tfp.bijectors.Glow

Implements the Glow Bijector from Kingma & Dhariwal (2018)[1].

Inherits From: Chain, Composition, Bijector

Overview: Glow is a chain of bijectors which transforms a rank-1 tensor (vector) into a rank-3 tensor (e.g. an RGB image). Glow does this by chaining together an alternating series of "Blocks," "Squeezes," and "Exits" which are each themselves special chains of other bijectors. The intended use of Glow is as part of a tfp.distributions.TransformedDistribution, in which the base distribution over the vector space is used to generate samples in the image space. In the paper, an Independent Normal distribution is used as the base distribution.

A "Block" (implemented as the GlowBlock Bijector) performs much of the transformations which allow glow to produce sophisticated and complex mappings between the image space and the latent space and therefore achieve rich image generation performance. A Block is composed of num_steps_per_block steps, which are each implemented as a Chain containing an ActivationNormalization (ActNorm) bijector, followed by an (invertible) OneByOneConv bijector, and finally a coupling bijector. The coupling bijector is an instance of a RealNVP bijector, and uses the coupling_bijector_fn function to instantiate the coupling bijector function which is given to the RealNVP. This function returns a bijector which defines the coupling (e.g. Shift(Scale) for affine coupling or Shift for additive coupling).

A "Squeeze" converts spatial features into channel features. It is implemented using the Expand bijector. The difference in names is due to the fact that the forward function from glow is meant to ultimately correspond to sampling from a tfp.util.TransformedDistribution object, which would use Expand (Squeeze is just Invert(Expand)). The Expand bijector takes a tensor with shape [H, W, C] and returns a tensor with shape [2H, 2W, C / 4], such that each 2x2x1 spatial tile in the output is composed from a single 1x1x4 tile in the input tensor, as depicted in the figure below.

                     Forward pass (Expand)
                    ______        __________
                    \     \       \    \    \
                    \\     \ ----> \  1 \  2 \
                    \\\__1__\       \____\____\
                    \\\__2__\        \    \    \
                     \\__3__\  <----  \  3 \  4 \
                      \__4__\          \____\____\
                         Inverse pass (Squeeze)

This is implemented using a chain of Reshape -> Transpose -> Reshape bijectors. Note that on an inverse pass through the bijector, each Squeeze will cause the width/height of the image to decrease by a factor of 2. Therefore, the input image must be evenly divisible by 2 at least num_glow_blocks times, since it will pass through a Squeeze step that many times.

An "Exit" is simply a junction at which some of the tensor "exits" from the glow bijector and therefore avoids any further alteration. Each exit is implemented as a Blockwise bijector, where some channels are given to the rest of the glow model, and the rest are given to a bypass implemented using the Identity bijector. The fraction of channels to be removed at each exit is determined by the grab_after_block arg, indicates the fraction of remaining channels which join the identity bypass. The fraction is converted to an integer number of channels by multiplying by the remaining number of channels and rounding.

Additionally, at each exit, glow couples the tensor exiting the highway to the tensor continuing onward. This makes small scale features in the image dependent on larger scale features, since the larger scale features dictate the mean and scale of the distribution over the smaller scale features. This coupling is done similarly to the Coupling bijector in each step of the flow (i.e. using a RealNVP bijector). However for the exit bijector, the coupling is instantiated using exit_bijector_fn rather than coupling bijector fn, allowing for different behaviors between standard coupling and exit coupling. Also note that because the exit utilizes a coupling bijector, there are two special cases (all channels exiting and no channels exiting).

The full Glow bijector consists of num_glow_blocks Blocks each of which contains num_steps_per_block steps. Each step implements a coupling using bijector_coupling_fn. Between blocks, glow converts between spatial pixels and channels using the Expand Bijector, and splits channels out of the bijector using the Exit Bijector. The channels which have exited continue onward through Identity bijectors and those which have not exited are given to the next block. After passing through all Blocks, the tensor is reshaped to a rank-1 tensor with the same number of elements. This is where the distribution will be defined.

A schematic diagram of Glow is shown below. The forward function of the bijector starts from the bottom and goes upward, while the inverse function starts from the top and proceeds downward.

