Reproducing kernel space embeddings and metrics on probability measures

摘要: The notion of Hilbert space embedding probability measures has recently been used in various statistical applications like dimensionality reduction, homogeneity testing, independence etc. This represents any measure as a mean element reproducing kernel (RKHS). A pseudometric on the can be defined distance between distribution embeddings: we denote this γ k, indexed by positive definite (pd) function k that defines inner product RKHS. In dissertation, theoretical properties and associated RKHS are presented. First, order for to useful practice, it is essential metric not just pseudometric. Therefore, easily checkable characterizations have obtained so γk (such referred characteristic kernels), contrast previously published which either difficult check or may apply only restricted circumstances (e.g., compact domains). Second, relation kernels richness RKHS--how well an approximates some target space--and other common notions pd strictly (spd), integrally spd, conditionally etc., studied. Third, question nature topology induced studied wherein shown with spd kernels--a stronger than kernel--metrize weak* (weak-star) measures. Fourth, compared integral metrics (IPMs) p-divergences, empirical estimator simple compute exhibits fast rate convergence those IPMs p-divergences. These make more applicable practice these families distances. Finally, novel into Banach (RKBS) proposed its It generalize their counterparts, thereby resulting richer probabilities.