# Source code for picos.expressions.exp_entropy

# ------------------------------------------------------------------------------
# Copyright (C) 2019 Maximilian Stahlberg
# Based on the original picos.expressions module by Guillaume Sagnol.
#
# This file is part of PICOS.
#
# PICOS is free software: you can redistribute it and/or modify it under the
# Foundation, either version 3 of the License, or (at your option) any later
# version.
#
# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY
# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
# A PARTICULAR PURPOSE.  See the GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License along with
# this program.  If not, see <http://www.gnu.org/licenses/>.
# ------------------------------------------------------------------------------

"""Implements :class:Entropy and :class:NegativeEntropy."""

# TODO: Common base class for Entropy and NegativeEntropy.

import math
import operator
from collections import namedtuple

import cvxopt
import numpy

from .. import glyphs
from ..apidoc import api_end, api_start
from ..caching import cached_selfinverse_unary_operator, cached_unary_operator
from ..constraints import KullbackLeiblerConstraint
from .data import convert_and_refine_arguments, convert_operands, cvx2np
from .exp_affine import AffineExpression
from .expression import Expression, refine_operands, validate_prediction

_API_START = api_start(globals())
# -------------------------------

[docs]class Entropy(Expression):
r"""Entropy or negative relative entropy of an affine expression.

Negative relative entropy is also known as the perspective of the logarithm.

:Definition:

Let :math:x be an :math:n-dimensional real affine expression.

1.  If no additional expression :math:y is given, this is the entropy

.. math::

-\sum_{i = 1}^n \operatorname{vec}(x)_i
\log(\operatorname{vec}(x)_i).

2.  If an additional affine expression :math:y of same shape as :math:x
is given, this is the negative relative entropy (or logarithmic
perspective)

.. math::

-\sum_{i = 1}^n \operatorname{vec}(x)_i
\log\left(
\frac{\operatorname{vec}(x)_i}{\operatorname{vec}(y)_i}
\right)
&= -\sum_{i = 1}^n \operatorname{vec}(x)_i
\left[
\log(\operatorname{vec}(x)_i) - \log(\operatorname{vec}(y)_i)
\right] \\
&= \sum_{i = 1}^n \operatorname{vec}(x)_i
\left[
\log(\operatorname{vec}(y)_i) - \log(\operatorname{vec}(x)_i
\right] \\
&= \sum_{i = 1}^n \operatorname{vec}(x)_i
\log\left(
\frac{\operatorname{vec}(y)_i}{\operatorname{vec}(x)_i}
\right).

.. warning::

When you pose a lower bound on this expression, then PICOS enforces
:math:x \geq 0 through an auxiliary constraint during solution search.
When an additional expression :math:y is given, PICOS enforces
:math:y \geq 0 as well.
"""

# --------------------------------------------------------------------------
# Initialization and factory methods.
# --------------------------------------------------------------------------

[docs]    @convert_and_refine_arguments("x", "y", allowNone=True)
def __init__(self, x, y=None):
"""Construct an :class:Entropy.

:param x: The affine expression :math:x.
:type x: ~picos.expressions.AffineExpression
:param y: An additional affine expression :math:y. If necessary, PICOS
will attempt to reshape or broadcast it to the shape of :math:x.
:type y: ~picos.expressions.AffineExpression
"""
if not isinstance(x, AffineExpression):
raise TypeError("Can only take the elementwise logarithm of a real "
"affine expression, not of {}.".format(type(x).__name__))

if y is not None:
if not isinstance(y, AffineExpression):
raise TypeError("The additional parameter y must be a real "
"affine expression, not {}.".format(type(x).__name__))
elif x.shape != y.shape:

if y.is1:
y = None

self._x = x
self._y = y

if y is None:
typeStr = "Entropy"
if len(x) == 1:
symbStr = glyphs.neg(glyphs.mul(x.string, glyphs.log(x.string)))
else:
symbStr = glyphs.neg(
glyphs.dotp(x.string, glyphs.log(x.string)))
else:
typeStr = "Logarithmic Perspective"
if len(x) == 1:
symbStr = glyphs.mul(
x.string, glyphs.log(glyphs.div(y.string, x.string)))
else:
symbStr = glyphs.dotp(x.string,
glyphs.sub(glyphs.log(y.string), glyphs.log(x.string)))

Expression.__init__(self, typeStr, symbStr)

# --------------------------------------------------------------------------
# Abstract method implementations and method overridings, except _predict.
# --------------------------------------------------------------------------

def _get_refined(self):
if self._x.constant and (self._y is None or self._y.constant):
return AffineExpression.from_constant(self.value, 1, self._symbStr)
else:
return self

