proximal_convex_conj_kl¶

odl.solvers.nonsmooth.proximal_operators.proximal_convex_conj_kl(space, lam=1, g=None)[source]¶

Proximal operator factory of the convex conjugate of the KL divergence.

Function returning the proximal operator of the convex conjugate of the functional F where F is the entropy-type Kullback-Leibler (KL) divergence:

F(x) = sum_i (x_i - g_i + g_i ln(g_i) - g_i ln(pos(x_i))) + ind_P(x)

with x and g elements in the linear space X, and g non-negative. Here, pos denotes the nonnegative part, and ind_P is the indicator function for nonnegativity.

Parameters:

spaceTensorSpace: Space X which is the domain of the functional F
lampositive float, optional: Scaling factor.
gspace element, optional: Data term, positive. If None it is take as the one-element.

Returns:

prox_factoryfunction: Factory for the proximal operator to be initialized.

See also

proximal_convex_conj_kl_cross_entropy: proximal for releated functional

Notes

The functional is given by the expression

$F(x) = \sum_i (x_i - g_i + g_i \ln(g_i) - g_i \ln(pos(x_i))) + I_{x \geq 0}(x)$

The indicator function $I_{x \geq 0}(x)$ is used to restrict the domain of $F$ such that $F$ is defined over whole space $X$ . The non-negativity thresholding $pos$ is used to define $F$ in the real numbers.

Note that the functional is not well-defined without a prior g. Hence, if g is omitted this will be interpreted as if g is equal to the one-element.

The convex conjugate $F^*$ of $F$ is

$F^*(p) = \sum_i (-g_i \ln(\text{pos}({1_X}_i - p_i))) + I_{1_X - p \geq 0}(p)$

where $p$ is the variable dual to $x$ , and $1_X$ is an element of the space $X$ with all components set to 1.

The proximal operator of the convex conjugate of F is

$\mathrm{prox}_{\sigma (\lambda F)^*}(x) = \frac{\lambda 1_X + x - \sqrt{(x - \lambda 1_X)^2 + 4 \lambda \sigma g}}{2}$

where $\sigma$ is the step size-like parameter, and $\lambda$ is the weighting in front of the function $F$ .

KL based objectives are common in MLEM optimization problems and are often used when data noise governed by a multivariate Poisson probability distribution is significant.

The intermediate image estimates can have negative values even though the converged solution will be non-negative. Non-negative intermediate image estimates can be enforced by adding an indicator function ind_P the primal objective.

This functional $F$ , described above, is related to the Kullback-Leibler cross entropy functional. The KL cross entropy is the one described in this Wikipedia article, and the functional $F$ is obtained by switching place of the prior and the varialbe in the KL cross entropy functional. See the See Also section.