tf.linalg.LinearOperatorInversion

TensorFlow 1 version View source on GitHub

LinearOperator representing the inverse of another operator.

Inherits From: LinearOperator

This operator represents the inverse of another operator.

# Create a 2 x 2 linear operator.
operator = LinearOperatorFullMatrix([[1., 0.], [0., 2.]])
operator_inv = LinearOperatorInversion(operator)

operator_inv.to_dense()
==> [[1., 0.]
     [0., 0.5]]

operator_inv.shape
==> [2, 2]

operator_inv.log_abs_determinant()
==> - log(2)

x = ... Shape [2, 4] Tensor
operator_inv.matmul(x)
==> Shape [2, 4] Tensor, equal to operator.solve(x)

Performance

The performance of LinearOperatorInversion depends on the underlying operators performance: solve and matmul are swapped, and determinant is inverted.

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.

operator LinearOperator object. If operator.is_non_singular == False, an exception is raised. We do allow operator.is_non_singular == None, in which case this operator will have is_non_singular == None. Similarly for is_self_adjoint and is_positive_definite.
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.
name A name for this LinearOperator. Default is operator.name + "_inv".

ValueError If operator.is_non_singular is False.

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.
operator The operator before inversion.
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

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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

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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

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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

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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.