==============================================================================
                         Glow Schematic Diagram

Input Image     ########################   shape = [H, W, C]

                \                      /<- Expand Bijector turns spatial
                 \                    /    dimensions into channels.
                _
               |  XXXXXXXXXXXXXXXXXXXX
               |  XXXXXXXXXXXXXXXXXXXX
               |  XXXXXXXXXXXXXXXXXXXX     A single step of the flow consists
 Glow Block  - |  XXXXXXXXXXXXXXXXXXXX  <- of ActNorm -> 1x1Conv -> Coupling.
               |  XXXXXXXXXXXXXXXXXXXX     there are num_steps_per_block
               |  XXXXXXXXXXXXXXXXXXXX     steps of the flow in each block.
               |_ XXXXXXXXXXXXXXXXXXXX

                  \                  / <-- Expand bijectors follow each glow
                   \                /      block

                    XXXXXXXX\\\\\\\\   <-- Exit Bijector removes channels
                _                    _     from additional alteration.
               |    XXXXXXXX !  |  !
               |    XXXXXXXX !  |  !
               |    XXXXXXXX !  |  !       After exiting, channels are passed
 Glow Block  - |    XXXXXXXX !  |  !  <--- downward using the Blockwise and
               |    XXXXXXXX !  |  !       Identify bijectors.
               |    XXXXXXXX !  |  !
               |_   XXXXXXXX !  |  !

                    \              / <---- Expand Bijector
                     \            /

                      XXX\\\    | !  <---- Exit Bijector
                _
               |      XXX ! |   | !
               |      XXX ! |   | !
               |      XXX ! |   | !
 Glow Block  - |      XXX ! |   | !
               |      XXX ! |   | !
               |      XXX ! |   | !
               |_     XXX ! |   | !

                      XX\ ! |   | ! <----- (Optional) Exit Bijector

                       |    |   |
                       v    v   v
Output Distribution    ##########          shape = [H * W * C]
                                                   _________________________
                                                  |         Legend          |
                                                  | XX  = Step of flow      |
                                                  | X\  = Exit bijector     |
                                                  | \/  = Expand bijector   |
                                                  | !|! = Identity bijector |
                                                  |                         |
                                                  | up  = Forward pass      |
                                                  | dn  = Inverse pass      |
                                                  |_________________________|

==============================================================================

The default configuration for glow is meant to replicate the architecture in [1] for generating images from CIFAR-10.

Example usage:


from functools import reduce
from operator import mul
import tensorflow as tf
import tensorflow_datasets as tfds
import tensorflow_probability as tfp
tfb = tfp.bijectors
tfd = tfp.distributions

data, info = tfds.load('cifar10', with_info=True)
train_data, test_data = data['train'], data['test']

preprocess = lambda x: tf.cast(x['image'], tf.float32)
train_data = train_data.batch(4).map(preprocess)
test_data = test_data.batch(4).map(preprocess)

x = next(iter(train_data))

glow = tfb.Glow(output_shape=info.features['image'].shape,
                coupling_bijector_fn=tfb.GlowDefaultNetwork,
                exit_bijector_fn=tfb.GlowDefaultExitNetwork)

z_shape = glow.inverse_event_shape(info.features['image'].shape)

pz = tfd.Sample(tfd.Normal(0., 1.), z_shape)

# Calling glow on distribution p(z) creates our glow distribution over images.
px = glow(pz)

# Take samples from the distribution to get images from your dataset
images = px.sample(4)

# Map images to positions in the distribution
z = glow.inverse(x)

# Get the z's corresponding to each spatial scale. To do this, we have to
# find out how many zs are passed through blockwise at each stage that were
# not passed at the previous stage. This is encoded in the second element of
# each list of blockwise splits. However because the bijector iteratively
# converts spatial pixels to channels, we also need to multiply the size of
# that second element by the number of spatial-to-channel conversions that the
# tensor receives after exiting (including after any alteration).
ztake = [bs[1] * 4**(i+2) for i, bs in enumerate(glow.blockwise_splits)]
total_z_taken = sum(ztake)
split_sizes = [z_shape.as_list()[0]-total_z_taken] + ztake
zsplits = tf.split(z, num_or_size_splits=split_sizes, axis=-1)

References:

[1]: Diederik P Kingma, Prafulla Dhariwal, Glow: Generative Flow with Invertible 1x1 Convolutions. In Neural Information Processing Systems, 2018. https://arxiv.org/abs/1807.03039

