We study Benamou's domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback--Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient implementation, such as adaptive sparsity, a coarse-to-fine scheme and parallelization, paving the way to numerically solving large-scale optimal transport problems. We demonstrate efficient numerical performance for computing the Wasserstein-2 distance between 2D images and observe that, even without parallelization, domain decomposition compares favorably to applying a single efficient implementation of the Sinkhorn algorithm in terms of runtime, memory and solution quality.
翻译:我们研究贝纳穆的域分解算法,以优化在对流正常环境下的运输。 关键观察是, 常规变方在非常温和的假设下会与全球最佳解决办法相融合。 我们证明算法在Kullback- Leibertr差异方面的线性趋同, 并以数字例子来说明( 可能非常慢的) 率。 关于足够的几何结构( 如瓦塞斯坦图像之间的距离) 的问题, 我们期望更快的趋同。 然后我们讨论计算高效实施的重要方面, 比如适应性宽度、粗略到软化的计划和平行化, 为从数字上解决大规模最佳运输问题铺平道路。 我们展示了计算2D图像之间瓦塞斯坦-2距离的高效数字性表现, 并观察到即使没有平行化, 领域分解也比在运行时间、 记忆和解决方案质量方面对辛克霍恩算法的单一有效实施要好得多。