proximal_convex_conj_l1¶

odl.solvers.nonsmooth.proximal_operators.proximal_convex_conj_l1(space, lam=1, g=None)[source]¶

Proximal operator factory of the L1 norm/distance convex conjugate.

Implements the proximal operator of the convex conjugate of the functional

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

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

Parameters:

spaceLinearSpace or ProductSpace of LinearSpace spaces: Domain of the functional F
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_l2: isotropic variant for vector-valued functions
proximal_l1: proximal without convex conjugate

Notes

The convex conjugate $F^*$ of the functional

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

is in the case of scalar-valued functions given by

$F^*(y) = \iota_{B_\infty} \big( \lambda^{-1}\, y \big) + \left\langle \lambda^{-1}\, y,\: g \right\rangle,$

where $\iota_{B_\infty}$ is the indicator function of the unit ball with respect to $\|\cdot\|_\infty$ . For vector-valued functions, the convex conjugate is

$F^*(y) = \sum_{k=1}^d F^*(y_k)$

due to separability of the (non-isotropic) 1-norm.

For a step size $\sigma$ , the proximal operator of $\sigma F^*$ is given by

$\mathrm{prox}_{\sigma F^*}(y) = \frac{\lambda (y - \sigma g)}{ \max(\lambda, |y - \sigma g|)}$

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 $x$ .