tfp.bijectors.MaskedAutoregressiveFlow

Affine MaskedAutoregressiveFlow bijector.

Inherits From: Bijector

tfp.bijectors.MaskedAutoregressiveFlow(
    shift_and_log_scale_fn=None,
    bijector_fn=None,
    is_constant_jacobian=False,
    validate_args=False,
    unroll_loop=False,
    event_ndims=1,
    name=None
)

The affine autoregressive flow [(Papamakarios et al., 2016)][3] provides a relatively simple framework for user-specified (deep) architectures to learn a distribution over continuous events. Regarding terminology,

'Autoregressive models decompose the joint density as a product of conditionals, and model each conditional in turn. Normalizing flows transform a base density (e.g. a standard Gaussian) into the target density by an invertible transformation with tractable Jacobian.' [(Papamakarios et al., 2016)][3]

In other words, the 'autoregressive property' is equivalent to the decomposition, p(x) = prod{ p(x[perm[i]] | x[perm[0:i]]) : i=0, ..., d } where perm is some permutation of {0, ..., d}. In the simple case where the permutation is identity this reduces to: p(x) = prod{ p(x[i] | x[0:i]) : i=0, ..., d }.

In TensorFlow Probability, 'normalizing flows' are implemented as tfp.bijectors.Bijectors. The forward 'autoregression' is implemented using a tf.while_loop and a deep neural network (DNN) with masked weights such that the autoregressive property is automatically met in the inverse.

A TransformedDistribution using MaskedAutoregressiveFlow(...) uses the (expensive) forward-mode calculation to draw samples and the (cheap) reverse-mode calculation to compute log-probabilities. Conversely, a TransformedDistribution using Invert(MaskedAutoregressiveFlow(...)) uses the (expensive) forward-mode calculation to compute log-probabilities and the (cheap) reverse-mode calculation to compute samples. See 'Example Use' [below] for more details.

Given a shift_and_log_scale_fn, the forward and inverse transformations are (a sequence of) affine transformations. A 'valid' shift_and_log_scale_fn must compute each shift (aka loc or 'mu' in [Germain et al. (2015)][1]) and log(scale) (aka 'alpha' in [Germain et al. (2015)][1]) such that each are broadcastable with the arguments to forward and inverse, i.e., such that the calculations in forward, inverse [below] are possible.

For convenience, tfp.bijectors.AutoregressiveNetwork is offered as a possible shift_and_log_scale_fn function. It implements the MADE architecture [(Germain et al., 2015)][1]. MADE is a feed-forward network that computes a shift and log(scale) using masked dense layers in a deep neural network. Weights are masked to ensure the autoregressive property. It is possible that this architecture is suboptimal for your task. To build alternative networks, either change the arguments to tfp.bijectors.AutoregressiveNetwork or use some other architecture, e.g., using tf.keras.layers.

Assuming shift_and_log_scale_fn has valid shape and autoregressive semantics, the forward transformation is

def forward(x):
  y = zeros_like(x)
  event_size = x.shape[-event_dims:].num_elements()
  for _ in range(event_size):
    shift, log_scale = shift_and_log_scale_fn(y)
    y = x * tf.exp(log_scale) + shift
  return y

and the inverse transformation is

def inverse(y):
  shift, log_scale = shift_and_log_scale_fn(y)
  return (y - shift) / tf.exp(log_scale)

Notice that the inverse does not need a for-loop. This is because in the forward pass each calculation of shift and log_scale is based on the y calculated so far (not x). In the inverse, the y is fully known, thus is equivalent to the scaling used in forward after event_size passes, i.e., the 'last' y used to compute shift, log_scale. (Roughly speaking, this also proves the transform is bijective.)

The bijector_fn argument allows specifying a more general coupling relation, such as the LSTM-inspired activation from [4], or Neural Spline Flow [5]. It must logically operate on each element of the input individually, and still obey the 'autoregressive property' described above. The forward transformation is

def forward(x):
  y = zeros_like(x)
  event_size = x.shape[-event_dims:].num_elements()
  for _ in range(event_size):
    bijector = bijector_fn(y)
    y = bijector.forward(x)
  return y

and inverse transformation is

def inverse(y):
    bijector = bijector_fn(y)
    return bijector.inverse(y)

Examples

tfd = tfp.distributions
tfb = tfp.bijectors

dims = 2

# A common choice for a normalizing flow is to use a Gaussian for the base
# distribution.  (However, any continuous distribution would work.) Here, we
# use `tfd.Sample` to create a joint Gaussian distribution with diagonal
# covariance for the base distribution (note that in the Gaussian case,
# `tfd.MultivariateNormalDiag` could also be used.)
maf = tfd.TransformedDistribution(
    distribution=tfd.Sample(
        tfd.Normal(loc=0., scale=1.), sample_shape=[dims]),
    bijector=tfb.MaskedAutoregressiveFlow(
        shift_and_log_scale_fn=tfb.AutoregressiveNetwork(
            params=2, hidden_units=[512, 512])))

x = maf.sample()  # Expensive; uses `tf.while_loop`, no Bijector caching.
maf.log_prob(x)   # Almost free; uses Bijector caching.
# Cheap; no `tf.while_loop` despite no Bijector caching.
maf.log_prob(tf.zeros(dims))

