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：切片瓦瑟斯坦距离因使用独立的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切片瓦瑟斯坦距离（$K\geq 1$）寻求最具判别性的正交投影方向。尽管能减少投影数量，但由于优化的非最优性，实践中无法保证Max-K-SW的度量性。此外，正交约束的计算成本高昂且可能效率低下。为此，我们提出一类新的SW距离——马尔可夫切片瓦瑟斯坦距离，该距离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、转移分布及燃烧与稀疏化技术），我们探讨了MSW的多种变体。同时研究了MSW的理论性质，包括拓扑性质（度量性、弱收敛性及其与其他距离的关联）、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，我们在梯度流、颜色传输和深度生成建模等多种应用中对比MSW距离与先前SW变体的性能，验证了MSW的优越表现。