tf.linalg.LinearOperatorIdentity

TensorFlow 1 version

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LinearOperator acting like a [batch] square identity matrix.

Inherits From: LinearOperator, Module

View aliases

Compat aliases for migration

See Migration guide for more details.

tf.compat.v1.linalg.LinearOperatorIdentity

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.

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}`.
`TypeError`	If `num_rows` or `batch_shape` is ref-type (e.g. Variable).

Attributes
`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 `Tensor`s handled by this `LinearOperator`.
`graph_parents`	List of graph dependencies of this `LinearOperator`. (deprecated) Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version. Instructions for updating: Do not call `graph_parents`.
`is_non_singular`
`is_positive_definite`
`is_self_adjoint`
`is_square`	Return `True/False` depending on if this operator is square.
`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(
    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`.

`adjoint`

View source

adjoint(
    name='adjoint'
)

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='cholesky'
)

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='inverse'
)

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

tf.linalg.LinearOperatorIdentity

View aliases

Shape compatibility

Performance

Matrix property hints

Args

Raises

Attributes

Methods

add_to_tensor

adjoint

assert_non_singular

assert_positive_definite

assert_self_adjoint

batch_shape_tensor

cholesky

cond

determinant

diag_part

domain_dimension_tensor

eigvals

inverse

log_abs_determinant

`add_to_tensor`

`adjoint`

`assert_non_singular`

`assert_positive_definite`

`assert_self_adjoint`

`batch_shape_tensor`

`cholesky`

`cond`

`determinant`

`diag_part`

`domain_dimension_tensor`

`eigvals`

`inverse`

`log_abs_determinant`