Relationship between the Bregman divergence and beta-divergence and their Applications

The Bregman divergence have been the subject of several studies. We do not go to do an exhaustive study of its subclasses, but propose a proof that shows that the \b{eta}-divergence are subclasses of the Bregman divergences. It is in this order of idea that we will make a proposition of demonstration which shows that the \b{eta}-divergence are particular cases of the Bregman divergence. And also we will propose algorithms and their applications to show the consistency of our approach. This is of interest for numerous applications since these divergences are widely used for instant non-negative matrix factorization (NMF).