Functional Bregman divergence

To characterize the differences between two positive functions or two distributions, a class of distortion functions has recently been defined termed the functional Bregman divergences. The class generalizes the standard Bregman divergence defined for vectors, and includes total squared difference and relative entropy. Recently a key property was discovered for the vector Bregman divergence: that the mean minimizes the average Bregman divergence for a finite set of vectors. In this paper the analog result is proven: that the mean function minimizes the average Bregman divergence for a set of positive functions that can be parameterized by a finite number of parameters. In addition, the relationship of the functional Bregman divergence to the vector Bregman divergence and pointwise Bregman divergence is stated, as well as some important properties.