Markovian Sliced Wasserstein Distances: Beyond Independent Projections

TL;DR

This paper introduces Markovian sliced Wasserstein (MSW) distances, a new family of metrics that improve upon existing sliced Wasserstein methods by imposing a Markov structure on projection directions, enhancing theoretical properties and practical performance.

Contribution

The paper proposes MSW distances with a Markov structure on projections, providing better metricity, theoretical guarantees, and computational efficiency compared to prior SW variants.

Findings

MSW distances satisfy metricity and weak convergence.

MSW demonstrates improved performance in generative modeling and color transfer.

Theoretical analysis shows favorable sample complexity and computational properties.

Abstract

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.