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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 )

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.

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)`

.

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.

`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.

`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]

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.

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

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

FILL THIS IN

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 | |
---|---|

`dtype` | The type of the this `LinearOperator` . Arguments to `matmul` and `solve` will have to be this type. |

`graph_parents` | 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` . |

Raises | |
---|---|

`ValueError` | If any member of graph_parents is `None` or not a `Tensor` . |

`ValueError` | If hints are set incorrectly. |

Attributes | |
---|---|

`H` | Returns the adjoint of the current `LinearOperator` . Given |

`batch_shape` | `TensorShape` of batch dimensions of this `LinearOperator` . If this operator acts like the batch matrix |

`domain_dimension` | Dimension (in the sense of vector spaces) of the domain of this operator. If this operator acts like the batch matrix |

`dtype` | The `DType` of `Tensor` s 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. |

`range_dimension` | Dimension (in the sense of vector spaces) of the range of this operator. If this operator acts like the batch matrix |

`shape` | `TensorShape` of this `LinearOperator` . If this operator acts like the batch matrix |

`tensor_rank` | Rank (in the sense of tensors) of matrix corresponding to this operator. If this operator acts like the batch matrix |

`add_to_tensor`

add_to_tensor( x, name='add_to_tensor' )

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`

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`

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` |

`cholesky`

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. |

`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.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`

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` |

`inverse`

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`

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` | `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` . |

Returns | |
---|---|

A `LinearOperator` or `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.

# 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.

# 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` . |

© 2020 The TensorFlow Authors. All rights reserved.

Licensed under the Creative Commons Attribution License 3.0.

Code samples licensed under the Apache 2.0 License.

https://www.tensorflow.org/versions/r1.15/api_docs/python/tf/linalg/LinearOperator