BregmanDistance

class odl.functionals.functional.BregmanDistance(*args, **kwargs)[source]

Bases: Functional

The Bregman distance functional.

The Bregman distance, also refered to as the Bregman divergence, is similar to a metric but satisfies neither the triangle inequality nor symmetry. Nevertheless, the Bregman distance is used in variational regularization of inverse problems, see, e.g., [Bur2016].

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.

__init__(functional, point, subgrad)[source]

Initialize a new instance.

Parameters

functionalFunctional: Functional on which to base the Bregman distance.
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).

Examples

Example of initializing the Bregman distance functional:

>>> space = odl.uniform_discr(0, 1, 10)
>>> l2_squared = odl.functionals.L2NormSquared(space)
>>> point = space.one()
>>> subgrad = l2_squared.gradient(point)
>>> bregman_dist = odl.functionals.BregmanDistance(
...     l2_squared, point, subgrad)

This is gives squared L2 distance to the given point, ||x - 1||^2:

>>> expected_functional = l2_squared.translated(point)
>>> bregman_dist(space.zero()) == expected_functional(space.zero())
True