tfp.bijectors.FFJORD

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

Inherits From: Bijector

tfp.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): Differential equation to be solved.
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. arxiv.org.abs/1810.01367

Args:

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.

Attributes:

dtype: dtype of Tensors transformable by this distribution.
forward_min_event_ndims: Returns the minimal number of dimensions bijector.forward operates on.
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.
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.
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
assert list(a.submodules) == [b, c]
assert list(b.submodules) == [c]
assert list(c.submodules) == []

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.

Methods

`call`

View source

__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:

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

Args:

value: A tfd.Distribution, tfb.Bijector, or a 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 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.])

`forward`

View source

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

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

Args:

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

Returns:

Tensor.

Raises:

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

`forward_event_shape`

View source

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 indicating event-portion shape passed into forward function.

Returns:

forward_event_shape_tensor: TensorShape indicating event-portion shape after applying forward. Possibly unknown.

`forward_event_shape_tensor`

View source

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 indicating event-portion shape passed into forward function.
name: name to give to the op

Returns:

forward_event_shape_tensor: Tensor, int32 vector indicating event-portion shape after applying forward.

`forward_log_det_jacobian`

View source

forward_log_det_jacobian(
    x, event_ndims, name='forward_log_det_jacobian', **kwargs
)

Returns both the forward_log_det_jacobian.

Args:

x: Tensor. 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.
name: The name to give this op.
**kwargs: Named arguments forwarded to subclass implementation.

Returns:

Tensor, if this bijector is injective. If not injective this is not implemented.

Raises:

TypeError: if self.dtype is specified and y.dtype is not self.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

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

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

Args:

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

Returns:

Tensor, 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 self.dtype is specified and y.dtype is not self.dtype.
NotImplementedError: if _inverse is not implemented.

`inverse_event_shape`

View source

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 indicating event-portion shape passed into inverse function.

Returns:

inverse_event_shape_tensor: TensorShape indicating event-portion shape after applying inverse. Possibly unknown.

`inverse_event_shape_tensor`

View source

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 indicating event-portion shape passed into inverse function.
name: name to give to the op

Returns:

inverse_event_shape_tensor: Tensor, int32 vector indicating event-portion shape after applying inverse.

`inverse_log_det_jacobian`

View source

inverse_log_det_jacobian(
    y, event_ndims, 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. 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.
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 self.dtype is specified and y.dtype is not self.dtype.
NotImplementedError: if _inverse_log_det_jacobian is not implemented.

`with_name_scope`

@classmethod
with_name_scope(
    cls, method
)

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], 64]))
    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([8, 32]))
# ==> <tf.Tensor: ...>
mod.w
# ==> <tf.Variable ...'my_module/w:0'>

Args:

method: The method to wrap.

Returns:

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

tfp.bijectors.FFJORD

Custom ode_solve_fn and trace_augmentation_fn examples:

Usage example:

References

Args:

Attributes:

Methods

__call__

Args:

Returns:

Examples

forward

Args:

Returns:

Raises:

forward_event_shape

Args:

Returns:

forward_event_shape_tensor

Args:

Returns:

forward_log_det_jacobian

Args:

Returns:

Raises:

inverse

Args:

Returns:

Raises:

inverse_event_shape

Args:

Returns:

inverse_event_shape_tensor

Args:

Returns:

inverse_log_det_jacobian

Args:

Returns:

Raises:

with_name_scope

Args:

Returns:

Custom `ode_solve_fn` and `trace_augmentation_fn` examples:

`call`

`forward`

`forward_event_shape`

`forward_event_shape_tensor`

`forward_log_det_jacobian`

`inverse`

`inverse_event_shape`

`inverse_event_shape_tensor`

`inverse_log_det_jacobian`

`with_name_scope`