tf.linalg.LinearOperatorBlockDiag

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

Inherits From: LinearOperator, Module

tf.linalg.LinearOperatorBlockDiag(
    operators,
    is_non_singular=None,
    is_self_adjoint=None,
    is_positive_definite=None,
    is_square=True,
    name=None
)

Used in the notebooks

Used in the tutorials

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.

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

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.

Methods

add_to_tensor

View source

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.

adjoint

View source

adjoint(
    name: str = 'adjoint'
) -> 'LinearOperator'

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

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

View source

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

View source

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

View source

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

cholesky

View source

cholesky(
    name: str = 'cholesky'
) -> 'LinearOperator'

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

cond(
    name='cond'
)

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(
    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

View source

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.linalg.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

View source

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

eigvals

View source

eigvals(
    name='eigvals'
)

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via is_self_adjoint) this computation can be more efficient.

Args
name A name for this Op.

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

inverse

View source

inverse(
    name: str = 'inverse'
) -> 'LinearOperator'

Returns the Inverse of this LinearOperator.

Given A representing this LinearOperator, return a LinearOperator representing A^-1.

Args
name A name scope to use for ops added by this method.

Returns
LinearOperator representing inverse of this matrix.

Raises
ValueError When the LinearOperator is not hinted to be non_singular.

log_abs_determinant

View source

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

View source

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 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].

matvec

View source

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, 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.

range_dimension_tensor

View source

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

View source

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

View source

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, 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.

Raises
NotImplementedError If self.is_non_singular or is_square is False.

solvevec

View source

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, 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.
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

View source

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

View source

to_dense(
    name='to_dense'
)

Return a dense (batch) matrix representing this operator.

trace

View source

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.

__getitem__

View source

__getitem__(
    slices
)

__matmul__

View source

__matmul__(
    other
)