proximal_arg_scaling¶

odl.solvers.nonsmooth.proximal_operators.proximal_arg_scaling(prox_factory, scaling)[source]¶

Calculate the proximal of function F(x * scaling).

Parameters

prox_factorycallable: A factory function that, when called with a step size, returns the proximal operator of F
scalingfloat or sequence of floats or space element: Scaling parameter. The permissible types depent on the stepsizes accepted by prox_factory. It may not contain any nonzero imaginary parts. If it is a scalar, it may be zero, in which case the resulting proxmial operator is the identity. If not a scalar, it may not contain any zero components.

Returns

prox_factoryfunction: Factory for the proximal operator to be initialized

Notes

Given a functional $F$ , and scaling factor $\alpha$ this is calculated according to the rule

$\mathrm{prox}_{\sigma F(\alpha \, \cdot)}(x) = \frac{1}{\alpha} \mathrm{prox}_{\sigma \alpha^2 F(\cdot) }(\alpha x)$

where $\sigma$ is the step size.

For reference on the identity used, see [CP2011c].

References

[CP2011c] Combettes, P L, and Pesquet, J-C. Proximal splitting methods in signal processing. In: Bauschke, H H, Burachik, R S, Combettes, P L, Elser, V, Luke, D R, and Wolkowicz, H. Fixed-point algorithms for inverse problems in science and engineering, Springer, 2011.