Distributional Sliced-Wasserstein and Applications to Generative Modeling
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Sliced-Wasserstein distance (SWD) and its variation, Max Sliced-Wasserstein distance (Max-SWD), have been widely used in the recent years due to their fast computation and scalability when the probability measures lie in very high dimension. However, these distances still have their weakness, SWD requires a lot of projection samples because it uses the uniform distribution to sample projecting directions, Max-SWD uses only one projection, causing it to lose a large amount of information. In this paper, we propose a novel distance that finds optimal penalized probability measure over the slices, which is named Distributional Sliced-Wasserstein distance (DSWD). We show that the DSWD is a generalization of both SWD and Max-SWD, and the proposed distance could be found by searching for the push-forward measure over a set of measures satisfying some certain constraints. Moreover, similar to SWD, we can extend Generalized Sliced-Wasserstein distance (GSWD) to Distributional Generalized Sliced-Wasserstein distance (DGSWD). Finally, we carry out extensive experiments to demonstrate the favorable generative modeling performances of our distances over the previous sliced-based distances in large-scale real datasets.
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