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In this article we study the field of Hilbertian metrics and positive definit(pd) kernels on probability measures, they have a real interest in kernelmethods. Firstly we will make a study based on the Alpha-Beta-divergence tohave a Hilbercan metric by proposing an improvement of this divergence byconstructing it so that its is symmetrical the Alpha-Beta-Symmetric-divergence(ABS-divergence) and also do some studies on these properties but also proposethe kernels associated with this divergence. Secondly we will do mumericalstudies incorporating all proposed metrics/kernels into support vector machine(SVM). Finally we presented a algorithm for image classification by using ourdivergence.

The Alpha-Beta-Symetric Divergence and their Positive Definite Kernel