tfp.substrates.numpy.bijectors.CorrelationCholesky

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Maps unconstrained reals to Cholesky-space correlation matrices.

Inherits From: Bijector

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Main aliases

tfp.experimental.substrates.numpy.bijectors.CorrelationCholesky

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

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.
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.
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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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.

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, 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	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. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
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.

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 forward 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.

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

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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 (structure). 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. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
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.

__call__

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__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.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.])

__eq__

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__eq__(
    other
)

Return self==value.

__ne__

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__ne__(
    other
)

Return self!=value.