"Bregman" is a term that refers to a vector space in mathematics. Specifically, it is a type of norm defined on a real or complex vector space, which measures the distance between two points within that space. Introduced by L. M. Bregman in the early 1960s, this norm is often used in optimization and information theory.
In mathematical terms, a Bregman divergence measures the difference between a point and a reference point from the perspective of a given convex function. It is defined as the value of the convex function at the given point, minus the value of the convex function at the reference point, as well as the inner product of their difference. Bregman divergences are non-negative and satisfy the triangle inequality, making them a suitable measure of dissimilarity between two points.
One useful property of Bregman divergences is that they are invariant to affine transformations and do not require the vector space to be Euclidean. This makes them applicable to a wide range of problems, including data clustering, machine learning, and pattern recognition.
The concept of Bregman divergences has further been extended to define Bregman projections, operators that find the closest point to a given one with respect to a convex function. These projections have found applications in various fields, such as image and signal processing, as well as game theory, where they are used to model decision-making and information asymmetry.
