tf.linalg.LinearOperatorComposition

Class LinearOperatorComposition

Inherits From: LinearOperator

Aliases:

• Class tf.contrib.linalg.LinearOperatorComposition
• Class tf.linalg.LinearOperatorComposition

Defined in tensorflow/python/ops/linalg/linear_operator_composition.py.

See the guide: Linear Algebra (contrib) > LinearOperator

Composes one or more LinearOperators.

This operator composes one or more linear operators [op1,...,opJ], building a new LinearOperator with action defined by:

op_composed(x) := op1(op2(...(opJ(x)...))

If opj acts like [batch] matrix Aj, then op_composed acts like the [batch] matrix formed with the multiplication A1 A2...AJ.

If opj has shape batch_shape_j + [M_j, N_j], then we must have N_j = M_{j+1}, in which case the composed operator has shape equal to broadcast_batch_shape + [M_1, N_J], where broadcast_batch_shape is the mutual broadcast of batch_shape_j, j = 1,...,J, assuming the intermediate batch shapes broadcast. Even if the composed shape is well defined, the composed operator's methods may fail due to lack of broadcasting ability in the defining operators' methods.

# Create a 2 x 2 linear operator composed of two 2 x 2 operators.
operator_1 = LinearOperatorFullMatrix([[1., 2.], [3., 4.]])
operator_2 = LinearOperatorFullMatrix([[1., 0.], [0., 1.]])
operator = LinearOperatorComposition([operator_1, operator_2])

operator.to_dense()
==> [[1., 2.]
     [3., 4.]]

operator.shape
==> [2, 2]

operator.log_abs_determinant()
==> scalar Tensor

x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> Shape [2, 4] Tensor

# Create a [2, 3] batch of 4 x 5 linear operators.
matrix_45 = tf.random_normal(shape=[2, 3, 4, 5])
operator_45 = LinearOperatorFullMatrix(matrix)

# Create a [2, 3] batch of 5 x 6 linear operators.
matrix_56 = tf.random_normal(shape=[2, 3, 5, 6])
operator_56 = LinearOperatorFullMatrix(matrix_56)

# Compose to create a [2, 3] batch of 4 x 6 operators.
operator_46 = LinearOperatorComposition([operator_45, operator_56])

# Create a shape [2, 3, 6, 2] vector.
x = tf.random_normal(shape=[2, 3, 6, 2])
operator.matmul(x)
==> Shape [2, 3, 4, 2] Tensor

Performance

The performance of LinearOperatorComposition 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.

Properties

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.get_shape()[:-2]

Returns:

TensorShape, statically determined, may be undefined.

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.

Returns:

Dimension object.

dtype

The DType of Tensors 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.

name

Name prepended to all ops created by this LinearOperator.

operators

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.

Returns:

Dimension object.

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.get_shape().

Returns:

TensorShape, statically determined, may be undefined.

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.

Args:

• name: A name for this `Op.

Returns:

Python integer, or None if the tensor rank is undefined.

Methods

__init__

__init__(
    operators,
    is_non_singular=None,
    is_self_adjoint=None,
    is_positive_definite=None,
    is_square=None,
    name=None
)

Initialize a LinearOperatorComposition.

LinearOperatorComposition is initialized with a list of operators [op_1,...,op_J]. For the matmul method to be well defined, the composition op_i.matmul(op_{i+1}(x)) must be defined. Other methods have similar constraints.

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.
• 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.

add_to_tensor

add_to_tensor(
    x,
    name='add_to_tensor'
)

Add matrix represented by this operator to x. Equivalent to A + x.

Args:

• x: Tensor with same dtype and shape broadcastable to self.shape.
• name: A name to give this Op.

Returns:

A Tensor with broadcast shape and same dtype as self.

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

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.matrix_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

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: 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 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.

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.

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.

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.
• 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.