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