Accelerated Primal-Dual Coordinate Descent for Computational Optimal Transport
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We propose and analyze a novel accelerated primal-dual coordinate descent framework for computing the optimal transport (OT) distance between two discrete probability distributions. First, we introduce the accelerated primal-dual randomized coordinate descent (APDRCD) algorithm for computing OT. Then we provide a complexity upper bound O(n^5/2/ε) for the APDRCD method for approximating OT distance, where n stands for the number of atoms of these probability measures and ε > 0 is the desired accuracy. This upper bound matches the best known complexities of adaptive primal-dual accelerated gradient descent (APDAGD) and adaptive primal-dual accelerate mirror descent (APDAMD) algorithms while it is better than those of Sinkhorn and Greenkhorn algorithms, which are of the order O(n^2/ε^2), in terms of the desired accuracy ε > 0. Furthermore, we propose a greedy version of APDRCD algorithm that we refer to as the accelerated primal-dual greedy coordinate descent (APDGCD) algorithm and demonstrate that it has a better practical performance than the APDRCD algorithm. Extensive experimental studies demonstrate the favorable performance of the APDRCD and APDGCD algorithms over state-of-the-art primal-dual algorithms for OT in the literature.
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