LinearOperatorLowerTriangular
Inherits From: LinearOperator
tf.contrib.linalg.LinearOperatorLowerTriangular
tf.linalg.LinearOperatorLowerTriangular
Defined in tensorflow/python/ops/linalg/linear_operator_lower_triangular.py
.
See the guide: Linear Algebra (contrib) > 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.
# 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)
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.
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
.
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:
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.is_X == False
, callers should expect the operator to not have X
.is_X == None
(the default), callers should have no expectation either way.batch_shape
TensorShape
of batch dimensions of this LinearOperator
.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns TensorShape([B1,...,Bb])
, equivalent to A.get_shape()[:-2]
TensorShape
, statically determined, may be undefined.
domain_dimension
Dimension (in the sense of vector spaces) of the domain of this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns N
.
Dimension
object.
dtype
The DType
of Tensor
s handled by this LinearOperator
.
graph_parents
List of graph dependencies of this LinearOperator
.
is_non_singular
is_positive_definite
is_self_adjoint
is_square
Return True/False
depending on if this operator is square.
name
Name prepended to all ops created by this LinearOperator
.
range_dimension
Dimension (in the sense of vector spaces) of the range of this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns M
.
Dimension
object.
shape
TensorShape
of this LinearOperator
.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns TensorShape([B1,...,Bb, M, N])
, equivalent to A.get_shape()
.
TensorShape
, statically determined, may be undefined.
tensor_rank
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns b + 2
.
name
: A name for this `Op.Python integer, or None if the tensor rank is undefined.
__init__
__init__( tril, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, is_square=None, name='LinearOperatorLowerTriangular' )
Initialize a LinearOperatorLowerTriangular
.
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. Allowed dtypes: float16
, float32
, float64
.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_matricesis_square
: Expect that this operator acts like square [batch] matrices.name
: A name for this LinearOperator
.TypeError
: If diag.dtype
is not an allowed type.ValueError
: If is_square
is False
.add_to_tensor
add_to_tensor( x, name='add_to_tensor' )
Add matrix represented by this operator to x
. Equivalent to A + x
.
x
: Tensor
with same dtype
and shape broadcastable to self.shape
.name
: A name to give this Op
.A Tensor
with broadcast shape and same dtype
as self
.
assert_non_singular
assert_non_singular(name='assert_non_singular')
Returns an Op
that asserts this operator is non singular.
This operator is considered non-singular if
ConditionNumber < max{100, range_dimension, domain_dimension} * eps, eps := np.finfo(self.dtype.as_numpy_dtype).eps
name
: A string name to prepend to created ops.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is singular.
assert_positive_definite
assert_positive_definite(name='assert_positive_definite')
Returns an Op
that asserts this operator is positive definite.
Here, positive definite means that 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.
name
: A name to give this Op
.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is not positive definite.
assert_self_adjoint
assert_self_adjoint(name='assert_self_adjoint')
Returns an Op
that asserts this operator is self-adjoint.
Here we check that this operator is exactly equal to its hermitian transpose.
name
: A string name to prepend to created ops.An Assert
Op
, that, when run, will raise an InvalidArgumentError
if the operator is not self-adjoint.
batch_shape_tensor
batch_shape_tensor(name='batch_shape_tensor')
Shape of batch dimensions of this operator, determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns a Tensor
holding [B1,...,Bb]
.
name
: A name for this `Op.int32
Tensor
determinant
determinant(name='det')
Determinant for every batch member.
name
: A name for this `Op.Tensor
with shape self.batch_shape
and same dtype
as self
.
NotImplementedError
: If self.is_square
is False
.diag_part
diag_part(name='diag_part')
Efficiently get the [batch] diagonal part of this operator.
If this operator has shape [B1,...,Bb, M, N]
, this returns a Tensor
diagonal
, of shape [B1,...,Bb, min(M, N)]
, where diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i]
.
my_operator = LinearOperatorDiag([1., 2.]) # Efficiently get the diagonal my_operator.diag_part() ==> [1., 2.] # Equivalent, but inefficient method tf.matrix_diag_part(my_operator.to_dense()) ==> [1., 2.]
name
: A name for this Op
.diag_part
: A Tensor
of same dtype
as self.domain_dimension_tensor
domain_dimension_tensor(name='domain_dimension_tensor')
Dimension (in the sense of vector spaces) of the domain of this operator.
Determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns N
.
name
: A name for this Op
.int32
Tensor
log_abs_determinant
log_abs_determinant(name='log_abs_det')
Log absolute value of determinant for every batch member.
name
: A name for this `Op.Tensor
with shape self.batch_shape
and same dtype
as self
.
