LinearOperatorLowRankUpdate - LostTech.TensorFlow Documentation

Type LinearOperatorLowRankUpdate

Namespace tensorflow.linalg

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(
                diag_update=[1., 2., 3.], is_non_singular=True, is_self_adjoint=True,
                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

Methods

NewDyn

Properties

Public static methods

LinearOperatorLowRankUpdate NewDyn(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

object base_operator get;

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

object base_operator_dyn get;

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

object batch_shape get;

object batch_shape_dyn get;

object diag_operator get;

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

object diag_operator_dyn get;

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

object diag_update get;

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

object diag_update_dyn get;

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

Dimension domain_dimension get;

object domain_dimension_dyn get;

object dtype get;

object dtype_dyn get;

IList<object> graph_parents get;

object graph_parents_dyn get;

Nullable<bool> is_diag_update_positive get;

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

object is_diag_update_positive_dyn get;

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

Nullable<bool> is_non_singular get;

object is_non_singular_dyn get;

object is_positive_definite get;

object is_positive_definite_dyn get;

object is_self_adjoint get;

object is_self_adjoint_dyn get;

Nullable<bool> is_square get;

object is_square_dyn get;

object name get;

object name_dyn get;

object name_scope get;

object name_scope_dyn get;

object PythonObject get;

Dimension range_dimension get;

object range_dimension_dyn get;

TensorShape shape get;

object shape_dyn get;

ValueTuple<object> submodules get;

object submodules_dyn get;

Nullable<int> tensor_rank get;

object tensor_rank_dyn get;

object trainable_variables get;

object trainable_variables_dyn get;

object u get;

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

object u_dyn get;