切片Orlicz-Wasserstein距离
Sliced Orlicz-Wasserstein
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中文总结 AI 辅助
本文提出切片Orlicz-Wasserstein（SOW）距离，推广切片Wasserstein距离，用Luxemburg范数替代$L^p$范数，证明其度量性、拓扑性质及样本复杂度最优性，并展示其在计算效率和分布检测上的优势。
中文摘要 AI 辅助
我们提出了切片Orlicz-Wasserstein（SOW）距离，它是切片Wasserstein（SW）距离的推广。SOW将SW中的$L^p$范数替换为由Orlicz函数$\phi$诱导的Luxemburg范数代价。首先，我们证明了SOW距离在具有有限Orlicz范数的测度空间上是一个度量，并表明当Orlicz函数为$\phi(x)=x^p$时，它恢复为SW距离。接下来，我们推导了SOW距离的拓扑性质。特别地，我们证明了在SOW下的收敛蕴含弱收敛，并且在紧支撑条件下其逆命题也成立。然后，我们给出了估计SOW距离的理论结果。我们推导了距离本身及其幂泛函的样本复杂度，并证明了它们的极小极大最优性。此外，我们讨论了通过蒙特卡洛估计和二分搜索近似SOW距离的计算算法，以及相关的近似误差和计算复杂度分析。我们的实验结果表明，与Orlicz-Wasserstein（OW）距离相比，SOW具有优越的计算效率。同时，在实验中，我们通过比较它们在两样本检验中的性能以及在图像数据集上评估生成模型，展示了SOW距离相对于SW在检测分布差异方面的良好灵活性。
英文摘要
We propose sliced Orlicz-Wasserstein (SOW) distance which is a generalization of sliced Wasserstein (SW) distance. SOW replaces the $L^p$ norm in SW with a Luxemburg norm cost induced by an Orlicz function $ϕ$. First, we prove that SOW distance is a metric on the space of measures with finite Orlicz norm, and show that it recovers the SW distance when the Orlicz function is $ϕ(x)=x^p$. Next, we derive the topological properties of the SOW distance. In particular, we show that convergence under SOW implies weak convergence, and the converse is true under the compact support condition. We then present the theoretical results for estimating the SOW distance. We derive sample complexity for both the distance itself and the powered functional of the distance, and prove their minimax optimality. In addition, we discuss the computational algorithm for approximating the SOW distance by Monte-Carlo estimation and bisection search, as well as the associated approximation error and computational complexity analysis. Our experimental results reveal the superior computational efficiency of SOW compared with Orlicz-Wasserstein (OW) distance. Also, in the experiments, we demonstrate the favorable flexibility of SOW distance over SW in detecting differences between distributions by comparing their performance in two-sample tests and evaluating generative models on image datasets.
发表机构
- Université Gustave Eiffel（古斯塔夫·埃菲尔大学）
- Texas A&M University（得克萨斯农工大学）
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