transforms.py
# Copyright 2016 James Hensman, alexggmatthews
#
# 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.
from __future__ import absolute_import
import numpy as np
import tensorflow as tf
import itertools
from . import settings
from .misc import vec_to_tri
from .core.base import ITransform
class Transform(ITransform): # pylint: disable=W0223
def __call__(self, other_transform):
"""
Calling a Transform with another Transform results in a Chain of both.
The following are equivalent:
>>> Chain(t1, t2)
>>> t1(t2)
"""
if not isinstance(other_transform, Transform):
raise TypeError("transforms can only be chained with other transforms: "
"perhaps you want t.forward(x)")
return Chain(self, other_transform)
class Identity(Transform):
"""
The identity transform: y = x
"""
def forward_tensor(self, x):
return tf.identity(x)
def backward_tensor(self, y):
return tf.identity(y)
def forward(self, x):
return x
def backward(self, y):
return y
def log_jacobian_tensor(self, x):
return tf.zeros((1,), settings.float_type)
def __str__(self):
return '(none)'
class Chain(Transform):
"""
Chain two transformations together:
.. math::
y = t_1(t_2(x))
where y is the natural parameter and x is the free state
"""
def __init__(self, t1, t2):
self.t1 = t1
self.t2 = t2
def forward_tensor(self, x):
return self.t1.forward_tensor(self.t2.forward_tensor(x))
def backward_tensor(self, y):
return self.t2.backward_tensor(self.t1.backward_tensor(y))
def forward(self, x):
return self.t1.forward(self.t2.forward(x))
def backward(self, y):
return self.t2.backward(self.t1.backward(y))
def log_jacobian_tensor(self, x):
return self.t1.log_jacobian_tensor(self.t2.forward_tensor(x)) +\
self.t2.log_jacobian_tensor(x)
def __str__(self):
return "{} {}".format(self.t1.__str__(), self.t2.__str__())
class Exp(Transform):
"""
The exponential transform:
y = \exp(x) + \epsilon
x is a free variable, y is always positive. The epsilon value (self.lower)
prevents the optimizer reaching numerical zero.
"""
def __init__(self, lower=1e-6):
self._lower = lower
def forward_tensor(self, x):
return tf.exp(x) + self._lower
def backward_tensor(self, y):
return tf.log(y - self._lower)
def forward(self, x):
return np.exp(x) + self._lower
def backward(self, y):
return np.log(y - self._lower)
def log_jacobian_tensor(self, x):
return tf.reduce_sum(x)
def __str__(self):
return 'Exp'
class Log1pe(Transform):
"""
A transform of the form
.. math::
y = \log(1 + \exp(x))
x is a free variable, y is always positive.
This function is known as 'softplus' in tensorflow.
"""
def __init__(self, lower=1e-6):
"""
lower is a float that defines the minimum value that this transform can
take, default 1e-6. This helps stability during optimization, because
aggressive optimizers can take overly-long steps which lead to zero in
the transformed variable, causing an error.
"""
self._lower = lower
def forward(self, x):
"""
Implementation of softplus. Overflow avoided by use of the logaddexp function.
self._lower is added before returning.
"""
return np.logaddexp(0, x) + self._lower
def forward_tensor(self, x):
return tf.nn.softplus(x) + self._lower
def backward_tensor(self, y):
ys = tf.maximum(y - self._lower, tf.as_dtype(settings.float_type).min)
return ys + tf.log(-tf.expm1(-ys))
def log_jacobian_tensor(self, x):
return tf.negative(tf.reduce_sum(tf.nn.softplus(tf.negative(x))))
def backward(self, y):
"""
Inverse of the softplus transform:
.. math::
x = \log( \exp(y) - 1)
The bound for the input y is [self._lower. inf[, self._lower is
subtracted prior to any calculations. The implementation avoids overflow
explicitly by applying the log sum exp trick:
.. math::
\log ( \exp(y) - \exp(0)) &= ys + \log( \exp(y-ys) - \exp(-ys)) \\
&= ys + \log( 1 - \exp(-ys)
ys = \max(0, y)
As y can not be negative, ys could be replaced with y itself.
However, in case :math:`y=0` this results in np.log(0). Hence the zero is
replaced by a machine epsilon.
.. math::
ys = \max( \epsilon, y)
"""
ys = np.maximum(y - self._lower, np.finfo(settings.float_type).eps)
return ys + np.log(-np.expm1(-ys))
def __str__(self):
return '+ve'
class Logistic(Transform):
"""
The logistic transform, useful for keeping variables constrained between the limits a and b:
.. math::
y = a + (b-a) s(x)
s(x) = 1 / (1 + \exp(-x))
"""
def __init__(self, a=0., b=1.):
if a >= b:
raise ValueError("a must be smaller than b")
self.a, self.b = float(a), float(b)
def forward_tensor(self, x):
ex = tf.exp(-x)
return self.a + (self.b - self.a) / (1. + ex)
def forward(self, x):
ex = np.exp(-x)
return self.a + (self.b - self.a) / (1. + ex)
def backward_tensor(self, y):
return -tf.log((self.b - self.a) / (y - self.a) - 1.)
def backward(self, y):
return -np.log((self.b - self.a) / (y - self.a) - 1.)
def log_jacobian_tensor(self, x):
return tf.reduce_sum(x - 2. * tf.log(tf.exp(x) + 1.) + np.log(self.b - self.a))
def __str__(self):
return "[{}, {}]".format(self.a, self.b)
class Rescale(Transform):
"""
A transform that can linearly rescale parameters:
.. math::
y = factor * x
Use `Chain` to combine this with another transform such as Log1pe:
`Chain(Rescale(), otherTransform())` results in
y = factor * t(x)
`Chain(otherTransform(), Rescale())` results in
y = t(factor * x)
This is useful for avoiding overly large or small scales in optimization/MCMC.
