LostTech.TensorFlow : API Documentation

Type LinearOperatorLowRankUpdate

Namespace tensorflow.linalg

Parent LinearOperator

Interfaces ILinearOperatorLowRankUpdate

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

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. ### 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.
Show Example
```# 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 ```

Public static methods

LinearOperatorLowRankUpdateNewDyn(object base_operator, object u, object diag_update, object v, object is_diag_update_positive, object is_non_singular, object is_self_adjoint, object is_positive_definite, object is_square, ImplicitContainer<T> name)

Initialize a `LinearOperatorLowRankUpdate`.

This creates a `LinearOperator` of the form `A = L + U D V^H`, with `L` a `LinearOperator`, `U, V` both [batch] matrices, and `D` a [batch] diagonal matrix.

If `L` is non-singular, solves and determinants are available. Solves/determinants both involve a solve/determinant of a `K x K` system. In the event that L and D are self-adjoint positive-definite, and U = V, this can be done using a Cholesky factorization. The user should set the `is_X` matrix property hints, which will trigger the appropriate code path.
Parameters
`object` base_operator
Shape `[B1,...,Bb, M, N]`.
`object` u
Shape `[B1,...,Bb, M, K]` `Tensor` of same `dtype` as `base_operator`. This is `U` above.
`object` 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.
`object` 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.
`object` is_diag_update_positive
Python `bool`. If `True`, expect `diag_update > 0`.
`object` 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).
`object` 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`.
`object` 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`.
`object` is_square
Expect that this operator acts like square [batch] matrices.
`ImplicitContainer<T>` name
A name for this `LinearOperator`.

Public properties

objectbase_operator get;

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

objectbase_operator_dyn get;

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

objectdiag_operator get;

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

objectdiag_operator_dyn get;

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

objectdiag_update get;

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

objectdiag_update_dyn get;

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

Nullable<bool>is_diag_update_positive get;

If this operator is `A = L + U D V^H`, this hints `D > 0` elementwise.

objectis_diag_update_positive_dyn get;

If this operator is `A = L + U D V^H`, this hints `D > 0` elementwise.

objectu get;

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

objectu_dyn get;

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

objectv get;

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

objectv_dyn get;

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