NotImplementedError
: If self.is_square
is False
.matmul
matmul( x, adjoint=False, adjoint_arg=False, name='matmul' )
Transform [batch] matrix x
with left multiplication: x --> Ax
.
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] X = ... # shape [..., N, R], batch matrix, R > 0. Y = operator.matmul(X) Y.shape ==> [..., M, R] Y[..., :, r] = sum_j A[..., :, j] X[j, r]
x
: Tensor
with compatible shape and same dtype
as self
. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, left multiply by the adjoint: A^H x
.adjoint_arg
: Python bool
. If True
, compute A x^H
where x^H
is the hermitian transpose (transposition and complex conjugation).name
: A name for this `Op.A Tensor
with shape [..., M, R]
and same dtype
as self
.
matvec
matvec( x, adjoint=False, name='matvec' )
Transform [batch] vector x
with left multiplication: x --> Ax
.
# Make an operator acting like batch matric A. Assume A.shape = [..., M, N] operator = LinearOperator(...) X = ... # shape [..., N], batch vector Y = operator.matvec(X) Y.shape ==> [..., M] Y[..., :] = sum_j A[..., :, j] X[..., j]
x
: Tensor
with compatible shape and same dtype
as self
. x
is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, left multiply by the adjoint: A^H x
.name
: A name for this `Op.A Tensor
with shape [..., M]
and same dtype
as self
.
range_dimension_tensor
range_dimension_tensor(name='range_dimension_tensor')
Dimension (in the sense of vector spaces) of the range of this operator.
Determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns M
.
name
: A name for this Op
.int32
Tensor
shape_tensor
shape_tensor(name='shape_tensor')
Shape of this LinearOperator
, determined at runtime.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns a Tensor
holding [B1,...,Bb, M, N]
, equivalent to tf.shape(A)
.
name
: A name for this `Op.int32
Tensor
solve
solve( rhs, adjoint=False, adjoint_arg=False, name='solve' )
Solve (exact or approx) R
(batch) systems of equations: A X = rhs
.
The returned Tensor
will be close to an exact solution if A
is well conditioned. Otherwise closeness will vary. See class docstring for details.
Examples:
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] # Solve R > 0 linear systems for every member of the batch. RHS = ... # shape [..., M, R] X = operator.solve(RHS) # X[..., :, r] is the solution to the r'th linear system # sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r] operator.matmul(X) ==> RHS
rhs
: Tensor
with same dtype
as this operator and compatible shape. rhs
is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.adjoint
: Python bool
. If True
, solve the system involving the adjoint of this LinearOperator
: A^H X = rhs
.adjoint_arg
: Python bool
. If True
, solve A X = rhs^H
where rhs^H
is the hermitian transpose (transposition and complex conjugation).name
: A name scope to use for ops added by this method.Tensor
with shape [...,N, R]
and same dtype
as rhs
.
NotImplementedError
: If self.is_non_singular
or is_square
is False.solvevec
solvevec( rhs, adjoint=False, name='solve' )
Solve single equation with best effort: A X = rhs
.
The returned Tensor
will be close to an exact solution if A
is well conditioned. Otherwise closeness will vary. See class docstring for details.
Examples:
# Make an operator acting like batch matrix A. Assume A.shape = [..., M, N] operator = LinearOperator(...) operator.shape = [..., M, N] # Solve one linear system for every member of the batch. RHS = ... # shape [..., M] X = operator.solvevec(RHS) # X is the solution to the linear system # sum_j A[..., :, j] X[..., j] = RHS[..., :] operator.matvec(X) ==> RHS
rhs
: Tensor
with same dtype
as this operator. rhs
is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.adjoint
: Python bool
. If True
, solve the system involving the adjoint of this LinearOperator
: A^H X = rhs
.name
: A name scope to use for ops added by this method.Tensor
with shape [...,N]
and same dtype
as rhs
.
NotImplementedError
: If self.is_non_singular
or is_square
is False.tensor_rank_tensor
tensor_rank_tensor(name='tensor_rank_tensor')
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix A
with A.shape = [B1,...,Bb, M, N]
, then this returns b + 2
.
name
: A name for this `Op.int32
Tensor
, determined at runtime.
to_dense
to_dense(name='to_dense')
Return a dense (batch) matrix representing this operator.
trace
trace(name='trace')
Trace of the linear operator, equal to sum of self.diag_part()
.
If the operator is square, this is also the sum of the eigenvalues.
name
: A name for this Op
.Shape [B1,...,Bb]
Tensor
of same dtype
as self
.
© 2018 The TensorFlow Authors. All rights reserved.
Licensed under the Creative Commons Attribution License 3.0.
Code samples licensed under the Apache 2.0 License.
https://www.tensorflow.org/api_docs/python/tf/linalg/LinearOperatorLowerTriangular