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LinearOperator
acting like a [batch] square identity matrix.
Inherits From: LinearOperator
, Module
tf.linalg.LinearOperatorIdentity(
num_rows, batch_shape=None, dtype=None, is_non_singular=True,
is_self_adjoint=True, is_positive_definite=True, is_square=True,
assert_proper_shapes=False, name='LinearOperatorIdentity'
)
This operator acts like a [batch] identity 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.
LinearOperatorIdentity
is initialized with num_rows
, and optionally
batch_shape
, and dtype
arguments. If batch_shape
is None
, this
operator efficiently passes through all arguments. If batch_shape
is
provided, broadcasting may occur, which will require making copies.
# Create a 2 x 2 identity matrix.
operator = LinearOperatorIdentity(num_rows=2, dtype=tf.float32)
operator.to_dense()
==> [[1., 0.]
[0., 1.]]
operator.shape
==> [2, 2]
operator.log_abs_determinant()
==> 0.
x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> Shape [2, 4] Tensor, same as x.
y = tf.random.normal(shape=[3, 2, 4])
# Note that y.shape is compatible with operator.shape because operator.shape
# is broadcast to [3, 2, 2].
# This broadcast does NOT require copying data, since we can infer that y
# will be passed through without changing shape. We are always able to infer
# this if the operator has no batch_shape.
x = operator.solve(y)
==> Shape [3, 2, 4] Tensor, same as y.
# Create a 2-batch of 2x2 identity matrices
operator = LinearOperatorIdentity(num_rows=2, batch_shape=[2])
operator.to_dense()
==> [[[1., 0.]
[0., 1.]],
[[1., 0.]
[0., 1.]]]
# Here, even though the operator has a batch shape, the input is the same as
# the output, so x can be passed through without a copy. The operator is able
# to detect that no broadcast is necessary because both x and the operator
# have statically defined shape.
x = ... Shape [2, 2, 3]
operator.matmul(x)
==> Shape [2, 2, 3] Tensor, same as x
# Here the operator and x have different batch_shape, and are broadcast.
# This requires a copy, since the output is different size than the input.
x = ... Shape [1, 2, 3]
operator.matmul(x)
==> Shape [2, 2, 3] Tensor, equal to [x, x]
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]
Performance
If batch_shape
initialization arg is None
:
operator.matmul(x)
isO(1)
operator.solve(x)
isO(1)
operator.determinant()
isO(1)
If batch_shape
initialization arg is provided, and static checks cannot
rule out the need to broadcast:
operator.matmul(x)
isO(D1*...*Dd*N*R)
operator.solve(x)
isO(D1*...*Dd*N*R)
operator.determinant()
isO(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 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 | |
---|---|
num_rows
|
Scalar non-negative integer Tensor . Number of rows in the
corresponding identity matrix.
|
batch_shape
|
Optional 1-D integer Tensor . The shape of the leading
dimensions. If None , this operator has no leading dimensions.
|
dtype
|
Data type of the matrix that this operator represents. |
is_non_singular
|
Expect that this operator is non-singular. |
is_self_adjoint
|
Expect that this operator is equal to its hermitian transpose. |
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. |
assert_proper_shapes
|
Python bool . If False , only perform static
checks that initialization and method arguments have proper shape.
If True , and static checks are inconclusive, add asserts to the graph.
|
name
|
A name for this LinearOperator
|
Raises | |
---|---|
ValueError
|
If num_rows is determined statically to be non-scalar, or
negative.
|
ValueError
|
If batch_shape is determined statically to not be 1-D, or
negative.
|
ValueError
|
If any of the following is not True :
{is_self_adjoint, is_non_singular, is_positive_definite} .
|
TypeError
|
If num_rows or batch_shape is ref-type (e.g. Variable).
|
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 |
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.
|
parameters
|
Dictionary of parameters used to instantiate 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 |
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(
mat, name='add_to_tensor'
)
Add matrix represented by this operator to mat
. Equiv to I + mat
.
Args | |
---|---|
mat
|
Tensor with same dtype and shape broadcastable to self .
|
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 | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
Shape [B1,...,Bb] Tensor of same dtype as self .
|
determinant
determinant(
name='det'
)
Determinant for every batch member.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
Tensor with shape self.batch_shape and same dtype as self .
|
Raises | |
---|---|
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.linalg.diag_part(my_operator.to_dense())
==> [1., 2.]
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
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
.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
int32 Tensor
|
eigvals
eigvals(
name='eigvals'
)
Returns the eigenvalues of this linear operator.
If the operator is marked as self-adjoint (via is_self_adjoint
)
this computation can be more efficient.
Args | |
---|---|
name
|
A name for this Op .
|
Returns | |
---|---|
Shape [B1,...,Bb, N] Tensor of same dtype as self .
|
inverse
inverse(
name='inverse'
)
Returns the Inverse of this LinearOperator
.
Given A
representing this LinearOperator
, return a LinearOperator
representing A^-1
.
Args | |
---|---|
name
|
A name scope to use for ops added by this method. |
Returns | |
---|---|
LinearOperator representing inverse of this matrix.
|
Raises | |
---|---|
ValueError
|
When the LinearOperator is not hinted to be non_singular .
|