tfp.bijectors.CorrelationCholesky

Maps unconstrained reals to Cholesky-space correlation matrices.

Inherits From: AutoCompositeTensorBijector, Bijector, AutoCompositeTensor

tfp.bijectors.CorrelationCholesky(
    validate_args=False, name='correlation_cholesky'
)

Mathematical Details

This bijector provides a change of variables from unconstrained reals to a parameterization of the CholeskyLKJ distribution. The CholeskyLKJ distribution [1] is a distribution on the set of Cholesky factors of positive definite correlation matrices. The CholeskyLKJ probability density function is obtained from the LKJ density on n x n matrices as follows:

1 = int p(A | eta) dA = int Z(eta) * det(A) ** (eta - 1) dA = int Z(eta) L_ii ** {(n - i - 1) + 2 * (eta - 1)} ^dL_ij (0 <= i < j < n)

where Z(eta) is the normalizer; the matrix L is the Cholesky factor of the correlation matrix A; and ^dL_ij denotes the wedge product (or differential) of the strictly lower triangular entries of L. The entries L_ij are constrained such that each entry lies in [-1, 1] and the norm of each row is

1. The norm includes the diagonal; which is not included in the wedge product. To preserve uniqueness, we further specify that the diagonal entries are positive.

The image of unconstrained reals under the CorrelationCholesky bijector is the set of correlation matrices which are positive definite. A correlation matrix can be characterized as a symmetric positive semidefinite matrix with 1s on the main diagonal.

For a lower triangular matrix L to be a valid Cholesky-factor of a positive definite correlation matrix, it is necessary and sufficient that each row of L have unit Euclidean norm [1]. To see this, observe that if L_i is the ith row of the Cholesky factor corresponding to the correlation matrix R, then the ith diagonal entry of R satisfies:

1 = R_i,i = L_i . L_i = ||L_i||^2

where '.' is the dot product of vectors and ||...|| denotes the Euclidean norm.

Furthermore, observe that R_i,j lies in the interval [-1, 1]. By the Cauchy-Schwarz inequality:

|R_i,j| = |L_i . L_j| <= ||L_i|| ||L_j|| = 1

This is a consequence of the fact that R is symmetric positive definite with 1s on the main diagonal.

We choose the mapping from x in R^{m} to R^{n^2} where m is the (n - 1)th triangular number; i.e. m = 1 + 2 + ... + (n - 1).

L_ij = x_i,j / s_i (for i < j) L_ii = 1 / s_i

where s_i = sqrt(1 + x_i,0^2 + xi,1^2 + ... + x(i,i-1)^2). We can check that the required constraints on the image are satisfied.

Examples

bijector.CorrelationCholesky().forward([2., 2., 1.])
# Result: [[ 1.        ,  0.        ,  0.        ],
           [ 0.70710678,  0.70710678,  0.        ],
           [ 0.66666667,  0.66666667,  0.33333333]]

bijector.CorrelationCholesky().inverse(
    [[ 1.        ,  0.        ,  0. ],
     [ 0.70710678,  0.70710678,  0.        ],
     [ 0.66666667,  0.66666667,  0.33333333]])
# Result: [2., 2., 1.]

References

[1] Stan Manual. Section 24.2. Cholesky LKJ Correlation Distribution. https://mc-stan.org/docs/2_18/functions-reference/cholesky-lkj-correlation-distribution.html [2] Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, "Generating random correlation matrices based on vines and extended onion method," Journal of Multivariate Analysis 100 (2009), pp 1989-2001.

Args
graph_parents	Python list of graph prerequisites of this Bijector.
is_constant_jacobian	Python bool indicating that the Jacobian matrix is not a function of the input.
validate_args	Python bool, default False. Whether to validate input with asserts. If validate_args is False, and the inputs are invalid, correct behavior is not guaranteed.
dtype	tf.dtype supported by this Bijector. None means dtype is not enforced. For multipart bijectors, this value is expected to be the same for all elements of the input and output structures.
forward_min_event_ndims	Python integer (structure) indicating the minimum number of dimensions on which forward operates.
inverse_min_event_ndims	Python integer (structure) indicating the minimum number of dimensions on which inverse operates. Will be set to forward_min_event_ndims by default, if no value is provided.
experimental_use_kahan_sum	Python bool. When True, use Kahan summation to aggregate log-det jacobians from independent underlying log-det jacobian values, which improves against the precision of a naive float32 sum. This can be noticeable in particular for large dimensions in float32. See CPU caveat on tfp.math.reduce_kahan_sum.
parameters	Python dict of parameters used to instantiate this Bijector. Bijector instances with identical types, names, and parameters share an input/output cache. parameters dicts are keyed by strings and are identical if their keys are identical and if corresponding values have identical hashes (or object ids, for unhashable objects).
name	The name to give Ops created by the initializer.

Raises
ValueError	If neither forward_min_event_ndims and inverse_min_event_ndims are specified, or if either of them is negative.
ValueError	If a member of graph_parents is not a Tensor.

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.

Methods

copy

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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

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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

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.

with_name_scope

@classmethod
with_name_scope(
    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], 3]))
    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([1, 2]))
<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>
mod.w
<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,
numpy=..., dtype=float32)>

Args
method	The method to wrap.

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

__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__

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__eq__(
    other
)

Return self==value.

__getitem__

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__getitem__(
    slices
)

__iter__

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__iter__()