Generalized Sliced Distances for Probability Distributions
This work addresses the lack of established convergence behavior for algorithms using probability metrics in statistics and machine learning, particularly for generative modeling, though it appears incremental as it builds upon existing concepts like MMD.
The paper introduces Generalized Sliced Probability Metrics (GSPMs), a family of probability metrics based on the generalized Radon transform, and shows that a subset of these metrics is equivalent to maximum mean discrepancy with novel kernels, enabling gradient flows for generative modeling that converge to the global optimum under mild assumptions.
Probability metrics have become an indispensable part of modern statistics and machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a practical setting, the convergence behavior of the algorithms built upon these distances have not been well established, except for a few specific cases. In this paper, we introduce a broad family of probability metrics, coined as Generalized Sliced Probability Metrics (GSPMs), that are deeply rooted in the generalized Radon transform. We first verify that GSPMs are metrics. Then, we identify a subset of GSPMs that are equivalent to maximum mean discrepancy (MMD) with novel positive definite kernels, which come with a unique geometric interpretation. Finally, by exploiting this connection, we consider GSPM-based gradient flows for generative modeling applications and show that under mild assumptions, the gradient flow converges to the global optimum. We illustrate the utility of our approach on both real and synthetic problems.