This operator combines one or more linear operators [op1,...,opJ],
building a new LinearOperator, whose underlying matrix representation
has each operator opi on the main diagonal, and zero's elsewhere.
Shape compatibility
If opj acts like a [batch] matrix Aj, then op_combined acts like
the [batch] matrix formed by having each matrix Aj on the main
diagonal.
Each opj is required to represent a matrix, and hence will have
shape batch_shape_j + [M_j, N_j].
If opj has shape batch_shape_j + [M_j, N_j], then the combined operator
has shape broadcast_batch_shape + [sum M_j, sum N_j], where
broadcast_batch_shape is the mutual broadcast of batch_shape_j,
j = 1,...,J, assuming the intermediate batch shapes broadcast.
Arguments to matmul, matvec, solve, and solvevec may either be single
Tensors or lists of Tensors that are interpreted as blocks. The jth
element of a blockwise list of Tensors must have dimensions that match
opj for the given method. If a list of blocks is input, then a list of
blocks is returned as well.
When the opj are not guaranteed to be square, this operator's methods might
fail due to the combined operator not being square and/or lack of efficient
methods.
# Create a 4 x 4 linear operator combined of two 2 x 2 operators.
operator_1 = LinearOperatorFullMatrix([[1., 2.], [3., 4.]])
operator_2 = LinearOperatorFullMatrix([[1., 0.], [0., 1.]])
operator = LinearOperatorBlockDiag([operator_1, operator_2])
operator.to_dense()
==> [[1., 2., 0., 0.],
[3., 4., 0., 0.],
[0., 0., 1., 0.],
[0., 0., 0., 1.]]
operator.shape
==> [4, 4]
operator.log_abs_determinant()
==> scalar Tensor
x1 = ... # Shape [2, 2] Tensor
x2 = ... # Shape [2, 2] Tensor
x = tf.concat([x1, x2], 0) # Shape [2, 4] Tensor
operator.matmul(x)
==> tf.concat([operator_1.matmul(x1), operator_2.matmul(x2)])
# Create a 5 x 4 linear operator combining three blocks.
operator_1 = LinearOperatorFullMatrix([[1.], [3.]])
operator_2 = LinearOperatorFullMatrix([[1., 6.]])
operator_3 = LinearOperatorFullMatrix([[2.], [7.]])
operator = LinearOperatorBlockDiag([operator_1, operator_2, operator_3])
operator.to_dense()
==> [[1., 0., 0., 0.],
[3., 0., 0., 0.],
[0., 1., 6., 0.],
[0., 0., 0., 2.]]
[0., 0., 0., 7.]]
operator.shape
==> [5, 4]
# Create a [2, 3] batch of 4 x 4 linear operators.
matrix_44 = tf.random.normal(shape=[2, 3, 4, 4])
operator_44 = LinearOperatorFullMatrix(matrix)
# Create a [1, 3] batch of 5 x 5 linear operators.
matrix_55 = tf.random.normal(shape=[1, 3, 5, 5])
operator_55 = LinearOperatorFullMatrix(matrix_55)
# Combine to create a [2, 3] batch of 9 x 9 operators.
operator_99 = LinearOperatorBlockDiag([operator_44, operator_55])
# Create a shape [2, 3, 9] vector.
x = tf.random.normal(shape=[2, 3, 9])
operator_99.matmul(x)
==> Shape [2, 3, 9] Tensor
# Create a blockwise list of vectors.
x = [tf.random.normal(shape=[2, 3, 4]), tf.random.normal(shape=[2, 3, 5])]
operator_99.matmul(x)
==> [Shape [2, 3, 4] Tensor, Shape [2, 3, 5] Tensor]
Performance
The performance of LinearOperatorBlockDiag on any operation is equal to
the sum of the individual operators' operations.
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
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.
If is_X == False, callers should expect the operator to not have X.
If is_X == None (the default), callers should have no expectation either
way.
Args
operators
Iterable of LinearOperator objects, each with
the same dtype and composable shape.
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.
This is true by default, and will raise a ValueError otherwise.
name
A name for this LinearOperator. Default is the individual
operators names joined with _o_.
Raises
TypeError
If all operators do not have the same dtype.
ValueError
If operators is empty or are non-square.
Attributes
H
Returns the adjoint of the current LinearOperator.
Given A representing this LinearOperator, return A*.
Note that calling self.adjoint() and self.H are equivalent.
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.shape[:-2]
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.
dtype
The DType of Tensors 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.
operators
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 A with
A.shape = [B1,...,Bb, M, N], then this returns M.
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.shape.
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.
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 AssertOp, that, when run, will raise an InvalidArgumentError if
the operator is not positive definite.
Efficiently get the [batch] diagonal part of this operator.
If this operator has shape [B1,...,Bb, M, N], this returns a
Tensordiagonal, 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.]
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, Tensor with compatible shape and same dtype as
self, or a blockwise iterable of LinearOperators or Tensors. See
class docstring for definition of shape 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, or if x is blockwise, a list of Tensors with shapes that
concatenate to [..., M, R].
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, or an
iterable of Tensors (for blockwise operators). Tensors are treated
a [batch] vectors, 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.
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).
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
Args
rhs
Tensor with same dtype as this operator and compatible shape,
or a list of Tensors (for blockwise operators). Tensors are treated
like a [batch] matrices 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.
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
Args
rhs
Tensor with same dtype as this operator, or list of Tensors
(for blockwise operators). Tensors are treated as [batch] vectors,
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.
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