Subtype = namedtuple("Subtype", ("argdim", "y"))

def _get_subtype(self):
return self.Subtype(len(self._x), self._y is not None)

def _get_value(self):
x = numpy.ravel(cvx2np(self._x._get_value()))

if self._y is None:
s = -numpy.dot(x, numpy.log(x))
else:
y = numpy.ravel(cvx2np(self._y._get_value()))
s = numpy.dot(x, numpy.log(y / x))

return cvxopt.matrix(s)

@cached_unary_operator
def _get_mutables(self):
if self._y is None:
return self._x._get_mutables()
else:
return self._x._get_mutables().union(self._y.mutables)

def _is_convex(self):
return False

def _is_concave(self):
return True

def _replace_mutables(self, mapping):
return self.__class__(self._x._replace_mutables(mapping),
None if self._y is None else self._y._replace_mutables(mapping))

def _freeze_mutables(self, freeze):
return self.__class__(self._x._freeze_mutables(freeze),
None if self._y is None else self._y._freeze_mutables(freeze))

# --------------------------------------------------------------------------
# Python special method implementations, except constraint-creating ones.
# --------------------------------------------------------------------------

[docs]    @cached_selfinverse_unary_operator
def __neg__(self):
return NegativeEntropy(self._x, self._y)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
other_str = other.string

if isinstance(other, AffineExpression):
if other.is0:
return self

other = Entropy(other, other * math.e)

if isinstance(other, Entropy):
if self._y is None and other._y is None:
entropy = Entropy(self._x.vec // other._x.vec)
elif self._y is not None and other._y is None:
one = AffineExpression.from_constant(1.0, (other.n, 1))
entropy = Entropy(
self._x.vec // other._x.vec, self._y.vec // one)
elif self._y is None and other._y is not None:
one = AffineExpression.from_constant(1.0, (self.n, 1))
entropy = Entropy(
self._x.vec // other._x.vec, one // other._y.vec)
else:
entropy = Entropy(
self._x.vec // other._x.vec, self._y.vec // other._y.vec)

return entropy

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
if isinstance(other, (AffineExpression, Entropy)):

if entropy is not self:

return entropy

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __sub__(self, other):
if isinstance(other, (AffineExpression, NegativeEntropy)):
return self + (-other)

return Expression.__sub__(self, other)

# --------------------------------------------------------------------------
# Methods and properties that return expressions.
# --------------------------------------------------------------------------

@property
def x(self):
"""The expression :math:x."""
return self._x

@property
def y(self):
"""The additional expression :math:y, or :obj:None."""
return self._y

# --------------------------------------------------------------------------
# Methods and properties that describe the expression.
# --------------------------------------------------------------------------

@property
def n(self):
"""Length of :attr:x."""
return len(self._x)

# --------------------------------------------------------------------------
# Constraint-creating operators, and _predict.
# --------------------------------------------------------------------------

@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)

if relation == operator.__ge__:
if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return KullbackLeiblerConstraint.make_type(
argdim=subtype.argdim)

return NotImplemented

[docs]    @convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __ge__(self, other):
if isinstance(other, AffineExpression):
return KullbackLeiblerConstraint(-self, -other)
else:
return NotImplemented

[docs]class NegativeEntropy(Expression):
r"""Negative or relative entropy of an affine expression.

Relative entropy is also known as the Kullback-Leibler divergence.

:Definition:

Let :math:x be an :math:n-dimensional real affine expression.

1.  If no additional expression :math:y is given, this is the negative
entropy

.. math::

\sum_{i = 1}^n \operatorname{vec}(x)_i
\log(\operatorname{vec}(x)_i).

2.  If an additional affine expression :math:y of same shape as :math:x
is given, this is the relative entropy (or Kullback-Leibler divergence)

.. math::

\sum_{i = 1}^n \operatorname{vec}(x)_i
\log\left(
\frac{\operatorname{vec}(x)_i}{\operatorname{vec}(y)_i}
\right)
= \sum_{i = 1}^n \operatorname{vec}(x)_i
\left[
\log(\operatorname{vec}(x)_i) - \log(\operatorname{vec}(y)_i)
\right].

.. warning::

When you pose an upper bound on this expression, then PICOS enforces
:math:x \geq 0 through an auxiliary constraint during solution search.
When an additional expression :math:y is given, PICOS enforces
:math:y \geq 0 as well.
"""

# --------------------------------------------------------------------------
# Initialization and factory methods.
# --------------------------------------------------------------------------

[docs]    @convert_and_refine_arguments("x", "y", allowNone=True)
def __init__(self, x, y=None):
"""Construct a :class:NegativeEntropy.

