  Although Sinkhorn divergences are now routinely used in data sciences to
compare probability distributions, the computational effort required to compute
them remains expensive, growing in general quadratically in the size $n$ of the
support of these distributions. Indeed, solving optimal transport (OT) with an
entropic regularization requires computing a $n\times n$ kernel matrix (the
neg-exponential of a $n\times n$ pairwise ground cost matrix) that is
repeatedly applied to a vector. We propose to use instead ground costs of the
form $c(x,y)=-\log\dotp{\varphi(x)}{\varphi(y)}$ where $\varphi$ is a map from
the ground space onto the positive orthant $\RR^r_+$, with $r\ll n$. This
choice yields, equivalently, a kernel $k(x,y)=\dotp{\varphi(x)}{\varphi(y)}$,
and ensures that the cost of Sinkhorn iterations scales as $O(nr)$. We show
that usual cost functions can be approximated using this form. Additionaly, we
take advantage of the fact that our approach yields approximation that remain
fully differentiable with respect to input distributions, as opposed to
previously proposed adaptive low-rank approximations of the kernel matrix, to
train a faster variant of OT-GAN \cite{salimans2018improving}.
