tf.linalg.LinearOperatorIdentity

LinearOperator acting like a [batch] square identity matrix.

Inherits From: LinearOperator, Module

tf.linalg.LinearOperatorIdentity(
    num_rows,
    batch_shape=None,
    dtype=None,
    is_non_singular=True,
    is_self_adjoint=True,
    is_positive_definite=True,
    is_square=True,
    assert_proper_shapes=False,
    name='LinearOperatorIdentity'
)

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.

num_rowsScalar non-negative integer Tensor. Number of rows in the corresponding identity matrix.
batch_shapeOptional 1-D integer Tensor. The shape of the leading dimensions. If None, this operator has no leading dimensions.
dtypeData type of the matrix that this operator represents.
is_non_singularExpect that this operator is non-singular.
is_self_adjointExpect that this operator is equal to its hermitian transpose.
is_positive_definiteExpect 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_squareExpect that this operator acts like square [batch] matrices.
assert_proper_shapesPython 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.
nameA name for this LinearOperator

ValueErrorIf num_rows is determined statically to be non-scalar, or negative.
ValueErrorIf batch_shape is determined statically to not be 1-D, or negative.
ValueErrorIf any of the following is not True: {is_self_adjoint, is_non_singular, is_positive_definite}.
TypeErrorIf num_rows or batch_shape is ref-type (e.g. Variable).

HReturns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

batch_shapeTensorShape 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_dimensionDimension (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.

dtypeThe DType of Tensors handled by this LinearOperator.
graph_parentsList of graph dependencies of this LinearOperator. (deprecated)

is_non_singular

is_positive_definite

is_self_adjoint

is_squareReturn True/False depending on if this operator is square.
parametersDictionary of parameters used to instantiate this LinearOperator.
range_dimensionDimension (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.

shapeTensorShape 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_rankRank (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(
    mat, name='add_to_tensor'
)

Add matrix represented by this operator to mat. Equiv to I + mat.

Args
matTensor with same dtype and shape broadcastable to self.
nameA 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
nameA 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
nameA 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
nameA 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
nameA 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
nameA 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
nameA name for this Op.

Returns
LinearOperator which represents the lower triangular matrix in the Cholesky decomposition.

Raises
ValueErrorWhen 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
nameA 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
nameA name for this Op.

Returns
Tensor with shape self.batch_shape and same dtype as self.

Raises
NotImplementedErrorIf 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
nameA name for this Op.

Returns
diag_partA 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
nameA 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
nameA 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
nameA name scope to use for ops added by this method.

Returns
LinearOperator representing inverse of this matrix.

Raises
ValueErrorWhen 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
nameA name for this Op.

Returns
Tensor with shape self.batch_shape and same dtype as self.

Raises
NotImplementedErrorIf 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
xLinearOperator or Tensor with compatible shape and same dtype as self. See class docstring for definition of compatibility.
adjointPython bool. If True, left multiply by the adjoint: A^H x.
adjoint_argPython bool. If True, compute A x^H where x^H is the hermitian transpose (transposition and complex conjugation).
nameA name for this Op.

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

matvec

View source

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

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
xTensor 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.
adjointPython bool. If True, left multiply by the adjoint: A^H x.
nameA 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
nameA 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
nameA 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
rhsTensor 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.
adjointPython bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
adjoint_argPython bool. If True, solve A X = rhs^H where rhs^H is the hermitian transpose (transposition and complex conjugation).
nameA name scope to use for ops added by this method.

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

Raises
NotImplementedErrorIf 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
rhsTensor 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.
adjointPython bool. If True, solve the system involving the adjoint of this LinearOperator: A^H X = rhs.
nameA name scope to use for ops added by this method.

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

Raises
NotImplementedErrorIf 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
nameA 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
nameA name for this Op.

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

__getitem__

View source

__getitem__(
    slices
)

__matmul__

View source

__matmul__(
    other
)