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LinearOperator
acting like a scaled [batch] identity matrix A = c I
.
tf.linalg.LinearOperatorScaledIdentity(
num_rows, multiplier, is_non_singular=None, is_self_adjoint=None,
is_positive_definite=None, is_square=True, assert_proper_shapes=False,
name='LinearOperatorScaledIdentity'
)
This operator acts like a scaled [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
a scaled version of the N x N
identity matrix.
LinearOperatorIdentity
is initialized with num_rows
, and a multiplier
(a Tensor
) of shape [B1,...,Bb]
. N
is set to num_rows
, and the
multiplier
determines the scale for each batch member.
# Create a 2 x 2 scaled identity matrix.
operator = LinearOperatorIdentity(num_rows=2, multiplier=3.)
operator.to_dense()
==> [[3., 0.]
[0., 3.]]
operator.shape
==> [2, 2]
operator.log_abs_determinant()
==> 2 * Log[3]
x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> 3 * 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].
x = operator.solve(y)
==> 3 * x
# Create a 2-batch of 2x2 identity matrices
operator = LinearOperatorIdentity(num_rows=2, multiplier=5.)
operator.to_dense()
==> [[[5., 0.]
[0., 5.]],
[[5., 0.]
[0., 5.]]]
x = ... Shape [2, 2, 3]
operator.matmul(x)
==> 5 * x
# Here the operator and x have different batch_shape, and are broadcast.
x = ... Shape [1, 2, 3]
operator.matmul(x)
==> 5 * 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
operator.matmul(x)
isO(D1*...*Dd*N*R)
operator.solve(x)
isO(D1*...*Dd*N*R)
operator.determinant()
isO(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 | |
---|---|
num_rows
|
Scalar non-negative integer Tensor . Number of rows in the
corresponding identity matrix.
|
multiplier
|
Tensor of shape [B1,...,Bb] , or [] (a scalar).
|
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.
|
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.
|
multiplier
|
The [batch] scalar Tensor , c in cI .
|
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
|