Bregman distances in exponential families of probability measures

The concept of Bregman distances for Euclidean space vectors was introduced by Bregman (1967) in the context of convex programming. In this setup, the Bregman method has been widely applied and adapted, especially for the design of regularization algorithms for …nding a good approximate solution of inverse problems e.g. in image processing (tomography etc.), see for instance Censor and Lent (1981), Eggermont (1993), Byrne (1999), Resmerita (2005), Silva Neto and Cella (2006), Resmerita and Scherzer (2007), Resmerita and Anderssen (2007), Xu and Osher (2007), Burger et al. (2008), Cai et al. (2008), Marquina and Osher (2008), Osher et al. (2008), Scherzer et al. (2008), and the references therein. Bregman distances for non-negative functions were treated in Csiszár (1995). In the context of information theory and statistical decision theory, Bregman distances were studied e.g. by Csiszár (1991, 1994) as well as Pardo and Vajda (1997, 2003) basically for discrete probability measures or related functional quantities; closely related contexts are also applied in machine learning, see e.g. in La¤erty et al. (1997), Kivinen and Warmuth (1999), La¤erty (1999), Collins et al. (2001), Della Pietra et al. (2002), Murata et al. (2004), as well as in Cesa-Bianchi and Lugosi (2006). Applications to statistical physics are e.g. given in Topsoe (2007).