Bregman divergence

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In mathematics Bregman divergence or Bregman distance is similar to a metric, but does not satisfy the triangle inequality nor symmetry. There are two ways in which Bregman divergences are important. Firstly, they generalize squared Euclidean distance to a class of distances that all share similar properties. Secondly, they bear a strong connection to exponential families of distributions; as has been shown by (Banerjee et al. 2005), there is a bijection between exponential families and Bregman divergences.

Bregman divergences are named after L. M. Bregman, who introduced the concept in 1967. More recently researchers in geometric algorithms have shown that many important algorithms can be generalized from Euclidean metrics to distances defined by Bregman divergence (Banerjee et al. 2005; Nielsen and Nock 2006; Nielsen et al. 2007).

1 Definition
2 Properties
3 Examples
4 Generalizing projective duality
5 References
6 External links

[edit] Definition

Let $F: \Delta \to \Re$ be a continuously-differentiable real-valued and strictly convex function defined on a closed convex set $Δ$

The Bregman distance associated with $F$ for points $p, q \in \Delta$ is:

$B_F(p \rVert q) = F(p)-F(q)-\langle \nabla F(q),(p-q)\rangle$

Intuitively this can be thought of as the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p.

[edit] Properties

Non-negativity: $B_F(p \rVert q) \ge 0$ for all p,q. This is a consequence of the convexity of F.
Convexity: $B_F(p \rVert q)$ is convex in its first argument, but not necessarily in the second argument.
Linearity: If we think of the Bregman distance as an operator on the function F, then it is linear with respect to non-negative coefficients. In other words, for $F 1, F 2$ strictly convex and differentiable, and $λ > 0$ ,

$B_{F_1 + \lambda F_2}(p \rVert q) = B_{F_1}(p \rVert q) + \lambda B_{F_2}(p \rVert q)$

Duality: The function F has a convex conjugate $F *$ . The Bregman distance defined with respect to $F *$ has an interesting relationship to $B_F(p \rVert q)$

$B_{F^*}(p^* \rVert q^*) = B_F(q \rVert p)$

Here, $p^* = \nabla F(p)$ is the dual point corresponding to p

[edit] Examples

Squared Euclidean distance $B_F(x,y) = \|x - y\|^2$ is the canonical example of a Bregman distance, generated by the convex function $F(x) = \|x\|^2$
The Kullback-Leibler divergence $B_F(p,q) = \sum p(i) \log \frac{p(i)}{q(i)}$ is generated by the convex function $F(p) = \sum_i p(i)\log p(i) - \sum p(i)$

[edit] 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 $p = (p_1, \ldots p_d)$ to the hyperplane $x_{d+1} = \sum_1^d 2p_i x_i$ . This mapping can be interpreted (identifying the hyperplane with its normal) as the convex conjugate mapping that takes the point p to its dual point $p^* = \nabla F(p)$ , where F defines the d-dimensional paraboloid $x_{d+1} = \sum x_i^2$ .

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 (Nielsen, Nock and Boissonnat, 2006)

[edit] References

Banerjee, Arindam; Merugu, Srujana; Dhillon, Inderjit S.; Ghosh, Joydeep (2005). "Clustering with Bregman divergences". Journal of Machine Learning Research 6: 1705–1749.

Bregman, L. M. (1967). "The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming". USSR Computational Mathematics and Mathematical Physics 7: 200–217. doi:10.1016/0041-5553(67)90040-7.

Nielsen, Frank; Boissonnat, Jean-Daniel; Nock, Richard (2007). "On Visualizing Bregman Voronoi diagrams". Proc. 23rd ACM Conference on Computational Geometry (video track).

Nielsen, Frank; Boissonnat, Jean-Daniel; Nock, Richard (2007). "On Bregman Voronoi Diagrams". Proc. 18th ACM-SIAM Symposium on Discrete Algorithms (SODA).

Nielsen, Frank; Nock, Richard (2006). "On approximating the smallest enclosing Bregman Balls". Proceedings of the 22nd Annual Symposium on Computational Geometry: 485–486. doi:10.1145/1137856.1137931.