Theory of the GMM Kernel

We 1 develop some theoreti al results for a robust similarity measure named generalized minmax (GMM). This similarity has dire t appli ations in ma hine learning as a positive de nite kernel and an be e iently omputed via probabilisti hashing. Owing to the dis rete nature, the hashed values an also be used for e ient near neighbor sear h. We prove the theoreti al limit of GMM and the onsisten y result, assuming that the data follow an ellipti al distribution, whi h is a very general family of distributions and in ludes the multivariate tdistribution as a spe ial ase. The onsisten y result holds as long as the data have bounded rst moment (an assumption whi h essentially holds for datasets ommonly en ountered in pra ti e). Furthermore, we establish the asymptoti normality of GMM. Compared to the osine similarity whi h is routinely adopted in urrent pra ti e in statisti s and ma hine learning, the onsisten y of GMM requires mu h weaker onditions. Interestingly, when the data follow the t-distribution with ν degrees of freedom, GMM typi ally provides a better measure of similarity than osine roughly when ν < 8 (ν = 8 means the distribution is already very lose to a normal). These theoreti al results will help explain the re ent su ess of the use of the GMM kernel [11, 12, 13℄ in ma hine learning tasks.