mrpro.operators.functionals.L1Norm

class mrpro.operators.functionals.L1Norm[source]

Bases: ElementaryProximableFunctional

Functional class for the L1 Norm.

This implements the functional given by \(f: C^N \rightarrow [0, \infty), x \rightarrow \| W (x-b)\|_1\), where W is a either a scalar or tensor that corresponds to a (block-) diagonal operator that is applied to the input.

In most cases, consider setting divide_by_n to true to be independent of input size.

The norm of the vector is computed along the dimensions given at initialization.

Initialize a Functional.

We assume that functionals are given in the form \(f(x) = \phi ( \mathrm{weight} ( x - \mathrm{target}))\) for some functional \(\phi\).

Parameters:

target (Tensor | None | complex, default: None) – target element - often data tensor (see above)
weight (Tensor | complex, default: 1.0) – weight parameter (see above)
dim (int | Sequence[int] | None, default: None) – dimension(s) over which functional is reduced. All other dimensions of weight ( x - target) will be treated as batch dimensions.
divide_by_n (bool, default: False) – if true, the result is scaled by the number of elements of the dimensions index by dim in the tensor weight ( x - target). If true, the functional is thus calculated as the mean, else the sum.
keepdim (bool, default: False) – if true, the dimension(s) of the input indexed by dim are maintained and collapsed to singletons, else they are removed from the result.

__call__(x: Tensor) → tuple[Tensor][source]

Compute the L1 norm of the input tensor.

Calculates \(|| W * (x - b) ||_1\), where \(W\) is weight and \(b\) is target. The norm is computed along dimensions specified by dim. If divide_by_n is true, the result is averaged over these dimensions; otherwise, it’s summed.

Parameters:: x (Tensor) – Input tensor.
Returns:: The L1 norm. If keepdim is true, the dimensions dim are retained with size 1; otherwise, they are reduced.

forward(x: Tensor) → tuple[Tensor][source]: Apply forward of L1Norm.

Note

Prefer calling the instance of the L1Norm as operator(x) over directly calling this method. See this PyTorch discussion.

prox(x: Tensor, sigma: Tensor | float = 1.0) → tuple[Tensor][source]

Proximal Mapping of the L1 Norm.

Compute the proximal mapping of the L1 norm.

Parameters:

x (Tensor) – input tensor
sigma (Tensor | float, default: 1.0) – scaling factor

Returns:

Proximal mapping applied to the input tensor

prox_convex_conj(x: Tensor, sigma: Tensor | float = 1.0) → tuple[Tensor][source]

Convex conjugate of the L1 Norm.

Compute the proximal mapping of the convex conjugate of the L1 norm.

Parameters:

x (Tensor) – input tensor
sigma (Tensor | float, default: 1.0) – scaling factor

Returns:

Proximal of the convex conjugate applied to the input tensor

__add__(other: Operator[Unpack[Tin], Tout]) → Operator[Unpack[Tin], Tout][source]

__add__(other: Tensor | complex) → Operator[Unpack[Tin], tuple[Unpack[Tin]]]

Operator addition.

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

__matmul__(other: Operator[Unpack[Tin2], tuple[Unpack[Tin]]] | Operator[Unpack[Tin2], tuple[Tensor, ...]]) → Operator[Unpack[Tin2], Tout][source]

Operator composition.

Returns lambda x: self(other(x))

__mul__(other: Tensor | complex) → Operator[Unpack[Tin], Tout][source]

Operator multiplication with tensor.

Returns lambda x: self(x*other)

__or__(other: ProximableFunctional) → ProximableFunctionalSeparableSum[Tensor, Tensor][source]

Create a ProximableFunctionalSeparableSum object from two proximable functionals.

Parameters:: other (ProximableFunctional) – second functional to be summed
Returns:: ProximableFunctionalSeparableSum object

__radd__(other: Tensor | complex) → Operator[Unpack[Tin], tuple[Unpack[Tin]]][source]

Operator right addition.

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

__rmul__(scalar: Tensor | complex) → ProximableFunctional[source]: Multiply functional with scalar.