  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.
