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LinearOperator
acting like a [batch] of toeplitz matrices.
Inherits From: LinearOperator
tf.linalg.LinearOperatorToeplitz(
col, row, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None,
is_square=None, name='LinearOperatorToeplitz'
)
This operator acts like a [batch] Toeplitz 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. This matrix A
is not materialized, but for
purposes of broadcasting this shape will be relevant.
Description in terms of toeplitz matrices
Toeplitz means that A
has constant diagonals. Hence, A
can be generated
with two vectors. One represents the first column of the matrix, and the
other represents the first row.
Below is a 4 x 4 example:
A = |a b c d|
|e a b c|
|f e a b|
|g f e a|
Example of a Toeplitz operator.
# Create a 3 x 3 Toeplitz operator.
col = [1., 2., 3.]
row = [1., 4., -9.]
operator = LinearOperatorToeplitz(col, row)
operator.to_dense()
==> [[1., 4., -9.],
[2., 1., 4.],
[3., 2., 1.]]
operator.shape
==> [3, 3]
operator.log_abs_determinant()
==> scalar Tensor
x = ... Shape [3, 4] Tensor
operator.matmul(x)
==> Shape [3, 4] Tensor
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 = [C1,...,Cc] + [N, R],
and [C1,...,Cc] broadcasts with [B1,...,Bb] to [D1,...,Dd]
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 propertyX
. 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 haveX
. - If
is_X == None
(the default), callers should have no expectation either way.
Args | |
---|---|
col
|
Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0 .
The first column of the operator. Allowed dtypes: float16 , float32 ,
float64 , complex64 , complex128 . Note that the first entry of
col is assumed to be the same as the first entry of row .
|
row
|
Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0 .
The first row of the operator. Allowed dtypes: float16 , float32 ,
float64 , complex64 , complex128 . Note that the first entry of
row is assumed to be the same as the first entry of col .
|
is_non_singular
|
Expect that this operator is non-singular. |
is_self_adjoint
|
Expect that this operator is equal to its hermitian
transpose. If diag.dtype is real, this is auto-set to True .
|
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 .
|
Attributes | |
---|---|
H
|
Returns the adjoint of the current LinearOperator .
Given |
batch_shape
|
TensorShape of batch dimensions of this LinearOperator .
If this operator acts like the batch matrix |
col
|
|
domain_dimension
|
Dimension (in the sense of vector spaces) of the domain of this operator.
If this operator acts like the batch matrix |
dtype
|
The DType of Tensor s handled by this LinearOperator .
|
graph_parents
|
List of graph dependencies of this LinearOperator . (deprecated)
|
is_non_singular
|
|
is_positive_definite
|
|
is_self_adjoint
|
|
is_square
|
Return True/False depending on if this operator is square.
|
range_dimension
|
Dimension (in the sense of vector spaces) of the range of this operator.
If this operator acts like the batch matrix |
row
|
|
shape
|
TensorShape of this LinearOperator .
If this operator acts like the batch matrix |
tensor_rank
|
Rank (in the sense of tensors) of matrix corresponding to this operator.
If this operator acts like the batch matrix |
Methods
add_to_tensor
add_to_tensor(
x, name='add_to_tensor'
)
Add matrix represented by this operator to x
. Equivalent to A + x
.
Args | |
---|---|
x
|
Tensor with same dtype and shape broadcastable to self.shape .
|
name
|
A name to give this Op .
|
Returns | |
---|---|
A Tensor with broadcast shape and same dtype as self .
|
adjoint
adjoint(
name='adjoint'
)
Returns the adjoint of the current LinearOperator
.
Given A
representing this LinearOperator
, return A*
.
Note that calling self.adjoint()
and self.H
are equivalent.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
LinearOperator which represents the adjoint of this LinearOperator .
|
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
Args | |
---|---|
name
|
A string name to prepend to created ops. |
Returns | |
---|---|
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.
Args | |
---|---|
name
|
A name to give this Op .
|
Returns | |
---|---|
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.
Args | |
---|---|
name
|
A string name to prepend to created ops. |
Returns | |
---|---|
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]
.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
int32 Tensor
|
cholesky
cholesky(
name='cholesky'
)
Returns a Cholesky factor as a LinearOperator
.
Given A
representing this LinearOperator
, if A
is positive definite
self-adjoint, return L
, where A = L L^T
, i.e. the cholesky
decomposition.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
LinearOperator which represents the lower triangular matrix
in the Cholesky decomposition.
|
Raises | |
---|---|
ValueError
|
When the LinearOperator is not hinted to be positive
definite and self adjoint.
|
cond
cond(
name='cond'
)
Returns the condition number of this linear operator.
Args | |
---|---|