tfp.substrates.numpy.bijectors.FFJORD

Implements a continuous normalizing flow X->Y defined via an ODE.

Inherits From: Bijector

tfp.substrates.numpy.bijectors.FFJORD(
    state_time_derivative_fn,
    ode_solve_fn=None,
    trace_augmentation_fn=trace_jacobian_hutchinson,
    initial_time=0.0,
    final_time=1.0,
    validate_args=False,
    dtype=tf.float32,
    name='ffjord'
)

This bijector implements a continuous dynamics transformation parameterized by a differential equation, where initial and terminal conditions correspond to domain (X) and image (Y) i.e.

d/dt[state(t)]=state_time_derivative_fn(t, state(t))
state(initial_time) = X
state(final_time) = Y

For this transformation the value of log_det_jacobian follows another differential equation, reducing it to computation of the trace of the jacbian along the trajectory

state_time_derivative = state_time_derivative_fn(t, state(t))
d/dt[log_det_jac(t)] = Tr(jacobian(state_time_derivative, state(t)))

FFJORD constructor takes two functions ode_solve_fn and trace_augmentation_fn arguments that customize integration of the differential equation and trace estimation.

Differential equation integration is performed by a call to ode_solve_fn. Custom ode_solve_fn must accept the following arguments:

  • ode_fn(time, state, **condition_kwargs): Differential equation to be solved. Custom ode_solve_fns may optionally support conditional inputs by accepting a constants dict arg and computing gradients wrt the provided values in **condition_kwargs.
  • initial_time: Scalar float or floating Tensor representing the initial time.
  • initial_state: Floating Tensor representing the initial state.
  • solution_times: 1D floating Tensor of solution times.

And return a Tensor of shape [solution_times.shape, initial_state.shape] representing state values evaluated at solution_times. In addition ode_solve_fn must support nested structures. For more details see the interface of tfp.math.ode.Solver.solve().

Trace estimation is computed simultaneously with state_time_derivative using augmented_state_time_derivative_fn that is generated by trace_augmentation_fn. trace_augmentation_fn takes state_time_derivative_fn, state.shape and state.dtype arguments and returns a augmented_state_time_derivative_fn callable that computes both state_time_derivative and unreduced trace_estimation.

Custom ode_solve_fn and trace_augmentation_fn examples:

# custom_solver_fn: `callable(f, t_initial, t_solutions, y_initial, ...)`
# custom_solver_kwargs: Additional arguments to pass to custom_solver_fn.
def ode_solve_fn(ode_fn, initial_time, initial_state, solution_times):
  results = custom_solver_fn(ode_fn, initial_time, solution_times,
                             initial_state, **custom_solver_kwargs)
  return results

ffjord = tfb.FFJORD(state_time_derivative_fn, ode_solve_fn=ode_solve_fn)

# state_time_derivative_fn: `callable(time, state)`
# trace_jac_fn: `callable(time, state)` unreduced jacobian trace function

def trace_augmentation_fn(ode_fn, state_shape, state_dtype):
  def augmented_ode_fn(time, state):
    return ode_fn(time, state), trace_jac_fn(time, state)
  return augmented_ode_fn

ffjord = tfb.FFJORD(state_time_derivative_fn,
                    trace_augmentation_fn=trace_augmentation_fn)

For more details on FFJORD and continous normalizing flows see [1], [2].

Usage example:

tfd = tfp.distributions
tfb = tfp.bijectors
# state_time_derivative_fn: `Callable(time, state)` -> state_time_derivative
# e.g. Neural network with inputs and outputs of the same shapes and dtypes.

bijector = tfb.FFJORD(state_time_derivative_fn=state_time_derivative_fn)
y = bijector.forward(x)  # forward mapping
x = bijector.inverse(y)  # inverse mapping
base = tfd.Normal(tf.zeros_like(x), tf.ones_like(x))  # Base distribution
transformed_distribution = tfd.TransformedDistribution(base, bijector)

References

[1]: Chen, T. Q., Rubanova, Y., Bettencourt, J., & Duvenaud, D. K. (2018). Neural ordinary differential equations. In Advances in neural information processing systems (pp. 6571-6583)

