# -*- coding: utf-8 -*-
r"""
This module implements function objects which are then passed to solvers. The
:class:`func` base class defines the interface whereas specialised classes who
inherit from it implement the methods. These classes include :
* :class:`dummy`: A dummy function object which returns 0 for the
:meth:`_eval`, :meth:`_prox` and :meth:`_grad` methods.
* :class:`norm`: Norm base class.
* :class:`norm_l1`: L1-norm who implements the :meth:`_eval` and
:meth:`_prox` methods.
* :class:`norm_l2`: L2-norm who implements the :meth:`_eval`, :meth:`_prox`
and :meth:`_grad` methods.
"""
import numpy as np
def _soft_threshold(z, T, handle_complex=True):
r"""
Return the soft thresholded signal.
Parameters
----------
z : array_like
Input signal (real or complex).
T : float or array_like
Threshold on the absolute value of `z`. There could be either a single
threshold for the entire signal `z` or one threshold per dimension.
Useful when you use weighted norms.
handle_complex : bool
Indicate that we should handle the thresholding of complex numbers,
which may be slower. Default is True.
Returns
-------
sz : ndarray
Soft thresholded signal.
Examples
--------
>>> import pyunlocbox
>>> pyunlocbox.functions._soft_threshold([-2, -1, 0, 1, 2], 1)
array([-1., -0., 0., 0., 1.])
"""
sz = np.maximum(np.abs(z)-T, 0)
if not handle_complex:
# This soft thresholding method only supports real signal.
sz = np.sign(z) * sz
else:
# This soft thresholding method supports complex complex signal.
# Transform to float to avoid integer division.
# In our case 0 divided by 0 should be 0, not NaN, and is not an error.
# It corresponds to 0 thresholded by 0, which is 0.
old_err_state = np.seterr(invalid='ignore')
sz = np.nan_to_num(np.float64(sz) / (sz+T) * z)
np.seterr(**old_err_state)
return sz
[docs]class func(object):
r"""
This class defines the function object interface.
It is intended to be a base class for standard functions which will
implement the required methods. It can also be instantiated by user code
and dynamically modified for rapid testing. The instanced objects are
meant to be passed to the :func:`pyunlocbox.solvers.solve` solving
function.
Parameters
----------
verbosity : {'none', 'low', 'high'}, optional
The log level : ``'none'`` for no log, ``'low'`` for resume at
convergence, ``'high'`` to for all steps. Default is ``'low'``.
Examples
--------
Lets define a parabola as an example of the manual implementation of a
function object :
>>> import pyunlocbox
>>> f = pyunlocbox.functions.func()
>>> f._eval = lambda x : x**2
>>> f._grad = lambda x : 2*x
>>> x = [1, 2, 3, 4]
>>> f.eval(x)
array([ 1, 4, 9, 16])
>>> f.grad(x)
array([2, 4, 6, 8])
"""
def __init__(self, verbosity='none'):
if verbosity not in ['none', 'low', 'high']:
raise ValueError('Verbosity should be either none, low or high.')
else:
self.verbosity = verbosity
[docs] def eval(self, x):
r"""
Function evaluation.
Parameters
----------
x : array_like
The evaluation point.
Returns
-------
z : float
The objective function evaluated at `x`.
Notes
-----
This method is required by the :func:`pyunlocbox.solvers.solve` solving
function to evaluate the objective function.
"""
sol = self._eval(np.array(x))
if self.verbosity in ['low', 'high']:
print('%s evaluation : %e' % (self.__class__.__name__, sol))
return sol
def _eval(self, x):
raise NotImplementedError("Class user should define this method.")
[docs] def prox(self, x, T):
r"""
Function proximal operator.
Parameters
----------
x : array_like
The evaluation point.
T : float
The regularization parameter.
Returns
-------
z : ndarray
The proximal operator evaluated at `x`.
Notes
-----
This method is required by some solvers.
The proximal operator is defined by
:math:`\operatorname{prox}_{f,\gamma}(x) = \min_z \frac{1}{2}
||x-z||_2^2 + \gamma f(z)`
"""
return self._prox(np.array(x), T)
def _prox(self, x, T):
raise NotImplementedError("Class user should define this method.")
[docs] def grad(self, x):
r"""
Function gradient.
Parameters
----------
x : array_like
The evaluation point.
Returns
-------
z : ndarray
The objective function gradient evaluated at `x`.
Notes
-----
This method is required by some solvers.
"""
return self._grad(np.array(x))
def _grad(self, x):
raise NotImplementedError("Class user should define this method.")
[docs]class dummy(func):
r"""
Dummy function object.
This can be used as a second function object when there is only one
function to minimize. The :meth:`eval`, :meth:`prox` and :meth:`grad`
methods then all return 0.
See generic attributes descriptions of the
:class:`pyunlocbox.functions.func` base class.
Examples
--------
>>> import pyunlocbox
>>> f = pyunlocbox.functions.dummy(verbosity='low')
>>> x = [1, 2, 3, 4]
>>> f.eval(x)
dummy evaluation : 0.000000e+00
0
>>> f.prox(x, 1)
array([ 0., 0., 0., 0.])
>>> f.grad(x)
array([ 0., 0., 0., 0.])
