mrpro.operators.LinearOperator

class mrpro.operators.LinearOperator[source]

Bases: Operator[Tensor, tuple[Tensor]]

General Linear Operator.

LinearOperators have exactly one input tensors and one output tensor, and fulfill \(f(a*x + b*y) = a*f(x) + b*f(y)\) with \(a\), \(b\) scalars and \(x\), \(y\) tensors.

LinearOperators can be composed, added, multiplied, applied to tensors. LinearOperators have an H property that returns the adjoint operator, and a gram property that returns the Gram operator.

Subclasses must implement the forward and adjoint methods. When subclassing, the adjoint_as_backward class attribute can be set to True:

class MyOperator(LinearOperator, adjoint_as_backward=True):
    ...

This will make pytorch use the adjoint method as the backward method of the forward, and the forward method as the backward method of the adjoint, avoiding the need to have differentiable forward and adjoint methods.

property H: LinearOperator[source]

Adjoint operator.

Obtains the adjoint of an instance of this operator as an AdjointLinearOperator, which itself is a an LinearOperator that can be applied to tensors.

Note: linear_operator.H.H == linear_operator

property gram: LinearOperator[source]

Gram operator.

For a LinearOperator \(A\), the self-adjoint Gram operator is defined as \(A^H A\).

Note

This is the inherited default implementation.

__call__(*args: Unpack) → Tout[source]

Apply the forward operator.

For more information, see forward.

Note

Prefer using operator_instance(*parameters), i.e. using __call__ over using forward.

abstract adjoint(x: Tensor) → tuple[Tensor][source]

Adjoint of the operator.

abstract forward(*args: Unpack) → Tout[source]

Apply forward operator.

operator_norm(initial_value: Tensor, dim: Sequence[int] | None, max_iterations: int = 20, relative_tolerance: float = 1e-4, absolute_tolerance: float = 1e-5, callback: Callable[[Tensor], None] | None = None) → Tensor[source]

Power iteration for computing the operator norm of the operator.

Parameters:

  • initial_value (Tensor) – initial value to start the iteration; must be element of the domain. if the initial value contains a zero-vector for one of the considered problems, the function throws an ValueError.

  • dim (Sequence[int] | None) –

    The dimensions of the tensors on which the operator operates. The choice of dim determines how the operator norm is inperpreted. For example, for a matrix-vector multiplication with a batched matrix tensor of shape (batch1, batch2, row, column) and a batched input tensor of shape (batch1, batch2, row):

    • If dim=None, the operator is considered as a block diagonal matrix with batch1*batch2 blocks and the result is a tensor containing a single norm value (shape (1, 1, 1)).

    • If dim=(-1), batch1*batch2 matrices are considered, and for each a separate operator norm is computed.

    • If dim=(-2,-1), batch1 matrices with batch2 blocks are considered, and for each matrix a separate operator norm is computed.

    Thus, the choice of dim determines implicitly determines the domain of the operator.

  • max_iterations (int, default: 20) – maximum number of iterations

  • relative_tolerance (float, default: 1e-4) – absolute tolerance for the change of the operator-norm at each iteration; if set to zero, the maximal number of iterations is the only stopping criterion used to stop the power iteration.

  • absolute_tolerance (float, default: 1e-5) – absolute tolerance for the change of the operator-norm at each iteration; if set to zero, the maximal number of iterations is the only stopping criterion used to stop the power iteration.

  • callback (Callable[[Tensor], None] | None, default: None) – user-provided function to be called at each iteration

Returns:

An estimaton of the operator norm. Shape corresponds to the shape of the input tensor initial_value with the dimensions specified in dim reduced to a single value. The pointwise multiplication of initial_value with the result of the operator norm will always be well-defined.

__add__(other: LinearOperator | Tensor) → LinearOperator[source]
__add__(other: Operator[Tensor, tuple[Tensor]]) → Operator[Tensor, tuple[Tensor]]

Operator addition.

Returns lambda x: self(x) + other(x) if other is a operator, lambda x: self(x) + other if other is a tensor

__and__(other: LinearOperator) → LinearOperatorMatrix[source]

Vertical stacking of two LinearOperators.

A&B is a LinearOperatorMatrix with two rows, with (A&B)(x) == (A(x), B(x)). See mrpro.operators.LinearOperatorMatrix for more information.

__matmul__(other: LinearOperator) → LinearOperator[source]
__matmul__(other: Operator[Unpack, tuple[Tensor]]) → Operator[Unpack, tuple[Tensor]]

Operator composition.

Returns lambda x: self(other(x))

__mul__(other: Tensor | complex) → LinearOperator[source]

Operator elementwise left multiplication with tensor/scalar.

Returns lambda x: self(x*other)

__or__(other: LinearOperator) → LinearOperatorMatrix[source]

Horizontal stacking of two LinearOperators.

A|B is a LinearOperatorMatrix with two columns, with (A|B)(x1,x2) == A(x1)+B(x2). See mrpro.operators.LinearOperatorMatrix for more information.

__radd__(other: Tensor) → LinearOperator[source]

Operator addition.

Returns lambda x: self(x) + other*x

__rmul__(other: Tensor | complex) → LinearOperator[source]

Operator elementwise right multiplication with tensor/scalar.

Returns lambda x: other*self(x)