Norm operators class hierarchy¶

Base class¶

class pyunlocbox.functions.norm(lambda_=1, w=1, **kwargs)[source]¶

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)[source]¶

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)[source]¶

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])

Nuclear-norm¶

class pyunlocbox.functions.norm_nuclear(**kwargs)[source]¶

Bases: pyunlocbox.functions.norm

Nuclear-norm function object.

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

Notes

The nuclear-norm of the matrix x is given by \(\lambda \| x \|_* = \lambda \operatorname{trace} (\sqrt{x^* x}) = \lambda \sum_{i=1}^N |e_i|\) where e_i are the eigenvalues of x.
The nuclear-norm proximal operator evaluated at x is given by \(\operatorname{arg\,min}\limits_z \frac{1}{2} \|x-z\|_2^2 + \gamma \| x \|_*\) where \(\gamma = \lambda \cdot T\), which is a soft-thresholding of the eigenvalues.

Examples

>>> import pyunlocbox
>>> f = pyunlocbox.functions.norm_nuclear()
>>> f.eval([[1, 2],[2, 3]])
4.4721359549995787
>>> f.prox([[1, 2],[2, 3]], 1)
array([[ 0.89442719,  1.4472136 ],
       [ 1.4472136 ,  2.34164079]])

TV-norm¶

class pyunlocbox.functions.norm_tv(dim=2, verbosity='LOW', **kwargs)[source]¶

Bases: pyunlocbox.functions.norm

TV Norm function object.

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

Notes

TODO

See [BT09b] for details about the algorithm.

Examples

>>> import pyunlocbox
>>> import numpy as np
>>> f = pyunlocbox.functions.norm_tv()
>>> x = np.arange(0, 16)
>>> x = x.reshape(4, 4)
>>> f.eval(x)
    norm_tv evaluation : 5.210795e+01
52.107950630558953