[2]: Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density Estimation using Real NVP. In International Conference on Learning Representations, 2017. https://arxiv.org/abs/1605.08803

output_shape A list of integers, specifying the event shape of the output, of the bijectors forward pass (the image). Specified as [H, W, C]. Default Value: (32, 32, 3)
num_glow_blocks An integer, specifying how many downsampling levels to include in the model. This must divide equally into both H and W, otherwise the bijector would not be invertible. Default Value: 3
num_steps_per_block An integer specifying how many Affine Coupling and 1x1 convolution layers to include at each level of the spatial hierarchy. Default Value: 32 (i.e. the value used in the original glow paper).
coupling_bijector_fn A function which takes the argument input_shape and returns a callable neural network (e.g. a keras.Sequential). The network should either return a tensor with the same event shape as input_shape (this will employ additive coupling), a tensor with the same height and width as input_shape but twice the number of channels (this will employ affine coupling), or a bijector which takes in a tensor with event shape input_shape, and returns a tensor with shape input_shape.
exit_bijector_fn Similar to coupling_bijector_fn, exit_bijector_fn is a function which takes the argument input_shape and output_chan and returns a callable neural network. The neural network it returns should take a tensor of shape input_shape as the input, and return one of three options: A tensor with output_chan channels, a tensor with 2 * output_chan channels, or a bijector. Additional details can be found in the documentation for ExitBijector.
grab_after_block A tuple of floats, specifying what fraction of the remaining channels to remove following each glow block. Glow will take the integer floor of this number multiplied by the remaining number of channels. The default is half at each spatial hierarchy. Default value: None (this will take out half of the channels after each block.
use_actnorm A bool deciding whether or not to use actnorm. Data-dependent initialization is used to initialize this layer. Default value: False
seed A seed to control randomness in the 1x1 convolution initialization. Default value: None (i.e., non-reproducible sampling).
validate_args Python bool indicating whether arguments should be checked for correctness. Default value: False
name Python str, name given to ops managed by this object. Default value: 'glow'.

bijectors

blockwise_splits

dtype

forward_min_event_ndims Returns the minimal number of dimensions bijector.forward operates on.

Multipart bijectors return structured ndims, which indicates the expected structure of their inputs. Some multipart bijectors, notably Composites, may return structures of None.

graph_parents Returns this Bijector's graph_parents as a Python list.
has_static_min_event_ndims Returns True if the bijector has statically-known min_event_ndims.
inverse_min_event_ndims Returns the minimal number of dimensions bijector.inverse operates on.

Multipart bijectors return structured event_ndims, which indicates the expected structure of their outputs. Some multipart bijectors, notably Composites, may return structures of None.

is_constant_jacobian Returns true iff the Jacobian matrix is not a function of x.

name Returns the string name of this Bijector.
name_scope Returns a tf.name_scope instance for this class.
non_trainable_variables Sequence of non-trainable variables owned by this module and its submodules.
parameters Dictionary of parameters used to instantiate this Bijector.
submodules Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
list(a.submodules) == [b, c]
True
list(b.submodules) == [c]
True
list(c.submodules) == []
True

trainable_variables Sequence of trainable variables owned by this module and its submodules.

validate_args Returns True if Tensor arguments will be validated.
validate_event_size

variables Sequence of variables owned by this module and its submodules.

Methods

forward

View source

Returns the forward Bijector evaluation, i.e., X = g(Y).

Args
x Tensor (structure). The input to the 'forward' evaluation.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure).

Raises
TypeError if self.dtype is specified and x.dtype is not self.dtype.
NotImplementedError if _forward is not implemented.

forward_dtype

View source

Returns the dtype returned by forward for the provided input.

forward_event_ndims

View source

Returns the number of event dimensions produced by forward.

forward_event_shape

View source

Shape of a single sample from a single batch as a TensorShape.

Same meaning as forward_event_shape_tensor. May be only partially defined.

Args
input_shape TensorShape (structure) indicating event-portion shape passed into forward function.

Returns
forward_event_shape_tensor TensorShape (structure) indicating event-portion shape after applying forward. Possibly unknown.

forward_event_shape_tensor

View source

Shape of a single sample from a single batch as an int32 1D Tensor.

Args
input_shape Tensor, int32 vector (structure) indicating event-portion shape passed into forward function.
name name to give to the op

Returns
forward_event_shape_tensor Tensor, int32 vector (structure) indicating event-portion shape after applying forward.

forward_log_det_jacobian

View source

Returns both the forward_log_det_jacobian.

Args
x Tensor (structure). The input to the 'forward' Jacobian determinant evaluation.
event_ndims Number of dimensions in the probabilistic events being transformed. Must be greater than or equal to self.forward_min_event_ndims. The result is summed over the final dimensions to produce a scalar Jacobian determinant for each event, i.e. it has shape rank(x) - event_ndims dimensions. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure), if this bijector is injective. If not injective this is not implemented.

