In the theory of statistical inference and data processing problems, similarity measures (dissimilarities) play a very important role, in particular, the divergences of Bregman. So, a wide variety of distortion functions have been used to show that the β-divergences are subclasses of the Bregman divergences. In this paper, we propose and analyze a Legendre transform based on a parametric function which depends on β and prove that Bregman divergence generalizes the β-divergence. Moreover, we establish that the distance structure induced by this transformation function is preserved when the domain of the divergence of beta extends to the negative region. Comparisons with established algorithms lead to results in favor of our approach to examples artificial as well as real data in non-negative matrix factorization (NMF).

Bregman-divergence, β-divergence, non-negative matrix factorization (NMF).