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) 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)
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
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 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.
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 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.
Transform [batch] vector x with left multiplication: x --> Ax.
# Make an operator acting like batch matrix 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.
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.
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.
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.
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.
View source on GitHub