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LinearOperator
acting like a [batch] square identity matrix.
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]
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]
If batch_shape
initialization arg is None
:
operator.matmul(x)
is O(1)
operator.solve(x)
is O(1)
operator.determinant()
is O(1)
If batch_shape
initialization arg is provided, and static checks cannot rule out the need to broadcast:
operator.matmul(x)
is O(D1*...*Dd*N*R)
operator.solve(x)
is O(D1*...*Dd*N*R)
operator.determinant()
is O(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.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 . |
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 |
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 |
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. |
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 |
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 . |
log_abs_determinant
log_abs_determinant( name='log_abs_det' )
Log absolute value of 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 . |
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]
Args | |
---|---|
x | LinearOperator or 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 . |
Returns | |
---|---|
A LinearOperator or 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]
Args | |
---|---|
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 . |
Returns | |
---|---|
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
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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)
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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.
# 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
Args | |
---|---|
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. |
Returns | |
---|---|
Tensor with shape [...,N, R] and same dtype as rhs . |
Raises | |
---|---|
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.
# 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
Args | |
---|---|
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. |
Returns | |
---|---|
Tensor with shape [...,N] and same dtype as rhs . |
Raises | |
---|---|
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
.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
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.
Args | |
---|---|
name | A name for this Op . |
Returns | |
---|---|
Shape [B1,...,Bb] Tensor of same dtype as self . |
© 2020 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/versions/r1.15/api_docs/python/tf/linalg/LinearOperatorIdentity