If you want a transform for a positive quantity of a given scale, you want
>>> Rescale(scale)(positive)
"""
def __init__(self, factor=1.0):
self.factor = float(factor)
def forward_tensor(self, x):
return x * self.factor
def forward(self, x):
return x * self.factor
def backward_tensor(self, y):
return y / self.factor
def backward(self, y):
return y / self.factor
def log_jacobian_tensor(self, x):
N = tf.cast(tf.reduce_prod(tf.shape(x)), dtype=settings.float_type)
factor = tf.cast(self.factor, dtype=settings.float_type)
log_factor = tf.log(factor)
return N * log_factor
def __str__(self):
return "{}*".format(self.factor)
class DiagMatrix(Transform):
"""
A transform to represent diagonal matrices.
The output of this transform is a N x dim x dim array of diagonal matrices.
The constructor argument `dim` specifies the size of the matrices.
To make a constraint over positive-definite diagonal matrices, chain this
transform with a positive transform. For example, to get posdef matrices of size 2x2:
t = DiagMatrix(2)(positive)
"""
def __init__(self, dim=1):
self.dim = dim
def forward(self, x):
# create diagonal matrices
m = np.zeros((x.size * self.dim)).reshape(-1, self.dim, self.dim)
x = x.reshape(-1, self.dim)
m[(np.s_[:],) + np.diag_indices(x.shape[1])] = x
return m
def backward(self, y):
# Return diagonals of matrices
if len(y.shape) not in (2, 3) or not (y.shape[-1] == y.shape[-2] == self.dim):
raise ValueError("shape of input does not match this transform")
return y.reshape((-1, self.dim, self.dim)).diagonal(offset=0, axis1=1, axis2=2).flatten()
def backward_tensor(self, y):
reshaped = tf.reshape(y, shape=(-1, self.dim, self.dim))
return tf.reshape(tf.matrix_diag_part(reshaped), shape=[-1])
def forward_tensor(self, x):
# create diagonal matrices
return tf.matrix_diag(tf.reshape(x, (-1, self.dim)))
def log_jacobian_tensor(self, x):
return tf.zeros((1,), settings.float_type)
def __str__(self):
return 'DiagMatrix'
class LowerTriangular(Transform):
"""
A transform of the form
tri_mat = vec_to_tri(x)
x is a free variable, y is always a list of lower triangular matrices sized
(D x N x N).
"""
def __init__(self, N, num_matrices=1, squeeze=False):
"""
Create an instance of LowerTriangular transform.
Args:
N the size of the final lower triangular matrices.
num_matrices: Number of matrices to be stored.
squeeze: If num_matrices == 1, drop the redundant axis.
"""
self.num_matrices = num_matrices # We need to store this for reconstruction.
self.squeeze = squeeze
self.N = N
def _validate_vector_length(self, length):
"""
Check whether the vector length is consistent with being a triangular
matrix and with `self.num_matrices`.
Args:
length: Length of the free state vector.
Returns: Length of the vector with the lower triangular elements.
"""
L = length / self.num_matrices
if int(((L * 8) + 1) ** 0.5) ** 2.0 != (L * 8 + 1):
raise ValueError("The free state must be a triangle number.")
return L
def forward(self, x):
"""
Transforms from the free state to the variable.
Args:
x: Free state vector. Must have length of `self.num_matrices` *
triangular_number.
Returns:
Reconstructed variable.
"""
L = self._validate_vector_length(len(x))
matsize = int((L * 8 + 1) ** 0.5 * 0.5 - 0.5)
xr = np.reshape(x, (self.num_matrices, -1))
var = np.zeros((self.num_matrices, matsize, matsize), settings.float_type)
for i in range(self.num_matrices):
indices = np.tril_indices(matsize, 0)
var[(np.zeros(len(indices[0])).astype(int) + i,) + indices] = xr[i, :]
return var.squeeze() if self.squeeze else var
def backward(self, y):
"""
Transforms from the variable to the free state.
Args:
y: Variable representation.
Returns:
Free state.
"""
N = int(np.sqrt(y.size / self.num_matrices))
reshaped = np.reshape(y, (self.num_matrices, N, N))
# return reshaped[np.tril_indices(N, 0)].T
return np.vstack([reshaped[i, :, :][np.tril_indices(N, 0)] for i in range(len(reshaped))])
def forward_tensor(self, x):
reshaped = tf.reshape(x, (self.num_matrices, -1))
fwd = vec_to_tri(reshaped, self.N)
return tf.squeeze(fwd) if self.squeeze else fwd
def backward_tensor(self, y):
"""
CAVEAT: Requires defined shape and can't work with unknown shape.
"""
size = np.prod(y.shape.as_list())
N = int(np.sqrt(size / self.num_matrices))
reshaped = tf.reshape(y, shape=(self.num_matrices, N, N))
indices = np.array([np.hstack(x) for x in
itertools.product(np.arange(self.num_matrices), np.dstack(np.tril_indices(N))[0])])
triangular = tf.reshape(tf.gather_nd(reshaped, indices), shape=[-1])
return triangular[None, :]
def log_jacobian_tensor(self, x):
return tf.zeros((1,), settings.float_type)
def __str__(self):
return "LoTri->vec"
positive = Log1pe()
def positiveRescale(scale):
"""
The appropriate joint transform for positive parameters of a given `scale`
This is a convenient shorthand for
constrained = scale * log(1 + exp(unconstrained))
"""
return Rescale(scale)(positive)