:param x: The affine expression :math:x.
:type x: ~picos.expressions.AffineExpression
:param y: An additional affine expression :math:y. If necessary, PICOS
will attempt to reshape or broadcast it to the shape of :math:x.
:type y: ~picos.expressions.AffineExpression
"""
if not isinstance(x, AffineExpression):
raise TypeError("Can only take the elementwise logarithm of a real "
"affine expression, not of {}.".format(type(x).__name__))

if y is not None:
if not isinstance(y, AffineExpression):
raise TypeError("The additional parameter y must be a real "
"affine expression, not {}.".format(type(x).__name__))
elif x.shape != y.shape:

if y.is1:
y = None

self._x = x
self._y = y

if y is None:
typeStr = "Negative Entropy"
if len(x) == 1:
symbStr = glyphs.mul(x.string, glyphs.log(x.string))
else:
symbStr = glyphs.dotp(x.string, glyphs.log(x.string))
else:
typeStr = "Relative Entropy"
if len(x) == 1:
symbStr = glyphs.mul(
x.string, glyphs.log(glyphs.div(x.string, y.string)))
else:
symbStr = glyphs.dotp(x.string,
glyphs.sub(glyphs.log(x.string), glyphs.log(y.string)))

Expression.__init__(self, typeStr, symbStr)

# --------------------------------------------------------------------------
# Abstract method implementations and method overridings, except _predict.
# --------------------------------------------------------------------------

def _get_refined(self):
if self._x.constant and (self._y is None or self._y.constant):
return AffineExpression.from_constant(self.value, 1, self._symbStr)
else:
return self

Subtype = namedtuple("Subtype", ("argdim", "y"))

def _get_subtype(self):
return self.Subtype(len(self._x), self._y is not None)

def _get_value(self):
x = numpy.ravel(cvx2np(self._x._get_value()))

if self._y is None:
s = numpy.dot(x, numpy.log(x))
else:
y = numpy.ravel(cvx2np(self._y._get_value()))
s = numpy.dot(x, numpy.log(x / y))

return cvxopt.matrix(s)

@cached_unary_operator
def _get_mutables(self):
if self._y is None:
return self._x._get_mutables()
else:
return self._x._get_mutables().union(self._y.mutables)

def _is_convex(self):
return True

def _is_concave(self):
return False

def _replace_mutables(self, mapping):
return self.__class__(self._x._replace_mutables(mapping),
None if self._y is None else self._y._replace_mutables(mapping))

def _freeze_mutables(self, freeze):
return self.__class__(self._x._freeze_mutables(freeze),
None if self._y is None else self._y._freeze_mutables(freeze))

# --------------------------------------------------------------------------
# Python special method implementations, except constraint-creating ones.
# --------------------------------------------------------------------------

[docs]    @cached_selfinverse_unary_operator
def __neg__(self):
return Entropy(self._x, self._y)

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
other_str = other.string

if isinstance(other, AffineExpression):
if other.is0:
return self

other = NegativeEntropy(other, other / math.e)

if isinstance(other, NegativeEntropy):
if self._y is None and other._y is None:
negent = NegativeEntropy(self._x.vec // other._x.vec)
elif self._y is not None and other._y is None:
one = AffineExpression.from_constant(1.0, (other.n, 1))
negent = NegativeEntropy(
self._x.vec // other._x.vec, self._y.vec // one)
elif self._y is None and other._y is not None:
one = AffineExpression.from_constant(1.0, (self.n, 1))
negent = NegativeEntropy(
self._x.vec // other._x.vec, one // other._y.vec)
else:
negent = NegativeEntropy(
self._x.vec // other._x.vec, self._y.vec // other._y.vec)

return negent

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
if isinstance(other, (AffineExpression, NegativeEntropy)):

if negent is not self:

return negent

[docs]    @convert_operands(scalarRHS=True)
@refine_operands()
def __sub__(self, other):
if isinstance(other, (AffineExpression, Entropy)):
return self + (-other)

return Expression.__sub__(self, other)

# --------------------------------------------------------------------------
# Methods and properties that return expressions.
# --------------------------------------------------------------------------

@property
def x(self):
"""The expression :math:x."""
return self._x

@property
def y(self):
"""The additional expression :math:y, or :obj:None."""
return self._y

# --------------------------------------------------------------------------
# Methods and properties that describe the expression.
# --------------------------------------------------------------------------

@property
def n(self):
"""Length of :attr:x."""
return len(self._x)

# --------------------------------------------------------------------------
# Constraint-creating operators, and _predict.
# --------------------------------------------------------------------------

@classmethod
def _predict(cls, subtype, relation, other):
assert isinstance(subtype, cls.Subtype)

if relation == operator.__le__:
if issubclass(other.clstype, AffineExpression) \
and other.subtype.dim == 1:
return KullbackLeiblerConstraint.make_type(
argdim=subtype.argdim)

return NotImplemented

[docs]    @convert_operands(scalarRHS=True)
@validate_prediction
@refine_operands()
def __le__(self, other):
if isinstance(other, AffineExpression):
return KullbackLeiblerConstraint(self, other)
else:
return NotImplemented

# --------------------------------------
__all__ = api_end(_API_START, globals())