Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm

Understand

We analyze two algorithms for approximating the general optimal transport (OT) distance between two discrete distributions of size $n$, up to accuracy $\varepsilon$.

• For the first algorithm, which is based on the celebrated Sinkhorn's algorithm, we prove the complexity bound $\widetilde{O}\left({n^2/\varepsilon^2}\right)$ arithmetic operations.
• For the second one, which is based on our novel Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, we prove the complexity bound $\widetilde{O}\left(\min\left\{n^{9/4}/\varepsilon, n^{2}/\varepsilon^2 \right\}\right)$ arithmetic operations.
• Both bounds have better dependence on $\varepsilon$ than the state-of-the-art result given by $\widetilde{O}\left({n^2/\varepsilon^3}\right)$.