Regular Self-Proximal Distances are Bregman

Bregman distances play a key role in generalized versions of the proximal algorithm. This paper proposes a new characterization of Bregman distances in terms of their gradient and Hessian matrix. Thanks to this characterization, we obtain two results: all the so called self-proximal distances are Bregman, and all the induced proximal distances, under some regularity assumptions, are Bregman functions