Efficient Optimal Transport Algorithm by Accelerated Gradient Descent

Optimal transport (OT) plays an essential role in various areas like machine learning and deep learning. However, computing discrete OT for large scale problems with adequate accuracy and efficiency is highly challenging. Recently, methods based on the Sinkhorn algorithm add an entropy regularizer to the prime problem and obtain a trade off between efficiency and accuracy. In this paper, we propose a novel algorithm based on Nesterov's smoothing technique to further improve the efficiency and accuracy in computing OT. Basically, the non-smooth c-transform of the Kantorovich potential is approximated by the smooth Log-Sum-Exp function, which smooths the original non-smooth Kantorovich dual functional. The smooth Kantorovich functional can be efficiently optimized by a fast proximal gradient method, the fast iterative shrinkage thresholding algorithm (FISTA). Theoretically, the computational complexity of the proposed method is given by O(n52logn∕ϵ), which is lower than current estimation of the Sinkhorn algorithm. Experimentally, compared with the Sinkhorn algorithm, our results demonstrate that the proposed method achieves faster convergence and better accuracy with the same parameter.