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correlation_cholesky.py
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correlation_cholesky.py
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# Copyright 2018 The TensorFlow Probability Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
# ============================================================================
"""CorrelationCholesky bijector."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
# Dependency imports
import tensorflow.compat.v2 as tf
from tensorflow_probability.python.bijectors import bijector
from tensorflow_probability.python.bijectors import fill_triangular
from tensorflow_probability.python.internal import prefer_static as ps
from tensorflow_probability.python.internal import tensorshape_util
__all__ = [
'CorrelationCholesky',
]
class CorrelationCholesky(bijector.Bijector):
"""Maps unconstrained reals to Cholesky-space correlation matrices.
#### 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](https://en.wikipedia.org/wiki/Correlation_and_dependence#Correlation_matrices)
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
`i`th row of the Cholesky factor corresponding to the correlation matrix `R`,
then the `i`th 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 + x_i,1^2 + ... + x_(i,i-1)^2). We can check that
the required constraints on the image are satisfied.
#### Examples
```python
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.
"""
def __init__(self, validate_args=False, name='correlation_cholesky'):
parameters = dict(locals())
with tf.name_scope(name) as name:
super(CorrelationCholesky, self).__init__(
validate_args=validate_args,
forward_min_event_ndims=1,
inverse_min_event_ndims=2,
parameters=parameters,
name=name)
def _forward_event_shape(self, input_shape):
if tensorshape_util.rank(input_shape) is None:
return input_shape
tril_shape = fill_triangular.FillTriangular().forward_event_shape(
input_shape)
n = tril_shape[-1]
if n is not None:
n += 1
return tril_shape[:-2].concatenate([n, n])
def _forward_event_shape_tensor(self, input_shape):
tril_shape = fill_triangular.FillTriangular().forward_event_shape_tensor(
input_shape)
n = tril_shape[-1] + 1
return tf.concat([tril_shape[:-2], [n, n]], axis=-1)
def _inverse_event_shape(self, input_shape):
if not input_shape.rank:
return input_shape
n = input_shape[-1]
if n is not None:
n -= 1
y_shape = input_shape[:-2].concatenate([n, n])
return fill_triangular.FillTriangular().inverse_event_shape(y_shape)
def _inverse_event_shape_tensor(self, input_shape):
n = input_shape[-1] - 1
y_shape = tf.concat([input_shape[:-2], [n, n]], axis=-1)
return fill_triangular.FillTriangular().inverse_event_shape_tensor(y_shape)
def _forward(self, x):
x = tf.convert_to_tensor(x, name='x')
batch_shape = ps.shape(x)[:-1]
# Pad zeros on the top row and right column.
y = fill_triangular.FillTriangular().forward(x)
rank = ps.rank(y)
paddings = ps.concat(
[ps.zeros([rank - 2, 2], dtype=tf.int32),
[[1, 0], [0, 1]]],
axis=0)
y = tf.pad(y, paddings)
# Set diagonal to 1s.
n = ps.shape(y)[-1]
diag = tf.ones(ps.concat([batch_shape, [n]], axis=-1), dtype=x.dtype)
y = tf.linalg.set_diag(y, diag)
# Normalize each row to have Euclidean (L2) norm 1.
y /= tf.norm(y, axis=-1)[..., tf.newaxis]
return y
def _inverse(self, y):
n = ps.shape(y)[-1]
batch_shape = ps.shape(y)[:-2]
# Extract the reciprocal of the row norms from the diagonal.
diag = tf.linalg.diag_part(y)[..., tf.newaxis]
# Set the diagonal to 0s.
y = tf.linalg.set_diag(
y, tf.zeros(ps.concat([batch_shape, [n]], axis=-1), dtype=y.dtype))
# Multiply with the norm (or divide by its reciprocal) to recover the
# unconstrained reals in the (strictly) lower triangular part.
x = y / diag
# Remove the first row and last column before inverting the FillTriangular
# transformation.
return fill_triangular.FillTriangular().inverse(x[..., 1:, :-1])
def _forward_log_det_jacobian(self, x):
# TODO(b/133442896): It should be possible to use the fallback
# implementation of _forward_log_det_jacobian in terms of
# _inverse_log_det_jacobian in the base Bijector class.
return -self._inverse_log_det_jacobian(self.forward(x))
def _inverse_log_det_jacobian(self, y):
# The inverse log det jacobian (ILDJ) of the entire mapping is the sum of
# the ILDJs of each row's mapping.
#
# To compute the ILDJ for each row's mapping, consider the forward mapping
# `f_k` restricted to the `k`th (0-indexed) row. It maps unconstrained reals
# in `R^k` to the unit disk in `R^k`. `f_k : R^k -> R^k` is:
#
# f(x_1, x_2, ... x_k) = (x_1/s, x_2/s, ..., x_k/s)
#
# where `s = norm(x_1, x_2, ..., x_k, 1)`.
#
# The change in infinitesimal `k`-dimensional volume is given by
# |det(J)|; where J is the `k x k` Jacobian matrix.
#
# Claim: |det(J)| = s^{-(k + 2)}.
#
# Proof: We compute the entries of the Jacobian matrix J:
#
# J_ij = (s^2 - x_i^2) / s^3 if i == j
# J_ij = -(x_i * x_j) / s^3 if i != j
#
# We multiply each row by s^3, which contributes a factor of s^{-3k} to
# det(J). The remaining matrix can be written as s^2 I - xx^T. By the
# matrix determinant lemma
# (https://en.wikipedia.org/wiki/Matrix_determinant_lemma),
# det(s^2 I - xx^T) = s^{2k} (1 - (x^Tx / s^2)) = s^{2k - 2}. The last
# equality follows from s^2 - x^Tx = s^2 - sum x_i^2 = 1. Hence,
# det(J) = s^{-3k} s^{2k - 2} = s^{-(k + 2)}.
#
n = ps.shape(y)[-1]
return -tf.reduce_sum(
tf.range(2, n + 2, dtype=y.dtype) * tf.math.log(tf.linalg.diag_part(y)),
axis=-1)