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|
Maps unconstrained reals to Cholesky-space correlation matrices.
Inherits From: AutoCompositeTensorBijector, Bijector
tfp.substrates.numpy.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
- 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.
|
Methods
copy
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
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
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
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
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
forward_dtype(
dtype=UNSPECIFIED, name='forward_dtype', **kwargs
)
Returns the dtype returned by forward for the provided input.
forward_event_ndims
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
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
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
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
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
inverse_dtype(
dtype=UNSPECIFIED, name='inverse_dtype', **kwargs
)
Returns the dtype returned by inverse for the provided input.
inverse_event_ndims
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
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
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
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
@classmethodparameter_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__
__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.Distributioninstance, returntfd.TransformedDistribution(distribution=input, bijector=self). - If the input is a
tfb.Bijectorinstance, returntfb.Chain([self, input]). - 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__
__eq__(
other
)
Return self==value.
__getitem__
__getitem__(
slices
)
__iter__
__iter__()