BregmanDistance

class odl.solvers.functional.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.

Attributes:
adjoint

Adjoint of this operator (abstract).

convex_conj

The convex conjugate

domain

Set of objects on which this operator can be evaluated.

functional

The functional used to define the Bregman distance.

grad_lipschitz

Lipschitz constant for the gradient of the functional.

gradient

Gradient operator of the functional.

inverse

Return the operator inverse.

is_functional

True if this operator's range is a Field.

is_linear

True if this operator is linear.

point

The point used to define the Bregman distance.

proximal

Return the proximal factory of the functional.

range

Set in which the result of an evaluation of this operator lies.

subgrad

The subgradient used to define the Bregman distance.

Methods

__call__(x[, out])

Return self(x[, out, **kwargs]).

bregman(point, subgrad)

Return the Bregman distance functional.

derivative(point)

Return the derivative operator in the given point.

norm([estimate])

Return the operator norm of this operator.

translated(shift)

Return a translation of the functional.

__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.solvers.L2NormSquared(space)
>>> point = space.one()
>>> subgrad = l2_squared.gradient(point)
>>> bregman_dist = odl.solvers.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