proximal_convex_conj_l2¶

odl.solvers.nonsmooth.proximal_operators.proximal_convex_conj_l2(space, lam=1, g=None)[source]¶

Proximal operator factory of the convex conj of the l2-norm/distance.

Function for the proximal operator of the convex conjugate of the functional F where F is the l2-norm (or distance to g, if given):

F(x) =  lam ||x - g||_2

with x and g elements in space, scaling factor lam, and given data g.

Parameters

spaceLinearSpace: Domain of F(x). Needs to be a Hilbert space. That is, have an inner product (LinearSpace.inner).
lampositive float, optional: Scaling factor or regularization parameter.
gspace element, optional: An element in space. Default: space.zero.

Returns

prox_factoryfunction: Factory for the proximal operator to be initialized

See also

proximal_l2: proximal without convex conjugate
proximal_convex_conj_l2_squared: proximal for squared norm/distance

Notes

Most problems are forumlated for the squared norm/distance, in that case use the proximal_convex_conj_l2_squared instead.

The $L_2$ -norm/distance $F$ is given by is given by

$F(x) = \lambda \|x - g\|_2$

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

$F^*(y) = \begin{cases} 0 & \text{if } \|y-g\|_2 \leq \lambda, \\ \infty & \text{else.} \end{cases}$

For a step size $\sigma$ , the proximal operator of $\sigma F^*$ is given by the projection onto the set of $y$ satisfying $\|y-g\|_2 \leq \lambda$ , i.e., by

$\mathrm{prox}_{\sigma F^*}(y) = \begin{cases} \lambda \frac{y - g}{\|y - g\|} & \text{if } \|y-g\|_2 > \lambda, \\ y & \text{if } \|y-g\|_2 \leq \lambda \end{cases}$

Note that the expression is independent of $\sigma$ .