L2

class deepinv.optim.L2(sigma=1.0)[source]

Bases: DataFidelity

Implementation of \(\distancename\) as the normalized \(\ell_2\) norm

\[f(x) = \frac{1}{2\sigma^2}\|\forw{x}-y\|^2\]

It can be used to define a log-likelihood function associated with additive Gaussian noise by setting an appropriate noise level \(\sigma\).

Parameters:: sigma (float) – Standard deviation of the noise to be used as a normalisation factor.

>>> import torch
>>> import deepinv as dinv
>>> # define a loss function
>>> fidelity = dinv.optim.L2()
>>>
>>> x = torch.ones(1, 1, 3, 3)
>>> mask = torch.ones_like(x)
>>> mask[0, 0, 1, 1] = 0
>>> physics = dinv.physics.Inpainting(tensor_size=(1, 3, 3), mask=mask)
>>> y = physics(x)
>>>
>>> # Compute the data fidelity f(Ax, y)
>>> fidelity(x, y, physics)
tensor([0.])
>>> # Compute the gradient of f
>>> fidelity.grad(x, y, physics)
tensor([[[[0., 0., 0.],
          [0., 0., 0.],
          [0., 0., 0.]]]])
>>> # Compute the proximity operator of f
>>> fidelity.prox(x, y, physics, gamma=1.0)
tensor([[[[1., 1., 1.],
          [1., 1., 1.],
          [1., 1., 1.]]]])

d(u, y)[source]

Computes the data fidelity distance \(\datafid{u}{y}\), i.e.

\[\datafid{u}{y} = \frac{1}{2\sigma^2}\|u-y\|^2\]

Parameters:

u (torch.Tensor) – Variable \(u\) at which the data fidelity is computed.
y (torch.Tensor) – Data \(y\).

Returns:

(torch.Tensor) data fidelity \(\datafid{u}{y}\) of size B with B the size of the batch.

grad_d(u, y)[source]

Computes the gradient of \(\distancename\), that is \(\nabla_{u}\distance{u}{y}\), i.e.

\[\nabla_{u}\distance{u}{y} = \frac{1}{\sigma^2}(u-y)\]

Parameters:

u (torch.Tensor) – Variable \(u\) at which the gradient is computed.
y (torch.Tensor) – Data \(y\).

Returns:

(torch.Tensor) gradient of the distance function \(\nabla_{u}\distance{u}{y}\).

prox(x, y, physics, gamma=1.0)[source]

Proximal operator of \(\gamma \datafid{Ax}{y} = \frac{\gamma}{2\sigma^2}\|Ax-y\|^2\).

Computes \(\operatorname{prox}_{\gamma \datafidname}\), i.e.

\[\operatorname{prox}_{\gamma \datafidname} = \underset{u}{\text{argmin}} \frac{\gamma}{2\sigma^2}\|Au-y\|_2^2+\frac{1}{2}\|u-x\|_2^2\]

Parameters:

x (torch.Tensor) – Variable \(x\) at which the proximity operator is computed.
y (torch.Tensor) – Data \(y\).
physics (deepinv.physics.Physics) – physics model.
gamma (float) – stepsize of the proximity operator.

Returns:

(torch.Tensor) proximity operator \(\operatorname{prox}_{\gamma \datafidname}(x)\).

prox_d(x, y, gamma=1.0)[source]

Proximal operator of \(\gamma \distance{x}{y} = \frac{\gamma}{2\sigma^2}\|x-y\|^2\).

Computes \(\operatorname{prox}_{\gamma \distancename}\), i.e.

\[\operatorname{prox}_{\gamma \distancename} = \underset{u}{\text{argmin}} \frac{\gamma}{2\sigma^2}\|u-y\|_2^2+\frac{1}{2}\|u-x\|_2^2\]

Parameters:

x (torch.Tensor) – Variable \(x\) at which the proximity operator is computed.
y (torch.Tensor) – Data \(y\).
gamma (float) – thresholding parameter.

Returns: