# Copyright 2019 United Kingdom Research and Innovation
# Copyright 2019 The University of Manchester
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
#
# Authors:
# CIL Developers, listed at: https://github.com/TomographicImaging/CIL/blob/master/NOTICE.txt
import warnings
from numbers import Number
import numpy as np
from functools import reduce
from cil.utilities.errors import InPlaceError
[docs]
class Function(object):
r""" Abstract class representing a function
Parameters
----------
L: number, positive, default None
Lipschitz constant of the gradient of the function F(x), when it is differentiable.
Note
-----
The Lipschitz of the gradient of the function is a positive real number, such that :math:`\|f'(x) - f'(y)\| \leq L \|x-y\|`, assuming :math:`f: IG \rightarrow \mathbb{R}`
"""
def __init__(self, L=None):
# overrides the type check to allow None as initial value
self._L = L
def __call__(self, x):
raise NotImplementedError
[docs]
def gradient(self, x, out=None):
r"""Returns the value of the gradient of function :math:`F` evaluated at :math:`x`, if it is differentiable
.. math:: F'(x)
Parameters
----------
x : DataContainer
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
--------
DataContainer, the value of the gradient of the function at x.
"""
raise NotImplementedError
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the proximal operator of function :math:`\tau F` evaluated at x
.. math:: \text{prox}_{\tau F}(x) = \underset{z}{\text{argmin}} \frac{1}{2}\|z - x\|^{2} + \tau F(z)
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the proximal operator of the function at x with scalar :math:`\tau`.
"""
raise NotImplementedError
[docs]
def convex_conjugate(self, x):
r""" Evaluation of the function F* at x, where F* is the convex conjugate of function F,
.. math:: F^{*}(x^{*}) = \underset{x}{\sup} \langle x^{*}, x \rangle - F(x)
Parameters
----------
x : DataContainer
Returns
-------
The value of the convex conjugate of the function at x.
"""
raise NotImplementedError
[docs]
def proximal_conjugate(self, x, tau, out=None):
r"""Returns the proximal operator of the convex conjugate of function :math:`\tau F` evaluated at :math:`x^{*}`
.. math:: \text{prox}_{\tau F^{*}}(x^{*}) = \underset{z^{*}}{\text{argmin}} \frac{1}{2}\|z^{*} - x^{*}\|^{2} + \tau F^{*}(z^{*})
Due to Moreau’s identity, we have an analytic formula to compute the proximal operator of the convex conjugate :math:`F^{*}`
.. math:: \text{prox}_{\tau F^{*}}(x) = x - \tau\text{prox}_{\tau^{-1} F}(\tau^{-1}x)
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the value of the proximal operator of the convex conjugate at point :math:`x` for scalar :math:`\tau` or None if `out`.
"""
if id(x) == id(out):
raise InPlaceError(
message="The proximal_conjugate of a CIL function cannot be used in place")
try:
tmp = x
x.divide(tau, out=tmp)
except TypeError:
tmp = x.divide(tau, dtype=np.float32)
val = self.proximal(tmp, 1.0/tau, out=out)
if id(tmp) == id(x):
x.multiply(tau, out=x)
val.sapyb(-tau, x, 1.0, out=val)
return val
# Algebra for Function Class
# Add functions
# Subtract functions
# Add/Substract with Scalar
# Multiply with Scalar
def __add__(self, other):
""" Returns the sum of the functions.
Cases: a) the sum of two functions :math:`(F_{1}+F_{2})(x) = F_{1}(x) + F_{2}(x)`
b) the sum of a function with a scalar :math:`(F_{1}+scalar)(x) = F_{1}(x) + scalar`
"""
if isinstance(other, Number):
return SumScalarFunction(self, other)
return SumFunction(self, other)
def __radd__(self, other):
""" Making addition commutative. """
return self + other
def __sub__(self, other):
""" Returns the subtraction of the functions."""
return self + (-1) * other
def __rmul__(self, scalar):
"""Returns a function multiplied by a scalar."""
return ScaledFunction(self, scalar)
def __mul__(self, scalar):
return self.__rmul__(scalar)
def __neg__(self):
""" Return the negative of the function """
return -1 * self
[docs]
def centered_at(self, center):
""" Returns a translated function, namely if we have a function :math:`F(x)` the center is at the origin.
TranslateFunction is :math:`F(x - b)` and the center is at point b.
Parameters
----------
center: DataContainer
The point to center the function at.
Returns
-------
The translated function.
"""
if center is None:
return self
else:
return TranslateFunction(self, center)
@property
def L(self):
r'''Lipschitz of the gradient of function f.
L is positive real number, such that :math:`\|f'(x) - f'(y)\| \leq L\|x-y\|`, assuming :math:`f: IG \rightarrow \mathbb{R}`'''
return self._L
# return self._L
@L.setter
def L(self, value):
'''Setter for Lipschitz constant'''
if isinstance(value, (Number,)) and value >= 0:
self._L = value
else:
raise TypeError('The Lipschitz constant is a real positive number')
[docs]
class SumFunction(Function):
r"""SumFunction represents the sum of :math:`n\geq2` functions
.. math:: (F_{1} + F_{2} + ... + F_{n})(\cdot) = F_{1}(\cdot) + F_{2}(\cdot) + ... + F_{n}(\cdot)
Parameters
----------
*functions : Functions
Functions to set up a :class:`.SumFunction`
Raises
------
ValueError
If the number of function is strictly less than 2.
Examples
--------
.. math:: F(x) = \|x\|^{2} + \frac{1}{2}\|x - 1\|^{2}
>>> from cil.optimisation.functions import L2NormSquared
>>> from cil.framework import ImageGeometry
>>> f1 = L2NormSquared()
>>> f2 = 0.5 * L2NormSquared(b = ig.allocate(1))
>>> F = SumFunction(f1, f2)
.. math:: F(x) = \sum_{i=1}^{50} \|x - i\|^{2}
>>> F = SumFunction(*[L2NormSquared(b=i) for i in range(50)])
"""
def __init__(self, *functions):
super(SumFunction, self).__init__()
if not len(functions):
raise IndexError('At least 1 function needed')
self.functions = functions
@property
def L(self):
"""Returns the Lipschitz constant for the SumFunction
.. math:: L = \sum_{i} L_{i}
where :math:`L_{i}` is the Lipschitz constant of the smooth function :math:`F_{i}`.
"""
L = 0.
for f in self.functions:
if f.L is not None:
L += f.L
else:
L = None
break
self._L = L
return self._L
@L.setter
def L(self, value):
# call base class setter
super(SumFunction, self.__class__).L.fset(self, value)
@property
def Lmax(self):
"""Returns the maximum Lipschitz constant for the SumFunction
.. math:: L = \max_{i}\{L_{i}\}
where :math:`L_{i}` is the Lipschitz constant of the smooth function :math:`F_{i}`.
"""
l = []
for f in self.functions:
if f.L is not None:
l.append(f.L)
else:
l = None
break
self._Lmax = max(l)
return self._Lmax
@Lmax.setter
def Lmax(self, value):
# call base class setter
super(SumFunction, self.__class__).Lmax.fset(self, value)
def __call__(self, x):
r"""Returns the value of the sum of functions evaluated at :math:`x`.
.. math:: (F_{1} + F_{2} + ... + F_{n})(x) = F_{1}(x) + F_{2}(x) + ... + F_{n}(x)
"""
ret = 0.
for f in self.functions:
ret += f(x)
return ret
[docs]
def gradient(self, x, out=None):
r"""Returns the value of the sum of the gradient of functions evaluated at :math:`x`, if all of them are differentiable.
.. math:: (F'_{1} + F'_{2} + ... + F'_{n})(x) = F'_{1}(x) + F'_{2}(x) + ... + F'_{n}(x)
Parameters
----------
x : DataContainer
Point to evaluate the gradient at.
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the value of the sum of the gradients evaluated at point :math:`x`.
"""
if out is not None and id(x) == id(out):
raise InPlaceError
for i, f in enumerate(self.functions):
if i == 0:
ret = f.gradient(x, out=out)
else:
ret += f.gradient(x)
return ret
def __add__(self, other):
""" Addition for the SumFunction.
* :code:`SumFunction` + :code:`SumFunction` is a :code:`SumFunction`.
* :code:`SumFunction` + :code:`Function` is a :code:`SumFunction`.
"""
if isinstance(other, SumFunction):
functions = list(self.functions) + list(other.functions)
return SumFunction(*functions)
elif isinstance(other, Function):
functions = list(self.functions)
functions.append(other)
return SumFunction(*functions)
else:
return super(SumFunction, self).__add__(other)
@property
def num_functions(self):
return len(self.functions)
[docs]
class ScaledFunction(Function):
r""" ScaledFunction represents the scalar multiplication with a Function.
Let a function F then and a scalar :math:`\alpha`.
If :math:`G(x) = \alpha F(x)` then:
1. :math:`G(x) = \alpha F(x)` ( __call__ method )
2. :math:`G'(x) = \alpha F'(x)` ( gradient method )
3. :math:`G^{*}(x^{*}) = \alpha F^{*}(\frac{x^{*}}{\alpha})` ( convex_conjugate method )
4. :math:`\text{prox}_{\tau G}(x) = \text{prox}_{(\tau\alpha) F}(x)` ( proximal method )
"""
def __init__(self, function, scalar):
super(ScaledFunction, self).__init__()
if not isinstance(scalar, Number):
raise TypeError('expected scalar: got {}'.format(type(scalar)))
self.scalar = scalar
self.function = function
@property
def L(self):
if self._L is None:
if self.function.L is not None:
self._L = abs(self.scalar) * self.function.L
else:
self._L = None
return self._L
@L.setter
def L(self, value):
# call base class setter
super(ScaledFunction, self.__class__).L.fset(self, value)
@property
def scalar(self):
return self._scalar
@scalar.setter
def scalar(self, value):
if isinstance(value, (Number, )):
self._scalar = value
else:
raise TypeError(
'Expecting scalar type as a number type. Got {}'.format(type(value)))
def __call__(self, x):
r"""Returns the value of the scaled function evaluated at :math:`x`.
.. math:: G(x) = \alpha F(x)
Parameters
----------
x : DataContainer
Returns
--------
DataContainer, the value of the scaled function.
"""
return self.scalar * self.function(x)
[docs]
def convex_conjugate(self, x):
r"""Returns the convex conjugate of the scaled function.
.. math:: G^{*}(x^{*}) = \alpha F^{*}(\frac{x^{*}}{\alpha})
Parameters
----------
x : DataContainer
Returns
-------
The value of the convex conjugate of the scaled function.
"""
try:
x.divide(self.scalar, out=x)
tmp = x
except TypeError:
tmp = x.divide(self.scalar, dtype=np.float32)
val = self.function.convex_conjugate(tmp)
if id(tmp) == id(x):
x.multiply(self.scalar, out=x)
return self.scalar * val
[docs]
def gradient(self, x, out=None):
r"""Returns the gradient of the scaled function evaluated at :math:`x`.
.. math:: G'(x) = \alpha F'(x)
Parameters
----------
x : DataContainer
Point to evaluate the gradient at.
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the value of the gradient of the scaled function evaluated at :math:`x`.
"""
res = self.function.gradient(x, out=out)
res *= self.scalar
return res
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the proximal operator of the scaled function, evaluated at :math:`x`.
.. math:: \text{prox}_{\tau G}(x) = \text{prox}_{(\tau\alpha) F}(x)
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the proximal operator of the scaled function evaluated at :math:`x` with scalar :math:`\tau`.
"""
return self.function.proximal(x, tau*self.scalar, out=out)
[docs]
def proximal_conjugate(self, x, tau, out=None):
r"""This returns the proximal conjugate operator for the function at :math:`x`, :math:`\tau`
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the proximal conjugate operator for the function evaluated at :math:`x` and :math:`\tau`.
"""
if out is not None and id(x) == id(out):
raise InPlaceError
try:
tmp = x
x.divide(tau, out=tmp)
except TypeError:
tmp = x.divide(tau, dtype=np.float32)
val = self.function.proximal(tmp, self.scalar/tau, out=out)
if id(tmp) == id(x):
x.multiply(tau, out=x)
val.sapyb(-tau, x, 1.0, out=val)
return val
[docs]
class SumScalarFunction(SumFunction):
""" SumScalarFunction represents the sum a function with a scalar.
.. math:: (F + scalar)(x) = F(x) + scalar
Although SumFunction has no general expressions for
i) convex_conjugate
ii) proximal
iii) proximal_conjugate
if the second argument is a ConstantFunction then we can derive the above analytically.
"""
def __init__(self, function, constant):
super(SumScalarFunction, self).__init__(
function, ConstantFunction(constant))
self.constant = constant
self.function = function
[docs]
def convex_conjugate(self, x):
r""" Returns the convex conjugate of a :math:`(F+scalar)`, evaluated at :math:`x`.
.. math:: (F+scalar)^{*}(x^{*}) = F^{*}(x^{*}) - scalar
Parameters
----------
x : DataContainer
Returns
-------
The value of the convex conjugate evaluated at :math:`x`.
"""
return self.function.convex_conjugate(x) - self.constant
[docs]
def proximal(self, x, tau, out=None):
""" Returns the proximal operator of :math:`F+scalar`
.. math:: \text{prox}_{\tau (F+scalar)}(x) = \text{prox}_{\tau F}
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the evaluation of the proximal operator evaluated at :math:`x` and :math:`\tau`.
"""
return self.function.proximal(x, tau, out=out)
@property
def L(self):
if self._L is None:
if self.function.L is not None:
self._L = self.function.L
else:
self._L = None
return self._L
@L.setter
def L(self, value):
# call base class setter
super(SumScalarFunction, self.__class__).L.fset(self, value)
[docs]
class ConstantFunction(Function):
r""" ConstantFunction: :math:`F(x) = constant, constant\in\mathbb{R}`
"""
def __init__(self, constant=0):
self.constant = constant
super(ConstantFunction, self).__init__(L=1)
def __call__(self, x):
""" Returns the value of the function, :math:`F(x) = constant`"""
return self.constant
[docs]
def gradient(self, x, out=None):
""" Returns the value of the gradient of the function, :math:`F'(x)=0`
Parameters
----------
x : DataContainer
Point to evaluate the gradient at.
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
A DataContainer of zeros, the same size as :math:`x`.
"""
if out is None:
return x * 0.
else:
out.fill(0)
return out
[docs]
def convex_conjugate(self, x):
r""" The convex conjugate of constant function :math:`F(x) = c\in\mathbb{R}` is
.. math::
F(x^{*})
=
\begin{cases}
-c, & if x^{*} = 0\\
\infty, & \mbox{otherwise}
\end{cases}
However, :math:`x^{*} = 0` only in the limit of iterations, so in fact this can be infinity.
We do not want to have inf values in the convex conjugate, so we have to penalise this value accordingly.
The following penalisation is useful in the PDHG algorithm, when we compute primal & dual objectives
for convergence purposes.
.. math:: F^{*}(x^{*}) = \sum \max\{x^{*}, 0\}
Parameters
----------
x : DataContainer
Returns
-------
The maximum of x and 0, summed over the entries of x.
"""
return x.maximum(0).sum()
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the proximal operator of the constant function, which is the same element, i.e.,
.. math:: \text{prox}_{\tau F}(x) = x
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, equal to :math:`x`.
"""
if out is None:
return x.copy()
else:
out.fill(x)
return out
@property
def constant(self):
return self._constant
@constant.setter
def constant(self, value):
if not isinstance(value, Number):
raise TypeError('expected scalar: got {}'.format(type(value)))
self._constant = value
@property
def L(self):
return 1.
def __rmul__(self, other):
'''defines the right multiplication with a number'''
if not isinstance(other, Number):
raise NotImplemented
constant = self.constant * other
return ConstantFunction(constant)
[docs]
class ZeroFunction(ConstantFunction):
""" ZeroFunction represents the zero function, :math:`F(x) = 0`
"""
def __init__(self):
super(ZeroFunction, self).__init__(constant=0.)
[docs]
class TranslateFunction(Function):
r""" TranslateFunction represents the translation of function F with respect to the center b.
Let a function F and consider :math:`G(x) = F(x - center)`.
Function F is centered at 0, whereas G is centered at point b.
If :math:`G(x) = F(x - b)` then:
1. :math:`G(x) = F(x - b)` ( __call__ method )
2. :math:`G'(x) = F'(x - b)` ( gradient method )
3. :math:`G^{*}(x^{*}) = F^{*}(x^{*}) + <x^{*}, b >` ( convex_conjugate method )
4. :math:`\text{prox}_{\tau G}(x) = \text{prox}_{\tau F}(x - b) + b` ( proximal method )
"""
def __init__(self, function, center):
try:
L = function.L
except NotImplementedError as nie:
L = None
super(TranslateFunction, self).__init__(L=L)
self.function = function
self.center = center
def __call__(self, x):
r"""Returns the value of the translated function.
.. math:: G(x) = F(x - b)
Parameters
----------
x : DataContainer
Returns
-------
The value of the translated function evaluated at :math:`x`.
"""
try:
x.subtract(self.center, out=x)
tmp = x
except TypeError:
tmp = x.subtract(self.center, dtype=np.float32)
val = self.function(tmp)
if id(tmp) == id(x):
x.add(self.center, out=x)
return val
[docs]
def gradient(self, x, out=None):
r"""Returns the gradient of the translated function.
.. math:: G'(x) = F'(x - b)
Parameters
----------
x : DataContainer
Point to evaluate the gradient at.
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the gradient of the translated function evaluated at :math:`x`.
"""
if id(x) == id(out):
raise InPlaceError
try:
x.subtract(self.center, out=x)
tmp = x
except TypeError:
tmp = x.subtract(self.center, dtype=np.float32)
val = self.function.gradient(tmp, out=out)
if id(tmp) == id(x):
x.add(self.center, out=x)
return val
[docs]
def proximal(self, x, tau, out=None):
r"""Returns the proximal operator of the translated function.
.. math:: \text{prox}_{\tau G}(x) = \text{prox}_{\tau F}(x-b) + b
Parameters
----------
x : DataContainer
tau: scalar
out: return DataContainer, if None a new DataContainer is returned, default None.
Returns
-------
DataContainer, the proximal operator of the translated function at :math:`x` and :math:`\tau`.
"""
if id(x) == id(out):
raise InPlaceError
try:
x.subtract(self.center, out=x)
tmp = x
except TypeError:
tmp = x.subtract(self.center, dtype=np.float32)
val = self.function.proximal(tmp, tau, out=out)
val.add(self.center, out=val)
if id(tmp) == id(x):
x.add(self.center, out=x)
return val
[docs]
def convex_conjugate(self, x):
r"""Returns the convex conjugate of the translated function.
.. math:: G^{*}(x^{*}) = F^{*}(x^{*}) + <x^{*}, b >
Parameters
----------
x : DataContainer
Returns
-------
The value of the convex conjugate of the translated function at :math:`x`.
"""
return self.function.convex_conjugate(x) + self.center.dot(x)