# [Papamakarios et al. (2016)][3] also describe an Inverse Autoregressive
# Flow [(Kingma et al., 2016)][2]:
iaf = tfd.TransformedDistribution(
    distribution=tfd.Sample(
        tfd.Normal(loc=0., scale=1.), sample_shape=[dims]),
    bijector=tfb.Invert(tfb.MaskedAutoregressiveFlow(
        shift_and_log_scale_fn=tfb.AutoregressiveNetwork(
            params=2, hidden_units=[512, 512]))))

x = iaf.sample()  # Cheap; no `tf.while_loop` despite no Bijector caching.
iaf.log_prob(x)   # Almost free; uses Bijector caching.
# Expensive; uses `tf.while_loop`, no Bijector caching.
iaf.log_prob(tf.zeros(dims))

# In many (if not most) cases the default `shift_and_log_scale_fn` will be a
# poor choice.  Here's an example of using a 'shift only' version and with a
# different number/depth of hidden layers.
made = tfb.AutoregressiveNetwork(params=1, hidden_units=[32])
maf_no_scale_hidden2 = tfd.TransformedDistribution(
    distribution=tfd.Sample(
        tfd.Normal(loc=0., scale=1.), sample_shape=[dims]),
    bijector=tfb.MaskedAutoregressiveFlow(
        lambda y: (made(y)[..., 0], None),
        is_constant_jacobian=True))
maf_no_scale_hidden2._made = made  # Ensure maf_no_scale_hidden2.trainable
# NOTE: The last line ensures that maf_no_scale_hidden2.trainable_variables
# will include all variables from `made`.

Variable Tracking

A tfb.MaskedAutoregressiveFlow instance saves a reference to the values passed as shift_and_log_scale_fn and bijector_fn to its constructor. Thus, for most values passed as shift_and_log_scale_fn or bijector_fn, variables referenced by those values will be found and tracked by the tfb.MaskedAutoregressiveFlow instance. Please see the tf.Module documentation for further details.

However, if the value passed to shift_and_log_scale_fn or bijector_fn is a Python function, then tfb.MaskedAutoregressiveFlow cannot automatically track variables used inside shift_and_log_scale_fn or bijector_fn. To get tfb.MaskedAutoregressiveFlow to track such variables, either:

Replace the Python function with a tf.Module, tf.keras.Layer, or other callable object through which tf.Module can find variables.

Or, add a reference to the variables to the tfb.MaskedAutoregressiveFlow instance by setting an attribute -- for example:

````
made1 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10])
made2 = tfb.AutoregressiveNetwork(params=1, hidden_units=[10, 10])
maf = tfb.MaskedAutoregressiveFlow(lambda y: (made1(y), made2(y) + 1.))
maf._made_variables = made1.variables + made2.variables
````

References

[1]: Mathieu Germain, Karol Gregor, Iain Murray, and Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. In International Conference on Machine Learning, 2015. https://arxiv.org/abs/1502.03509

[2]: Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[3]: George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

[4]: Diederik P Kingma, Tim Salimans, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. In Neural Information Processing Systems, 2016. https://arxiv.org/abs/1606.04934

[5]: Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows, 2019. http://arxiv.org/abs/1906.04032

Args
`shift_and_log_scale_fn`	Python `callable` which computes `shift` and `log_scale` from the inverse domain (`y`). Calculation must respect the 'autoregressive property' (see class docstring). Suggested default `tfb.AutoregressiveNetwork(params=2, hidden_layers=...)`. Typically the function contains `tf.Variables`. Returning `None` for either (both) `shift`, `log_scale` is equivalent to (but more efficient than) returning zero. If `shift_and_log_scale_fn` returns a single `Tensor`, the returned value will be unstacked to get the `shift` and `log_scale`: `tf.unstack(shift_and_log_scale_fn(y), num=2, axis=-1)`.
`bijector_fn`	Python `callable` which returns a `tfb.Bijector` which transforms event tensor with the signature `(input, **condition_kwargs) -> bijector`. The bijector must operate on scalar events and must not alter the rank of its input. The `bijector_fn` will be called with `Tensors` from the inverse domain (`y`). Calculation must respect the 'autoregressive property' (see class docstring).
`is_constant_jacobian`	Python `bool`. Default: `False`. When `True` the implementation assumes `log_scale` does not depend on the forward domain (`x`) or inverse domain (`y`) values. (No validation is made; `is_constant_jacobian=False` is always safe but possibly computationally inefficient.)
`validate_args`	Python `bool` indicating whether arguments should be checked for correctness.
`unroll_loop`	Python `bool` indicating whether the `tf.while_loop` in `_forward` should be replaced with a static for loop. Requires that the final dimension of `x` be known at graph construction time. Defaults to `False`.
`event_ndims`	Python `integer`, the intrinsic dimensionality of this bijector. 1 corresponds to a simple vector autoregressive bijector as implemented by the `tfp.bijectors.AutoregressiveNetwork`, 2 might be useful for a 2D convolutional `shift_and_log_scale_fn` and so on.
`name`	Python `str`, name given to ops managed by this object.

