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# tf.linalg.LinearOperatorLowRankUpdate

Perturb a `LinearOperator` with a rank `K` update.

Inherits From: `LinearOperator`

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.

`LinearOperatorLowRankUpdate` represents `A = L + U D V^H`, where

``````L, is a LinearOperator representing [batch] M x N matrices
U, is a [batch] M x K matrix.  Typically K << M.
D, is a [batch] K x K matrix.
V, is a [batch] N x K matrix.  Typically K << N.
V^H is the Hermitian transpose (adjoint) of V.
``````

If `M = N`, determinants and solves are done using the matrix determinant lemma and Woodbury identities, and thus require L and D to be non-singular.

Solves and determinants will be attempted unless the "is_non_singular" property of L and D is False.

In the event that L and D are positive-definite, and U = V, solves and determinants can be done using a Cholesky factorization.

``````# Create a 3 x 3 diagonal linear operator.
diag_operator = LinearOperatorDiag(
is_positive_definite=True)

# Perturb with a rank 2 perturbation
operator = LinearOperatorLowRankUpdate(
operator=diag_operator,
u=[[1., 2.], [-1., 3.], [0., 0.]],
diag_update=[11., 12.],
v=[[1., 2.], [-1., 3.], [10., 10.]])

operator.shape
==> [3, 3]

operator.log_abs_determinant()
==> scalar Tensor

x = ... Shape [3, 4] Tensor
operator.matmul(x)
==> Shape [3, 4] Tensor
``````

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

### Performance

Suppose `operator` is a `LinearOperatorLowRankUpdate` of shape `[M, N]`, made from a rank `K` update of `base_operator` which performs `.matmul(x)` on `x` having `x.shape = [N, R]` with `O(L_matmul*N*R)` complexity (and similarly for `solve`, `determinant`. Then, if `x.shape = [N, R]`,

• `operator.matmul(x)` is `O(L_matmul*N*R + K*N*R)`

and if `M = N`,

• `operator.solve(x)` is `O(L_matmul*N*R + N*K*R + K^2*R + K^3)`
• `operator.determinant()` is `O(L_determinant + L_solve*N*K + K^2*N + K^3)`

If instead `operator` and `x` have shape `[B1,...,Bb, M, N]` and `[B1,...,Bb, N, R]`, every operation increases in complexity by `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`, `diag_update_positive` and `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.

`base_operator` Shape `[B1,...,Bb, M, N]`.
`u` Shape `[B1,...,Bb, M, K]` `Tensor` of same `dtype` as `base_operator`. This is `U` above.
`diag_update` Optional shape `[B1,...,Bb, K]` `Tensor` with same `dtype` as `base_operator`. This is the diagonal of `D` above. Defaults to `D` being the identity operator.
`v` Optional `Tensor` of same `dtype` as `u` and shape `[B1,...,Bb, N, K]` Defaults to `v = u`, in which case the perturbation is symmetric. If `M != N`, then `v` must be set since the perturbation is not square.
`is_diag_update_positive` Python `bool`. If `True`, expect `diag_update > 0`.
`is_non_singular` Expect that this operator is non-singular. Default is `None`, unless `is_positive_definite` is auto-set to be `True` (see below).
`is_self_adjoint` Expect that this operator is equal to its hermitian transpose. Default is `None`, unless `base_operator` is self-adjoint and `v = None` (meaning `u=v`), in which case this defaults to `True`.
`is_positive_definite` Expect that this operator is positive definite. Default is `None`, unless `base_operator` is positive-definite `v = None` (meaning `u=v`), and `is_diag_update_positive`, in which case this defaults to `True`. Note that we say an operator is positive definite when the quadratic form `x^H A x` has positive real part for all nonzero `x`.
`is_square` Expect that this operator acts like square [batch] matrices.
`name` A name for this `LinearOperator`.

`ValueError` If `is_X` flags are set in an inconsistent way.

`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.

`base_operator` If this operator is `A = L + U D V^H`, this is the `L`.
`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]`

`diag_operator` If this operator is `A = L + U D V^H`, this is `D`.
`diag_update` If this operator is `A = L + U D V^H`, this is the diagonal of `D`.
`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`.
`is_diag_update_positive` If this operator is `A = L + U D V^H`, this hints `D > 0` elementwise.
`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 `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`.

`u` If this operator is `A = L + U D V^H`, this is the `U`.
`v` If this operator is `A = L + U D V^H`, this is the `V`.

## 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`

View source

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`

View source

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`

View source

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`

View source

Determinant for eve