My new column for SIGACT News is up (non pay-walled version here). In it, I discuss two different ways of defining distances between distributions.
The earthmover distance is a familar measure to TCS folk, and the kernel distance (or the maximum mean distortion, or the distance covariance) is a familiar character in machine learning and statistics.
It turns out that while statistically, these are not as different as one might think, they are different when it comes to computing them, and understanding their geometry.
The column describes both measures and their properties, and does a compare-contrast between them. More generally, I try to describe how one should think about the process of constructing a distance function between two distributions.