Raises
`ValueError`	If both or none of `shift_and_log_scale_fn` and `bijector_fn` are specified.

Attributes
`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.
`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. Note: Jacobian matrix is either constant for both forward and inverse or neither.
`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. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`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. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`validate_args`	Returns True if Tensor arguments will be validated.
`variables`	Sequence of variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.

Args
`x_event_ndims`	Optional Python `int` (structure) number of dimensions in a probabilistic event passed to `forward`; this must be greater than or equal to `self.forward_min_event_ndims`. If `None`, defaults to `self.forward_min_event_ndims`. Mutually exclusive with `y_event_ndims`. Default value: `None`.
`y_event_ndims`	Optional Python `int` (structure) number of dimensions in a probabilistic event passed to `inverse`; this must be greater than or equal to `self.inverse_min_event_ndims`. Mutually exclusive with `x_event_ndims`. Default value: `None`.

Args
`x`	`Tensor` (structure). The point at which to calculate the density.
`tangent_space`	`TangentSpace` or one of its subclasses. The tangent to the support manifold at `x`.
`backward_compat`	`bool` specifying whether to assume that the Bijector is dimension-preserving.
`**kwargs`	Optional keyword arguments forwarded to tangent space methods.

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

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

Args
`event_ndims`	Structure of Python and/or Tensor `int`s, and/or `None` values. The structure should match that of `self.forward_min_event_ndims`, and all non-`None` values must be greater than or equal to the corresponding value in `self.forward_min_event_ndims`.
`**kwargs`	Optional keyword arguments forwarded to nested bijectors.

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

Args
`x`	`Tensor` (structure). The input to the 'forward' Jacobian determinant evaluation.
`event_ndims`	Optional number of dimensions in the probabilistic events being transformed; this must be greater than or equal to `self.forward_min_event_ndims`. If `event_ndims` is specified, the log Jacobian determinant is summed to produce a scalar log-determinant for each event. Otherwise (if `event_ndims` is `None`), no reduction is performed. Multipart bijectors require structured event_ndims, such that the batch rank `rank(y[i]) - event_ndims[i]` is the same for all elements `i` of the structured input. In most cases (with the exception of `tfb.JointMap`) they further require that `event_ndims[i] - self.inverse_min_event_ndims[i]` is the same for all elements `i` of the structured input. Default value: `None` (equivalent to `self.forward_min_event_ndims`).
`name`	The name to give this op.
`**kwargs`	Named arguments forwarded to subclass implementation.

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.
`ValueError`	if the value of `event_ndims` is not valid for this bijector.

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

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

Raises
`TypeError`	if `x`'s dtype is incompatible with the expected inverse-dtype.
`NotImplementedError`	if `_inverse_log_det_jacobian` is not implemented.
`ValueError`	if the value of `event_ndims` is not valid for this bijector.

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`.

tfp.bijectors.MaskedAutoregressiveFlow

Examples

Variable Tracking

References

Args

Raises

Attributes

Methods

copy

experimental_batch_shape

experimental_batch_shape_tensor

experimental_compute_density_correction

forward

forward_dtype

forward_event_ndims

forward_event_shape

forward_event_shape_tensor

forward_log_det_jacobian

inverse

inverse_dtype

inverse_event_ndims

inverse_event_shape

inverse_event_shape_tensor

inverse_log_det_jacobian

parameter_properties

with_name_scope

__call__

Examples

__eq__

__getitem__

__iter__

`copy`

`experimental_batch_shape`

`experimental_batch_shape_tensor`

`experimental_compute_density_correction`

`forward`

`forward_dtype`

`forward_event_ndims`

`forward_event_shape`

`forward_event_shape_tensor`

`forward_log_det_jacobian`

`inverse`

`inverse_dtype`

`inverse_event_ndims`

`inverse_event_shape`

`inverse_event_shape_tensor`

`inverse_log_det_jacobian`

`parameter_properties`

`with_name_scope`

`call`

`eq`

`getitem`

`iter`