Projection operators class hierarchy

Base class

class pyunlocbox.functions.proj(epsilon=1, method='FISTA', **kwargs)

Bases: pyunlocbox.functions.func

Base class which defines the attributes of the proj objects.

See generic attributes descriptions of the pyunlocbox.functions.func base class.

Parameters:

epsilon : float, optional

The radius of the ball. Default is 1.

method : {‘FISTA’, ‘ISTA’}, optional

The method used to solve the problem. It can be ‘FISTA’ or ‘ISTA’. Default is ‘FISTA’.

Notes

• All indicator functions (projections) evaluate to zero by definition.

L2-ball

class pyunlocbox.functions.proj_b2(**kwargs)

Bases: pyunlocbox.functions.proj

L2-ball function object.

This function is the indicator function \(i_S(z)\) of the set S which is zero if z is in the set and infinite otherwise. The set S is defined by \(\left\{z \in \mathbb{R}^N \mid \|A(z)-y\|_2 \leq \epsilon \right\}\).

See generic attributes descriptions of the pyunlocbox.functions.proj base class. Note that the constructor takes keyword-only parameters.

Notes

• The tol parameter is defined as the tolerance for the projection on the L2-ball. The algorithm stops if \(\frac{\epsilon}{1-tol} \leq \|y-A(z)\|_2 \leq \frac{\epsilon}{1+tol}\).
• The evaluation of this function is zero.
• The L2-ball proximal operator evaluated at x is given by \(\operatorname{arg\,min}\limits_z \frac{1}{2} \|x-z\|_2^2 + i_S(z)\) which has an identical solution as \(\operatorname{arg\,min}\limits_z \|x-z\|_2^2\) such that \(\|A(z)-y\|_2 \leq \epsilon\). It is thus a projection of the vector x onto an L2-ball of diameter epsilon.

Examples

>>> import pyunlocbox
>>> f = pyunlocbox.functions.proj_b2(y=[1, 1])
>>> x = [3, 3]
>>> f.eval(x)
0
>>> f.prox(x, 0)
array([ 1.70710678,  1.70710678])