tf.linalg.LinearOperatorIdentity

Class `LinearOperatorIdentity`

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

LinearOperator acting like a [batch] square identity matrix.

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.

`init`

__init__(
    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'
)

Initialize a LinearOperatorIdentity.

The LinearOperatorIdentity is initialized with arguments defining dtype and shape.

This operator is able to broadcast the leading (batch) dimensions, which sometimes requires copying data. If batch_shape is None, the operator can take arguments of any batch shape without copying. See examples.

Args:

num_rows: Scalar non-negative integer Tensor. Number of rows in the corresponding identity matrix.
batch_shape: Optional 1-D integer Tensor. The shape of the leading dimensions. If None, this operator has no leading dimensions.
dtype: Data type of the matrix that this operator represents.
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.
assert_proper_shapes: Python 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.
name: A name for this LinearOperator

Raises:

ValueError: If num_rows is determined statically to be non-scalar, or negative.
ValueError: If batch_shape is determined statically to not be 1-D, or negative.
ValueError: If any of the following is not True: {is_self_adjoint, is_non_singular, is_positive_definite}.

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.

`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

`add_to_tensor`

add_to_tensor(
    mat,
    name='add_to_tensor'
)

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

Args:

mat: Tensor with same dtype and shape broadcastable to self.
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.

tf.linalg.LinearOperatorIdentity

Class LinearOperatorIdentity

Shape compatibility

Performance

Matrix property hints

__init__

Args:

Raises:

Properties

batch_shape

Returns:

domain_dimension

Returns:

dtype

graph_parents

is_non_singular

is_positive_definite

is_self_adjoint

is_square

name

range_dimension

Returns:

shape

Returns:

tensor_rank

Args:

Returns:

Methods

add_to_tensor

Args:

Returns:

assert_non_singular

Args:

Returns:

assert_positive_definite

Args:

Returns:

assert_self_adjoint

Args:

Returns:

batch_shape_tensor

Args:

Returns:

determinant

Args:

Returns:

Raises:

diag_part

Args:

Returns:

domain_dimension_tensor

Args:

Returns:

log_abs_determinant

Args:

Returns:

Raises:

matmul

Args:

Returns:

matvec

Args:

Returns:

range_dimension_tensor

Args:

Returns:

shape_tensor

Args:

Returns:

solve

Args:

Returns:

Raises:

solvevec

Args:

Returns:

Raises:

tensor_rank_tensor

Args:

Returns:

Class `LinearOperatorIdentity`

`init`

`batch_shape`

`domain_dimension`

`dtype`

`graph_parents`

`is_non_singular`

`is_positive_definite`

`is_self_adjoint`

`is_square`

`name`

`range_dimension`

`shape`

`tensor_rank`

`add_to_tensor`

`assert_non_singular`

`assert_positive_definite`

`assert_self_adjoint`

`batch_shape_tensor`

`determinant`

`diag_part`

`domain_dimension_tensor`

`log_abs_determinant`

`matmul`

`matvec`

`range_dimension_tensor`

`shape_tensor`

`solve`

`solvevec`

`tensor_rank_tensor`

`to_dense`

`trace`