Slicing Wasserstein Over Wasserstein Via Functional Optimal Transport
This provides a scalable and stable alternative for researchers and practitioners in machine learning and computer vision working with complex data like images and shapes, though it is incremental as it builds on existing sliced Wasserstein methods.
The paper tackles the computational cost and instability of the Wasserstein over Wasserstein (WoW) distance for comparing datasets or distributions by proposing the double-sliced Wasserstein (DSW) metric, which avoids unstable higher-order moments and achieves computational savings while maintaining equivalence to WoW for discretized meta-measures.
Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space $L_2([0,1])$. For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between 1d meta-measures via infinite-dimensional $L_2$-projections, parametrized by Gaussian processes. Combining this 1d construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.