Sliced Wasserstein (SW) distance suffers from redundant projections due to independent uniform random projecting directions. To partially overcome the issue, max K sliced Wasserstein (Max-K-SW) distance ($K\geq 1$), seeks the best discriminative orthogonal projecting directions. Despite being able to reduce the number of projections, the metricity of Max-K-SW cannot be guaranteed in practice due to the non-optimality of the optimization. Moreover, the orthogonality constraint is also computationally expensive and might not be effective. To address the problem, we introduce a new family of SW distances, named Markovian sliced Wasserstein (MSW) distance, which imposes a first-order Markov structure on projecting directions. We discuss various members of MSW by specifying the Markov structure including the prior distribution, the transition distribution, and the burning and thinning technique. Moreover, we investigate the theoretical properties of MSW including topological properties (metricity, weak convergence, and connection to other distances), statistical properties (sample complexity, and Monte Carlo estimation error), and computational properties (computational complexity and memory complexity). Finally, we compare MSW distances with previous SW variants in various applications such as gradient flows, color transfer, and deep generative modeling to demonstrate the favorable performance of MSW.

翻译：切片Wasserstein（SW）距离因使用独立均匀随机投影方向而存在冗余投影问题。为部分解决该问题，最大K切片Wasserstein（Max-K-SW）距离（$K\geq 1$）试图寻找最优判别性正交投影方向。尽管能减少投影数量，但由于优化非最优性，Max-K-SW在实践中的度量性无法保证。此外，正交约束计算成本高昂且可能效果有限。针对该问题，我们引入一类新的SW距离——马尔可夫切片Wasserstein（MSW）距离，其对投影方向施加一阶马尔可夫结构。通过指定马尔可夫结构（包括先验分布、转移分布以及燃烧与稀释技术），我们讨论了MSW的多种变体。同时，我们研究了MSW的理论性质，包括拓扑性质（度量性、弱收敛性及与其他距离的联系）、统计性质（样本复杂度与蒙特卡洛估计误差）及计算性质（计算复杂度与内存复杂度）。最后，在梯度流、颜色传递和深度生成建模等应用中，我们将MSW距离与已有的SW变体进行对比，证明其优越性能。