proximal_l1¶

odl.solvers.nonsmooth.proximal_operators.proximal_l1(space, lam=1, g=None)[source]¶

Proximal operator factory of the L1 norm/distance.

Implements the proximal operator of the functional

F(x) = lam ||x - g||_1

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

Parameters:

spaceLinearSpace or ProductSpace: Domain of the functional.
lampositive float, optional: Scaling factor or regularization parameter.
gspace element, optional: Element to which the L1 distance is taken. Default: space.zero.

Returns:

prox_factoryfunction: Factory for the proximal operator to be initialized

See also

proximal_convex_conj_l1: proximal for convex conjugate
proximal_l1_l2: isotropic variant of the group L1 norm proximal

Notes

For the functional

$F(x) = \lambda \|x - g\|_1,$

and a step size $\sigma$ , the proximal operator of $\sigma F$ is given as the "soft-shrinkage" operator

$\mathrm{prox}_{\sigma F}(x) = \begin{cases} g, & \text{where } |x - g| \leq \sigma\lambda, \\ x - \sigma\lambda \mathrm{sign}(x - g), & \text{elsewhere.} \end{cases}$

Here, all operations are to be read pointwise.

For vector-valued $x$ and $g$ , the (non-isotropic) proximal operator is the component-wise scalar proximal:

$\mathrm{prox}_{\sigma F}(x) = \left( \mathrm{prox}_{\sigma F}(x_1), \dots, \mathrm{prox}_{\sigma F}(x_d) \right),$

where $d$ is the number of components of $x$ .