Raises
TypeError if y's dtype is incompatible with the expected output dtype.
NotImplementedError if neither _forward_log_det_jacobian nor {_inverse, _inverse_log_det_jacobian} are implemented, or this is a non-injective bijector.

inverse

View source

Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).

Args
y Tensor (structure). The input to the 'inverse' evaluation.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure), if this bijector is injective. If not injective, returns the k-tuple containing the unique k points (x1, ..., xk) such that g(xi) = y.

Raises
TypeError if y's structured dtype is incompatible with the expected output dtype.
NotImplementedError if _inverse is not implemented.

inverse_dtype

View source

Returns the dtype returned by inverse for the provided input.

inverse_event_ndims

View source

Returns the number of event dimensions produced by inverse.

inverse_event_shape

View source

Shape of a single sample from a single batch as a TensorShape.

Same meaning as inverse_event_shape_tensor. May be only partially defined.

Args
output_shape TensorShape (structure) indicating event-portion shape passed into inverse function.

Returns
inverse_event_shape_tensor TensorShape (structure) indicating event-portion shape after applying inverse. Possibly unknown.

inverse_event_shape_tensor

View source

Shape of a single sample from a single batch as an int32 1D Tensor.

Args
output_shape Tensor, int32 vector (structure) indicating event-portion shape passed into inverse function.
name name to give to the op

Returns
inverse_event_shape_tensor Tensor, int32 vector (structure) indicating event-portion shape after applying inverse.

inverse_log_det_jacobian

View source

Returns the (log o det o Jacobian o inverse)(y).

Mathematically, returns: log(det(dX/dY))(Y). (Recall that: X=g^{-1}(Y).)

Note that forward_log_det_jacobian is the negative of this function, evaluated at g^{-1}(y).

Args
y Tensor (structure). The input to the 'inverse' Jacobian determinant evaluation.
event_ndims Number of dimensions in the probabilistic events being transformed. Must be greater than or equal to self.inverse_min_event_ndims. The result is summed over the final dimensions to produce a scalar Jacobian determinant for each event, i.e. it has shape rank(y) - event_ndims dimensions. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
ildj Tensor, if this bijector is injective. If not injective, returns the tuple of local log det Jacobians, log(det(Dg_i^{-1}(y))), where g_i is the restriction of g to the ith partition Di.

Raises
TypeError if x's dtype is incompatible with the expected inverse-dtype.
NotImplementedError if _inverse_log_det_jacobian is not implemented.

with_name_scope

Decorator to automatically enter the module name scope.

class MyModule(tf.Module):
  @tf.Module.with_name_scope
  def __call__(self, x):
    if not hasattr(self, 'w'):
      self.w = tf.Variable(tf.random.normal([x.shape[1], 3]))
    return tf.matmul(x, self.w)

Using the above module would produce tf.Variables and tf.Tensors whose names included the module name:

mod = MyModule()
mod(tf.ones([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Args
method The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

__call__

View source

Applies or composes the Bijector, depending on input type.

This is a convenience function which applies the Bijector instance in three different ways, depending on the input:

  1. If the input is a tfd.Distribution instance, return tfd.TransformedDistribution(distribution=input, bijector=self).
  2. If the input is a tfb.Bijector instance, return tfb.Chain([self, input]).
  3. Otherwise, return self.forward(input)

Args
value A tfd.Distribution, tfb.Bijector, or a (structure of) Tensor.
name Python str name given to ops created by this function.
**kwargs Additional keyword arguments passed into the created tfd.TransformedDistribution, tfb.Bijector, or self.forward.

Returns
composition A tfd.TransformedDistribution if the input was a tfd.Distribution, a tfb.Chain if the input was a tfb.Bijector, or a (structure of) Tensor computed by self.forward.

Examples

sigmoid = tfb.Reciprocal()(
    tfb.AffineScalar(shift=1.)(
      tfb.Exp()(
        tfb.AffineScalar(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.AffineScalar(shift=1.),
#         tfb.Exp(),
#         tfb.AffineScalar(scale=-1.),
#      ])`  # ie, `tfb.Sigmoid()`

log_normal = tfb.Exp()(tfd.Normal(0, 1))
# ==> `tfd.TransformedDistribution(tfd.Normal(0, 1), tfb.Exp())`

tfb.Exp()([-1., 0., 1.])
# ==> tf.exp([-1., 0., 1.])

__eq__

View source

Return self==value.