  Optimal transport (OT) plays an essential role in various areas like machine
learning and deep learning. However, computing discrete optimal transport plan
for large scale problems with adequate accuracy and efficiency is still highly
challenging. Recently, methods based on the Sinkhorn algorithm add an entropy
regularizer to the prime problem and get a trade off between efficiency and
accuracy. In this paper, we propose a novel algorithm to further improve the
efficiency and accuracy based on Nesterov's smoothing technique. Basically, the
non-smooth c-transform of the Kantorovich potential is approximated by the
smooth Log-Sum-Exp function, which finally smooths the original non-smooth
Kantorovich dual functional (energy). The smooth Kantorovich functional can be
optimized by the fast proximal gradient algorithm (FISTA) efficiently.
Theoretically, the computational complexity of the proposed method is given by
$O(n^{\frac{5}{2}} \sqrt{\log n} /\epsilon)$, which is lower than that of the
Sinkhorn algorithm. Empirically, compared with the Sinkhorn algorithm, our
experimental results demonstrate that the proposed method achieves faster
convergence and better accuracy with the same parameter.
