This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that for some finite-dimensional problems, the $W_2$ distance leads to optimization problems that have better convexity than the classical $L^2$ and $H^{-1}$ distances, making it a more preferred distance to use when solving such inverse matching problems.

中文翻译：

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用于逆向数据匹配的二次 Wasserstein 度量

这项工作从分析和数值上表征了二次 Wasserstein ($W_2$) 距离的两个主要影响，作为反问题计算解决方案中数据差异的度量。首先，我们表明，在无限维设置中，$W_2$ 距离对反演过程具有平滑效果，使其对数据中的高频噪声具有鲁棒性，但导致重建对象的分辨率降低给定的噪音水平。其次，我们证明了对于一些有限维问题，$W_2$ 距离导致优化问题的凸度比经典的 $L^2$ 和 $H^{-1}$ 距离更好，使其成为更优选的距离在解决此类逆匹配问题时使用。