In mathematics, a Bregman divergence or Bregman distance is similar to a metric, but does not satisfy the triangle inequality nor symmetry.
Contents
- Definition
- Properties
- Examples
- Generalizing projective duality
- Bregman divergence on other objects
- References
Bregman divergences are named after Lev M. Bregman, who introduced the concept in 1967.
Definition
Let
The Bregman distance associated with F for points
Properties
Examples
Generalizing projective duality
A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. There are numerous analytical forms of the projective dual: one common form maps the point
If we now replace the paraboloid by an arbitrary convex function, we obtain a different dual mapping that retains the incidence and above-below properties of the standard projective dual. This implies that natural dual concepts in computational geometry like Voronoi diagrams and Delaunay triangulations retain their meaning in distance spaces defined by an arbitrary Bregman divergence. Thus, algorithms from "normal" geometry extend directly to these spaces (Boissonnat, Nielsen and Nock, 2010)
Bregman divergence on other objects
Bregman divergences can also be defined between matrices, between functions, and between measures (distributions). Bregman divergences between matrices include the Stein's loss and von Neumann entropy. Bregman divergences between functions include total squared error, relative entropy, and squared bias; see the references by Frigyik et al. below for definitions and properties. Similarly Bregman divergences have also been defined over sets, through a submodular set function which is known as the discrete analog of a convex function. The submodular Bregman divergences subsume a number of discrete distance measures, like the Hamming distance, precision and recall, mutual information and some other set based distance measures (see Iyer & Bilmes, 2012) for more details and properties of the submodular Bregman.)
For a list of common matrix Bregman divergences, see Table 15.1 in.