tfp.experimental.substrates.jax.bijectors.FFJORD

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

Inherits From: Bijector

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. 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 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.

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

forward

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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_dtype

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Returns the dtype of the output of the forward transformation.

Args
dtype tf.dtype, or nested structure of tf.dtypes, of the input to forward.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
tf.dtype or nested structure of tf.dtypes of the output of forward.

forward_event_shape

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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

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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

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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

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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_dtype

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Returns the dtype of the output of the inverse transformation.

Args
dtype tf.dtype, or nested structure of tf.dtypes, of the input to inverse.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
tf.dtype or nested structure of tf.dtypes of the output of inverse.

inverse_event_shape

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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

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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

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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.

__call__

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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 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.])