tfp.substrates.numpy.bijectors.RationalQuadraticSpline

A piecewise rational quadratic spline, as developed in [1].

Inherits From: AutoCompositeTensorBijector, Bijector

View aliases

Main aliases

tfp.experimental.substrates.numpy.bijectors.RationalQuadraticSpline

tfp.substrates.numpy.bijectors.RationalQuadraticSpline(
    bin_widths,
    bin_heights,
    knot_slopes,
    range_min=-1,
    validate_args=False,
    name=None
)

This transformation represents a monotonically increasing piecewise rational quadratic function. Outside of the bounds of knot_x/knot_y, the transform behaves as an identity function.

Typically this bijector will be used as part of a chain, with splines for trailing x dimensions conditioned on some of the earlier x dimensions, and with the inverse then solved first for unconditioned dimensions, then using conditioning derived from those inverses, and so forth. For example, if we split a 15-D xs vector into 3 components, we may implement a forward and inverse as follows:

nsplits = 3

class SplineParams(tf.Module):

  def __init__(self, nbins=32, interval_width=2, range_min=-1,
               min_bin_width=1e-3, min_slope=1e-3):
    self._nbins = nbins
    self._interval_width = interval_width  # Sum of bin widths.
    self._range_min = range_min  # Position of first knot.
    self._min_bin_width = min_bin_width  # Bin width lower bound.
    self._min_slope = min_slope  # Lower bound for slopes at internal knots.
    self._built = False
    self._bin_widths = None
    self._bin_heights = None
    self._knot_slopes = None

  def __call__(self, x, nunits):
    if not self._built:
      def _bin_positions(x):
        out_shape = tf.concat((tf.shape(x)[:-1], (nunits, self._nbins)), 0)
        x = tf.reshape(x, out_shape)
        return tf.math.softmax(x, axis=-1) * (
              self._interval_width - self._nbins * self._min_bin_width
              ) + self._min_bin_width

      def _slopes(x):
        out_shape = tf.concat((
          tf.shape(x)[:-1], (nunits, self._nbins - 1)), 0)
        x = tf.reshape(x, out_shape)
        return tf.math.softplus(x) + self._min_slope

      self._bin_widths = tf.keras.layers.Dense(
        nunits * self._nbins, activation=_bin_positions, name='w')
      self._bin_heights = tf.keras.layers.Dense(
        nunits * self._nbins, activation=_bin_positions, name='h')
      self._knot_slopes = tf.keras.layers.Dense(
        nunits * (self._nbins - 1), activation=_slopes, name='s')
      self._built = True

    return tfb.RationalQuadraticSpline(
      bin_widths=self._bin_widths(x),
      bin_heights=self._bin_heights(x),
      knot_slopes=self._knot_slopes(x),
      range_min=self._range_min)

xs = np.random.randn(3, 15).astype(np.float32)  # Keras won't Dense(.)(vec).
splines = [SplineParams() for _ in range(nsplits)]

def spline_flow():
  stack = tfb.Identity()
  for i in range(nsplits):
    stack = tfb.RealNVP(5 * i, bijector_fn=splines[i])(stack)
  return stack

ys = spline_flow().forward(xs)
ys_inv = spline_flow().inverse(ys)  # ys_inv ~= xs

For a one-at-a-time autoregressive flow as in [1], it would be profitable to implement a mask over xs to parallelize either the inverse or the forward pass and implement the other using a tf.while_loop. See tfp.bijectors.MaskedAutoregressiveFlow for support doing so (paired with tfp.bijectors.Invert depending which direction should be parallel).

References

[1]: Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. arXiv preprint arXiv:1906.04032, 2019. https://arxiv.org/abs/1906.04032

Args
bin_widths	The widths of the spans between subsequent knot x positions, a floating point Tensor. Must be positive, and at least 1-D. Innermost axis must sum to the same value as bin_heights. The knot x positions will be a first at range_min, followed by knots at range_min + cumsum(bin_widths, axis=-1).
bin_heights	The heights of the spans between subsequent knot y positions, a floating point Tensor. Must be positive, and at least 1-D. Innermost axis must sum to the same value as bin_widths. The knot y positions will be a first at range_min, followed by knots at range_min + cumsum(bin_heights, axis=-1).
knot_slopes	The slope of the spline at each knot, a floating point Tensor. Must be positive. 1s are implicitly padded for the first and last implicit knots corresponding to range_min and range_min + sum(bin_widths, axis=-1). Innermost axis size should be 1 less than that of bin_widths/bin_heights, or 1 for broadcasting.
range_min	The x/y position of the first knot, which has implicit slope 1. range_max is implicit, and can be computed as range_min + sum(bin_widths, axis=-1). Scalar floating point Tensor.
validate_args	Toggles argument validation (can hurt performance).
name	Optional name scope for associated ops. (Defaults to 'RationalQuadraticSpline').

Attributes
bin_heights
bin_widths
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.
knot_slopes
name	Returns the string name of this Bijector.
parameters	Dictionary of parameters used to instantiate this Bijector.
range_min
trainable_variables
validate_args	Returns True if Tensor arguments will be validated.
variables

Methods

copy

View source

copy(
    **override_parameters_kwargs
)

Creates a copy of the bijector.

Note: the copy bijector may continue to depend on the original initialization arguments.

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

View source

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

View source

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

View source

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

View source

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

View source

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

Returns the dtype returned by forward for the provided input.

forward_event_ndims

View source

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

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 (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

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

View source

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

View source

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

View source

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

Returns the dtype returned by inverse for the provided input.

inverse_event_ndims

View source

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

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 (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

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

View source

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__

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:

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__

View source

__eq__(
    other
)

Return self==value.

__getitem__

View source

__getitem__(
    slices
)

__iter__

View source

__iter__()