A new distance for data sets (and probability measures) in a RKHS context

In this paper we define distance functions for data sets (and distributions) in a RKHS context. To this aim we introduce kernels for data sets that provide a metrization of the set of points sets (the power set). An interesting point in the proposed kernel distance is that it takes into account the underlying (data) generating probability distributions. In particular, we propose kernel distances that rely on the estimation of density level sets of the underlying distribution, and can be extended from data sets to probability measures. The performance of the proposed distances is tested on a variety of simulated distributions plus a couple of real pattern recognition problems