N^2$ pairwise distances between $N$ distributions. We demonstrate
the benefits of LOT on a number of distribution classification problems.
