tf.contrib.linalg.LinearOperatorComposition

class tf.contrib.linalg.LinearOperatorComposition

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 = LinearOperatorMatrix([[1., 2.], [3., 4.]])
operator_2 = LinearOperatorMatrix([[1., 0.], [0., 1.]])
operator = LinearOperatorComposition([operator_1, operator_2])

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

operator.shape
==> [2, 2]

operator.log_determinant()
==> scalar Tensor

x = ... Shape [2, 4] Tensor
operator.apply(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 = LinearOperatorMatrix(matrix)

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

# Compose to create a [2, 3] batch of 4 x 6 operators.
opeartor_46 = LinearOperatorComposition([operator_45, operator_56])

# Create a shape [2, 3, 6, 2] vector.
x = tf.random_normal(shape=[2, 3, 6, 2])
operator.apply(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. 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

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__(operators, is_non_singular=None, is_self_adjoint=None, is_positive_definite=None, name=None)

Initialize a LinearOperatorComposition.

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

Args:

• operators: Iterable of LinearOperator objects, each with the same dtype and composible 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 real part of all eigenvalues is positive. 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
• 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(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.

apply(x, adjoint=False, name='apply')

Transform x with left multiplication: x --> Ax.

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.
• name: A name for this `Op.

Returns:

A Tensor with shape [..., M, R] and same dtype as self.

assert_non_singular(name='assert_non_singular')

Returns an Op that asserts this operator is non singular.

assert_positive_definite(name='assert_positive_definite')

Returns an Op that asserts this operator is positive definite.

Here, positive definite means the real part of all eigenvalues is positive. We do not require the operator to be self-adjoint.

Args:

• name: A name to give this Op.

Returns:

An Op that asserts this operator is positive definite.

assert_self_adjoint(name='assert_self_adjoint')

Returns an Op that asserts this operator is self-adjoint.

batch_shape_dynamic(name='batch_shape_dynamic')

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(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.

domain_dimension_dynamic(name='domain_dimension_dynamic')

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(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.

range_dimension_dynamic(name='range_dimension_dynamic')

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_dynamic(name='shape_dynamic')

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(rhs, adjoint=False, name='solve')

Solve R (batch) systems of equations exactly: A X = rhs.

Examples:

# Create an operator acting like a 10 x 2 x 2 matrix.
operator = LinearOperator(...)
operator.shape # = 10 x 2 x 2

# Solve one linear system (R = 1) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 1
X = operator.solve(RHS)  # shape 10 x 2 x 1

# Solve five linear systems (R = 5) for every member of the length 10 batch.
RHS = ... # shape 10 x 2 x 5
X = operator.solve(RHS)
X[3, :, 2]  # Solution to the linear system A[3, :, :] X = RHS[3, :, 2]

Args:

• rhs: Tensor with same dtype as this operator and compatible shape. See class docstring for definition of compatibility.
• adjoint: Python bool. If True, solve the system involving the adjoint of this LinearOperator.
• name: A name scope to use for ops added by this method.

Returns:

Tensor with shape [...,N, R] and same dtype as rhs.

Raises:

• ValueError: If self.is_non_singular is False.

tensor_rank_dynamic(name='tensor_rank_dynamic')

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(name='to_dense')

Return a dense (batch) matrix representing this operator.

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