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# Copyright 2016 The TensorFlow Authors. All Rights Reserved.
#
# 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.
# ==============================================================================
"""`LinearOperator` acting like a lower triangular matrix."""
from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from tensorflow.python.framework import dtypes
from tensorflow.python.framework import ops
from tensorflow.python.ops import array_ops
from tensorflow.python.ops import math_ops
from tensorflow.python.ops.linalg import linalg_impl as linalg
from tensorflow.python.ops.linalg import linear_operator
from tensorflow.python.ops.linalg import linear_operator_util
from tensorflow.python.util.tf_export import tf_export
__all__ = [
"LinearOperatorLowerTriangular",
]
@tf_export("linalg.LinearOperatorLowerTriangular")
class LinearOperatorLowerTriangular(linear_operator.LinearOperator):
"""`LinearOperator` acting like a [batch] square lower triangular matrix.
This operator acts like a [batch] lower triangular matrix `A` with shape
`[B1,...,Bb, N, N]` for some `b >= 0`. The first `b` indices index a
batch member. For every batch index `(i1,...,ib)`, `A[i1,...,ib, : :]` is
an `N x N` matrix.
`LinearOperatorLowerTriangular` is initialized with a `Tensor` having
dimensions `[B1,...,Bb, N, N]`. The upper triangle of the last two
dimensions is ignored.
```python
# Create a 2 x 2 lower-triangular linear operator.
tril = [[1., 2.], [3., 4.]]
operator = LinearOperatorLowerTriangular(tril)
# The upper triangle is ignored.
operator.to_dense()
==> [[1., 0.]
[3., 4.]]
operator.shape
==> [2, 2]
operator.log_abs_determinant()
==> scalar Tensor
x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> Shape [2, 4] Tensor
# Create a [2, 3] batch of 4 x 4 linear operators.
tril = tf.random_normal(shape=[2, 3, 4, 4])
operator = LinearOperatorLowerTriangular(tril)
```
#### Shape compatibility
This operator acts on [batch] matrix with compatible shape.
`x` is a batch matrix with compatible shape for `matmul` and `solve` if
```
operator.shape = [B1,...,Bb] + [N, N], with b >= 0
x.shape = [B1,...,Bb] + [N, R], with R >= 0.
```
#### Performance
Suppose `operator` is a `LinearOperatorLowerTriangular` of shape `[N, N]`,
and `x.shape = [N, R]`. Then
* `operator.matmul(x)` involves `N^2 * R` multiplications.
* `operator.solve(x)` involves `N * R` size `N` back-substitutions.
* `operator.determinant()` involves a size `N` `reduce_prod`.
If instead `operator` and `x` have shape `[B1,...,Bb, N, N]` and
`[B1,...,Bb, N, R]`, every operation increases in complexity by `B1*...*Bb`.
#### Matrix property hints
This `LinearOperator` is initialized with boolean flags of the form `is_X`,
for `X = non_singular, self_adjoint, positive_definite, square`.
These have the following meaning:
* If `is_X == True`, callers should expect the operator to have the
property `X`. This is a promise that should be fulfilled, but is *not* a
runtime assert. For example, finite floating point precision may result
in these promises being violated.
* If `is_X == False`, callers should expect the operator to not have `X`.
* If `is_X == None` (the default), callers should have no expectation either
way.
"""
def __init__(self,
tril,
is_non_singular=None,
is_self_adjoint=None,
is_positive_definite=None,
is_square=None,
name="LinearOperatorLowerTriangular"):
r"""Initialize a `LinearOperatorLowerTriangular`.
Args:
tril: Shape `[B1,...,Bb, N, N]` with `b >= 0`, `N >= 0`.
The lower triangular part of `tril` defines this operator. The strictly
upper triangle is ignored.
is_non_singular: Expect that this operator is non-singular.
This operator is non-singular if and only if its diagonal elements are
all non-zero.
is_self_adjoint: Expect that this operator is equal to its hermitian
transpose. This operator is self-adjoint only if it is diagonal with
real-valued diagonal entries. In this case it is advised to use
`LinearOperatorDiag`.
is_positive_definite: Expect that this operator is positive definite,
meaning the quadratic form `x^H A x` has positive real part for all
nonzero `x`. Note that we do not require the operator to be
self-adjoint to be positive-definite. See:
https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
is_square: Expect that this operator acts like square [batch] matrices.
name: A name for this `LinearOperator`.
Raises:
ValueError: If `is_square` is `False`.
"""
if is_square is False:
raise ValueError(
"Only square lower triangular operators supported at this time.")
is_square = True
with ops.name_scope(name, values=[tril]):
self._tril = ops.convert_to_tensor(tril, name="tril")
self._check_tril(self._tril)
self._tril = array_ops.matrix_band_part(tril, -1, 0)
self._diag = array_ops.matrix_diag_part(self._tril)
super(LinearOperatorLowerTriangular, self).__init__(
dtype=self._tril.dtype,
graph_parents=[self._tril],
is_non_singular=is_non_singular,
is_self_adjoint=is_self_adjoint,
is_positive_definite=is_positive_definite,
is_square=is_square,
name=name)
def _check_tril(self, tril):
"""Static check of the `tril` argument."""
allowed_dtypes = [
dtypes.float16,
dtypes.float32,
dtypes.float64,
dtypes.complex64,
dtypes.complex128,
]
dtype = tril.dtype
if dtype not in allowed_dtypes:
raise TypeError(
"Argument tril must have dtype in %s. Found: %s"
% (allowed_dtypes, dtype))
if tril.get_shape().ndims is not None and tril.get_shape().ndims < 2:
raise ValueError(
"Argument tril must have at least 2 dimensions. Found: %s"
% tril)
def _shape(self):
return self._tril.get_shape()
def _shape_tensor(self):
return array_ops.shape(self._tril)
def _assert_non_singular(self):
return linear_operator_util.assert_no_entries_with_modulus_zero(
self._diag,
message="Singular operator: Diagonal contained zero values.")
def _matmul(self, x, adjoint=False, adjoint_arg=False):
return linear_operator_util.matmul_with_broadcast(
self._tril, x, adjoint_a=adjoint, adjoint_b=adjoint_arg)
def _determinant(self):
return math_ops.reduce_prod(self._diag, reduction_indices=[-1])
def _log_abs_determinant(self):
return math_ops.reduce_sum(
math_ops.log(math_ops.abs(self._diag)), reduction_indices=[-1])
def _solve(self, rhs, adjoint=False, adjoint_arg=False):
rhs = linalg.adjoint(rhs) if adjoint_arg else rhs
return linear_operator_util.matrix_triangular_solve_with_broadcast(
self._tril, rhs, lower=True, adjoint=adjoint)
def _to_dense(self):
return self._tril
def _add_to_tensor(self, x):
return self._tril + x
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