negative_func.bregman

negative_func.bregman(self, point, subgrad)

Return the Bregman distance functional.

Parameters
pointelement of functional.domain

Point from which to define the Bregman distance.

subgradelement of functional.domain

A subgradient of functional in point. If it exists, a valid option is functional.gradient(point).

Returns
outBregmanDistance

The Bregman distance functional.

Notes

Given a functional f, a point y, and a (sub)gradient p \in \partial f(y), the Bregman distance functional D_f^p(\cdot, y) in a point x is given by

D_f^p(x, y) = f(x) - f(y) - \langle p, x - y \rangle.

For mathematical details, see [Bur2016]. See also the Wikipedia article: https://en.wikipedia.org/wiki/Bregman_divergence

References

[Bur2016] Burger, M. Bregman Distances in Inverse Problems and Partial Differential Equation. In: Advances in Mathematical Modeling, Optimization and Optimal Control, 2016. p. 3-33.