"""
def _eval(self, x):
return 0
def _prox(self, x, T):
return np.zeros(np.shape(x))
def _grad(self, x):
return np.zeros(np.shape(x))
[docs]class norm(func):
r"""
Base class which defines the attributes of the norm objects.
See generic attributes descriptions of the
:class:`pyunlocbox.functions.func` base class.
Parameters
----------
lambda_ : float, optional
Regularization parameter :math:`\lambda`. Default is 1.
y : array_like, optional
Measurements. Default is 0.
w : array_like, optional
Weights for a weighted norm. Default is 1.
A : function or ndarray, optional
The forward operator. Default is the identity, :math:`A(x)=x`. If `A`
is an ``ndarray``, it will be converted to the operator form.
At : function or ndarray, optional
The adjoint operator. If `At` is an ``ndarray``, it will be converted
to the operator form. If `A` is an ``ndarray``, default is the
transpose of `A`. If `A` is a function, default is `A`,
:math:`At(x)=A(x)`.
tight : bool, optional
``True`` if `A` is a tight frame, ``False`` otherwise. Default is
``True``.
nu : float, optional
Bound on the norm of the operator `A`, i.e. :math:`||A(x)||^2 \leq \nu
||x||^2`. Default is 1.
"""
def __init__(self, lambda_=1, y=0, w=1, A=None, At=None,
tight=True, nu=1, *args, **kwargs):
super(norm, self).__init__(*args, **kwargs)
self.lambda_ = lambda_
self.y = np.array(y)
self.w = np.array(w)
if A is None:
self.A = lambda x: x
else:
if type(A) is np.ndarray:
# Transform matrix form to operator form.
self.A = lambda x: np.dot(A, x)
else:
self.A = A
if At is None:
if type(A) is np.ndarray:
self.At = lambda x: np.dot(np.transpose(A), x)
else:
self.At = self.A
else:
if type(At) is np.ndarray:
# Transform matrix form to operator form.
self.At = lambda x: np.dot(At, x)
else:
self.At = At
self.tight = tight
self.nu = nu
[docs]class norm_l1(norm):
r"""
L1-norm function object.
See generic attributes descriptions of the
:class:`pyunlocbox.functions.norm` base class.
Notes
-----
* The L-1 norm of the vector `x` is given by
:math:`\lambda ||w \cdot (A(x)-y)||_1`
* The L1-norm proximal operator evaluated at `x` is given by
:math:`\min_z \frac{1}{2} ||x-z||_2^2 + \gamma ||w \cdot (A(z)-y)||_1`
where :math:`\gamma = \lambda \cdot T`
This is simply a soft thresholding.
Examples
--------
>>> import pyunlocbox
>>> f = pyunlocbox.functions.norm_l1(verbosity='low')
>>> f.eval([1, 2, 3, 4])
norm_l1 evaluation : 1.000000e+01
10
>>> f.prox([1, 2, 3, 4], 1)
array([ 0., 1., 2., 3.])
"""
def _eval(self, x):
sol = self.A(np.array(x)) - self.y
sol = self.lambda_ * np.sum(np.abs(self.w * sol))
return sol
def _prox(self, x, T):
# Gamma is T in the matlab UNLocBox implementation.
gamma = self.lambda_ * T
if self.tight:
sol = self.A(x) - self.y
sol = _soft_threshold(sol, gamma*self.nu*self.w) - sol
sol = x + self.At(sol) / self.nu
else:
raise NotImplementedError('Not implemented for non tight frame.')
return sol
[docs]class norm_l2(norm):
r"""
L2-norm function object.
See generic attributes descriptions of the
:class:`pyunlocbox.functions.norm` base class.
Notes
-----
* The squared L-2 norm of the vector `x` is given by
:math:`\lambda ||w \cdot (A(x)-y)||_2^2`
* The squared L2-norm proximal operator evaluated at `x` is given by
:math:`\min_z \frac{1}{2} ||x-z||_2^2 + \gamma ||w \cdot (A(z)-y)||_2^2`
where :math:`\gamma = \lambda \cdot T`
* The squared L2-norm gradient evaluated at `x` is given by
:math:`2 \lambda \cdot At(w \cdot (A(x)-y))`
Examples
--------
>>> import pyunlocbox
>>> f = pyunlocbox.functions.norm_l2(verbosity='low')
>>> x = [1, 2, 3, 4]
>>> f.eval(x)
norm_l2 evaluation : 3.000000e+01
30
>>> f.prox(x, 1)
array([ 0.33333333, 0.66666667, 1. , 1.33333333])
>>> f.grad(x)
array([2, 4, 6, 8])
"""
def _eval(self, x):
sol = self.A(np.array(x)) - self.y
sol = self.lambda_ * np.sum((self.w * sol)**2)
return sol
def _prox(self, x, T):
# Gamma is T in the matlab UNLocBox implementation.
gamma = self.lambda_ * T
if self.tight:
sol = np.array(x) + 2. * gamma * self.At(self.y * self.w**2)
sol /= 1. + 2. * gamma * self.nu * self.w**2
else:
raise NotImplementedError('Not implemented for non tight frame.')
return sol
def _grad(self, x):
sol = self.A(np.array(x)) - self.y
return 2 * self.lambda_ * self.w * self.At(sol)