[2]: Grathwohl, W., Chen, R. T., Betterncourt, J., Sutskever, I., & Duvenaud, D. (2018). Ffjord: Free-form continuous dynamics for scalable reversible generative models. arXiv preprint arXiv:1810.01367. http://arxiv.org.abs/1810.01367

state_time_derivative_fn Python callable taking arguments time (a scalar representing time) and state (a Tensor representing the state at given time) returning the time derivative of the state at given time.
ode_solve_fn Python callable taking arguments ode_fn (same as state_time_derivative_fn above), initial_time (a scalar representing the initial time of integration), initial_state (a Tensor of floating dtype represents the initial state) and solution_times (1D Tensor of floating dtype representing time at which to obtain the solution) returning a Tensor of shape [time_axis, initial_state.shape]. Will take [final_time] as the solution_times argument and state_time_derivative_fn as ode_fn argument. For details on providing custom ode_solve_fn see class docstring. If None a DormandPrince solver from tfp.math.ode is used. Default value: None
trace_augmentation_fn Python callable taking arguments ode_fn ( python callable same as state_time_derivative_fn above), state_shape (TensorShape of a the state), dtype (same as dtype of the state) and returning a python callable taking arguments time (a scalar representing the time at which the function is evaluted), state (a Tensor representing the state at given time) that computes a tuple (ode_fn(time, state), jacobian_trace_estimation). jacobian_trace_estimation should represent trace of the jacobian of ode_fn with respect to state. state_time_derivative_fn will be passed as ode_fn argument. For details on providing custom trace_augmentation_fn see class docstring. Default value: tfp.bijectors.ffjord.trace_jacobian_hutchinson
initial_time Scalar float representing time to which the x value of the bijector corresponds to. Passed as initial_time to ode_solve_fn. For default solver can be Python float or floating scalar Tensor. Default value: 0.
final_time Scalar float representing time to which the y value of the bijector corresponds to. Passed as solution_times to ode_solve_fn. For default solver can be Python float or floating scalar Tensor. Default value: 1.
validate_args Python 'bool' indicating whether to validate input. Default value: False
dtype tf.DType to prefer when converting args to Tensors. Else, we fall back to a common dtype inferred from the args, finally falling back to float32.
name Python str name prefixed to Ops created by this function.

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.
name Returns the string name of this Bijector.
parameters Dictionary of parameters used to instantiate this Bijector.
trainable_variables

validate_args Returns True if Tensor arguments will be validated.
variables

Methods

copy

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copy(
    **override_parameters_kwargs
)

Creates a copy of the bijector.

Args
**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

Returns
bijector A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

experimental_batch_shape

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experimental_batch_shape(
    x_event_ndims=None, y_event_ndims=None
)

Returns the batch shape of this bijector for inputs of the given rank.

The batch shape of a bijector decribes the set of distinct transformations it represents on events of a given size. For example: the bijector tfb.Scale([1., 2.]) has batch shape [2] for scalar events (event_ndims = 0), because applying it to a scalar event produces two scalar outputs, the result of two different scaling transformations. The same bijector has batch shape [] for vector events, because applying it to a vector produces (via elementwise multiplication) a single vector output.

Bijectors that operate independently on multiple state parts, such as tfb.JointMap, must broadcast to a coherent batch shape. Some events may not be valid: for example, the bijector tfd.JointMap([tfb.Scale([1., 2.]), tfb.Scale([1., 2., 3.])]) does not produce a valid batch shape when event_ndims = [0, 0], since the batch shapes of the two parts are inconsistent. The same bijector does define valid batch shapes of [], [2], and [3] if event_ndims is [1, 1], [0, 1], or [1, 0], respectively.

Since transforming a single event produces a scalar log-det-Jacobian, the batch shape of a bijector with non-constant Jacobian is expected to equal the shape of forward_log_det_jacobian(x, event_ndims=x_event_ndims) or inverse_log_det_jacobian(y, event_ndims=y_event_ndims), for x or y of the specified ndims.

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.

Returns
batch_shape TensorShape batch shape of this bijector for a value with the given event rank. May be unknown or partially defined.

experimental_batch_shape_tensor

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experimental_batch_shape_tensor(
    x_event_ndims=None, y_event_ndims=None
)

Returns the batch shape of this bijector for inputs of the given rank.

The batch shape of a bijector decribes the set of distinct transformations it represents on events of a given size. For example: the bijector tfb.Scale([1., 2.]) has batch shape [2] for scalar events (event_ndims = 0), because applying it to a scalar event produces two scalar outputs, the result of two different scaling transformations. The same bijector has batch shape [] for vector events, because applying it to a vector produces (via elementwise multiplication) a single vector output.

