tf.linalg.LinearOperator

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Base class defining a [batch of] linear operator[s].

Inherits From: Module

tf.linalg.LinearOperator(
    dtype,
    graph_parents=None,
    is_non_singular=None,
    is_self_adjoint=None,
    is_positive_definite=None,
    is_square=None,
    name=None,
    parameters=None
)

Subclasses of LinearOperator provide access to common methods on a (batch) matrix, without the need to materialize the matrix. This allows:

• Matrix free computations
• Operators that take advantage of special structure, while providing a consistent API to users.

Subclassing

To enable a public method, subclasses should implement the leading-underscore version of the method. The argument signature should be identical except for the omission of name="...". For example, to enable matmul(x, adjoint=False, name="matmul") a subclass should implement _matmul(x, adjoint=False).

Performance contract

Subclasses should only implement the assert methods (e.g. assert_non_singular) if they can be done in less than O(N^3) time.

Class docstrings should contain an explanation of computational complexity. Since this is a high-performance library, attention should be paid to detail, and explanations can include constants as well as Big-O notation.

Shape compatibility

LinearOperator subclasses should operate on a [batch] matrix with compatible shape. Class docstrings should define what is meant by compatible shape. Some subclasses may not support batching.

Examples:

x is a batch matrix with compatible shape for matmul if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
x.shape =   [B1,...,Bb] + [N, R]

rhs is a batch matrix with compatible shape for solve if

operator.shape = [B1,...,Bb] + [M, N],  b >= 0,
rhs.shape =   [B1,...,Bb] + [M, R]

Example docstring for subclasses.

This operator acts like a (batch) matrix A with shape [B1,...,Bb, M, N] for some b >= 0. The first b indices index a batch member. For every batch index (i1,...,ib), A[i1,...,ib, : :] is an m x n matrix. Again, this matrix A may not be materialized, but for purposes of identifying and working with compatible arguments the shape is relevant.

Examples:

some_tensor = ... shape = ????
operator = MyLinOp(some_tensor)

operator.shape()
==> [2, 4, 4]

operator.log_abs_determinant()
==> Shape [2] Tensor

x = ... Shape [2, 4, 5] Tensor

operator.matmul(x)
==> Shape [2, 4, 5] Tensor

Shape compatibility

This operator acts on batch matrices with compatible shape. FILL IN WHAT IS MEANT BY COMPATIBLE SHAPE

Performance

FILL THIS IN

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.

Initialization parameters

All subclasses of LinearOperator are expected to pass a parameters argument to super().__init__(). This should be a dict containing the unadulterated arguments passed to the subclass __init__. For example, MyLinearOperator with an initializer should look like:

def __init__(self, operator, is_square=False, name=None):
   parameters = dict(
       operator=operator,
       is_square=is_square,
       name=name
   )
   ...
   super().__init__(..., parameters=parameters)

Users can then access my_linear_operator.parameters to see all arguments passed to its initializer.

dtype The type of the this LinearOperator. Arguments to matmul and solve will have to be this type. graph_parents (Deprecated) Python list of graph prerequisites of this LinearOperator Typically tensors that are passed during initialization is_non_singular Expect that this operator is non-singular. is_self_adjoint Expect that this operator is equal to its hermitian transpose. If dtype is real, this is equivalent to being symmetric. 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. parameters Python dict of parameters used to instantiate this LinearOperator.

ValueError If any member of graph_parents is None or not a Tensor. ValueError If hints are set incorrectly.

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_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(
    x, name='add_to_tensor'
)

Add matrix represented by this operator to x. Equivalent to A + x.

x Tensor with same dtype and shape broadcastable to self.shape. name A name to give this Op.

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.

name A name for this Op.

assert_non_singular

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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

name A string name to prepend to created ops.

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.

name A name to give this Op.

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.

name A string name to prepend to created ops.

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].

name A name for this Op.

cholesky

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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.

name A name for this Op.

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.

name A name for this Op.

determinant

View source

determinant(
    name='det'
)

Determinant for every batch member.

name A name for this Op.

NotImplementedError If self.is_square is False.

diag_part

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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.]

name A name for this Op.

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.

name A name for this Op.

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.

name A name for this Op.

inverse

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

Returns the Inverse of this LinearOperator.

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

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

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.

name A name for this Op.

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]

x LinearOperator or 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.

matvec

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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]

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.

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.

name A name for this Op.

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).

name A name for this Op.

solve

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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

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.

NotImplementedError If self.is_non_singular or is_square is False.

solvevec

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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

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.

NotImplementedError If self.is_non_singular or is_square is False.

tensor_rank_tensor

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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.

name A name for this Op.

to_dense

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to_dense(
    name='to_dense'
)

Return a dense (batch) matrix representing this operator.

trace

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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.

name A name for this Op.

__getitem__

View source

__getitem__(
    slices
)

__matmul__

View source

__matmul__(
    other
)