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# tf.linalg.LinearOperatorLowRankUpdate

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

Inherits From: `LinearOperator`, `Module`

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