tf.linalg.LinearOperatorBlockDiag

TensorFlow 1 version View source on GitHub

Combines one or more LinearOperators in to a Block Diagonal matrix.

Inherits From: LinearOperator

This operator combines one or more linear operators [op1,...,opJ], building a new LinearOperator, whose underlying matrix representation is square and has each operator opi on the main diagonal, and zero's elsewhere.

Shape compatibility

If opj acts like a [batch] square matrix Aj, then op_combined acts like the [batch] square matrix formed by having each matrix Aj on the main diagonal.

Each opj is required to represent a square matrix, and hence will have shape batch_shape_j + [M_j, M_j].

If opj has shape batch_shape_j + [M_j, M_j], then the combined operator has shape broadcast_batch_shape + [sum M_j, sum M_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 combined shape is well defined, the combined operator's methods may fail due to lack of broadcasting ability in the defining operators' methods.

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.

# 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 [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.

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_.

TypeError If all operators do not have the same dtype.
ValueError If operators is empty or are non-square.

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

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.

Methods

add_to_tensor

View source

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.

adjoint

View source

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

View source

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

View source

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

View source

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

View source

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

View source

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.

cond

View source

Returns the condition number of this linear operator.

Args
name A name for this Op.

Returns
Shape [B1,...,Bb] Tensor of same dtype as self.

determinant

View source

Determinant for every batch member.

Arg