Bijectors that operate independently on multiple state parts, such as tfb.JointMap, must broadcast to a coherent batch shape. Some events may not be valid: for example, the bijector tfd.JointMap([tfb.Scale([1., 2.]), tfb.Scale([1., 2., 3.])]) does not produce a valid batch shape when event_ndims = [0, 0], since the batch shapes of the two parts are inconsistent. The same bijector does define valid batch shapes of [], [2], and [3] if event_ndims is [1, 1], [0, 1], or [1, 0], respectively.

Since transforming a single event produces a scalar log-det-Jacobian, the batch shape of a bijector with non-constant Jacobian is expected to equal the shape of forward_log_det_jacobian(x, event_ndims=x_event_ndims) or inverse_log_det_jacobian(y, event_ndims=y_event_ndims), for x or y of the specified ndims.

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.

Returns
batch_shape_tensor integer Tensor batch shape of this bijector for a value with the given event rank.

experimental_compute_density_correction

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experimental_compute_density_correction(
    x, tangent_space, backward_compat=False, **kwargs
)

Density correction for this transformation wrt the tangent space, at x.

Subclasses of Bijector may call the most specific applicable method of TangentSpace, based on whether the transformation is dimension-preserving, coordinate-wise, a projection, or something more general. The backward-compatible assumption is that the transformation is dimension-preserving (goes from R^n to R^n).

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.

Returns
density_correction Tensor representing the density correction---in log space---under the transformation that this Bijector denotes.

Raises
TypeError if backward_compat is False but no method of TangentSpace has been called explicitly.

forward

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forward(
    x, name='forward', **kwargs
)

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

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forward_dtype(
    dtype=UNSPECIFIED, name='forward_dtype', **kwargs
)

Returns the dtype returned by forward for the provided input.

forward_event_ndims

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forward_event_ndims(
    event_ndims, **kwargs
)

Returns the number of event dimensions produced by forward.

Args
event_ndims Structure of Python and/or Tensor ints, 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.

Returns
forward_event_ndims Structure of integers and/or None values matching self.inverse_min_event_ndims. These are computed using 'prefer static' semantics: if any inputs are None, some or all of the outputs may be None, indicating that the output dimension could not be inferred (conversely, if all inputs are non-None, all outputs will be non-None). If all input event_ndims are Python ints, all of the (non-None) outputs will be Python ints; otherwise, some or all of the outputs may be Tensor ints.

forward_event_shape

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forward_event_shape(
    input_shape
)

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

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forward_event_shape_tensor(
    input_shape, name='forward_event_shape_tensor'
)

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

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forward_log_det_jacobian(
    x, event_ndims=None, name='forward_log_det_jacobian', **kwargs
)

Returns both the forward_log_det_jacobian.

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.

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

inverse

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inverse(
    y, name='inverse', **kwargs
)

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

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inverse_dtype(
    dtype=UNSPECIFIED, name='inverse_dtype', **kwargs
)

Returns the dtype returned by inverse for the provided input.

inverse_event_ndims

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inverse_event_ndims(
    event_ndims, **kwargs
)

Returns the number of event dimensions produced by inverse.

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

Returns
inverse_event_ndims Structure of integers and/or None values matching self.forward_min_event_ndims. These are computed using 'prefer static' semantics: if any inputs are None, some or all of the outputs may be None, indicating that the output dimension could not be inferred (conversely, if all inputs are non-None, all outputs will be non-None). If all input event_ndims are Python ints, all of the (non-None) outputs will be Python ints; otherwise, some or all of the outputs may be Tensor ints.

inverse_event_shape

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inverse_event_shape(
    output_shape
)

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

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inverse_event_shape_tensor(
    output_shape, name='inverse_event_shape_tensor'
)

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

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inverse_log_det_jacobian(
    y, event_ndims=None, name='inverse_log_det_jacobian', **kwargs
)

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 Optional number of dimensions in the probabilistic events being transformed; this must be greater than or equal to self.inverse_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.inverse_min_event_ndims).
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.
ValueError if the value of event_ndims is not valid for this bijector.

parameter_properties

View source

@classmethod
parameter_properties(
    dtype=tf.float32
)

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the bijector's Tensor-valued constructor arguments.

Args
dtype Optional float dtype to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., tfb.Softplus.low) must be instantiated with the same dtype as the values to be transformed.

Returns
parameter_properties A str ->tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

__call__

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__call__(
    value, name=None, **kwargs
)

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.Shift(shift=1.)(
      tfb.Exp()(
        tfb.Scale(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.Shift(shift=1.),
#         tfb.Exp(),
#         tfb.Scale(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__

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__eq__(
    other
)

Return self==value.

__getitem__

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__getitem__(
    slices
)

__iter__

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__iter__()