Norm operators class hierarchy

Base class

class pyunlocbox.functions.norm(lambda_=1, w=1, **kwargs)

Bases: pyunlocbox.functions.func

Base class which defines the attributes of the norm objects.

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

Parameters:

lambda_ : float, optional

Regularization parameter \(\lambda\). Default is 1.

w : array_like, optional

Weights for a weighted norm. Default is 1.

L1-norm

class pyunlocbox.functions.norm_l1(**kwargs)

Bases: pyunlocbox.functions.norm

L1-norm function object.

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

Notes

• The L1-norm of the vector x is given by \(\lambda \|w \cdot (A(x)-y)\|_1\).
• The L1-norm proximal operator evaluated at x is given by \(\operatorname{arg\,min}\limits_z \frac{1}{2} \|x-z\|_2^2 + \gamma \|w \cdot (A(z)-y)\|_1\) where \(\gamma = \lambda \cdot T\). This is simply a soft thresholding.

Examples

>>> import pyunlocbox
>>> f = pyunlocbox.functions.norm_l1()
>>> f.eval([1, 2, 3, 4])
10
>>> f.prox([1, 2, 3, 4], 1)
array([ 0.,  1.,  2.,  3.])

L2-norm

class pyunlocbox.functions.norm_l2(**kwargs)

Bases: pyunlocbox.functions.norm

L2-norm function object.

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

Notes

• The squared L2-norm of the vector x is given by \(\lambda \|w \cdot (A(x)-y)\|_2^2\).
• The squared L2-norm proximal operator evaluated at x is given by \(\operatorname{arg\,min}\limits_z \frac{1}{2} \|x-z\|_2^2 + \gamma \|w \cdot (A(z)-y)\|_2^2\) where \(\gamma = \lambda \cdot T\).
• The squared L2-norm gradient evaluated at x is given by \(2 \lambda \cdot At(w \cdot (A(x)-y))\).

Examples

>>> import pyunlocbox
>>> f = pyunlocbox.functions.norm_l2()
>>> x = [1, 2, 3, 4]
>>> f.eval(x)
30
>>> f.prox(x, 1)
array([ 0.33333333,  0.66666667,  1.        ,  1.33333333])
>>> f.grad(x)
array([2, 4, 6, 8])