DataFidelity

class deepinv.optim.data_fidelity.DataFidelity(d=None)

Bases: Potential

Base class for the data fidelity term \(\distance{A(x)}{y}\) where \(A\) is the forward operator, \(x\in\xset\) is a variable and \(y\in\yset\) is the data, and where \(d\) is a distance function, from the class deepinv.optim.Distance().

Parameters:

d (callable) – distance function \(d(x, y)\) between a variable \(x\) and an observation \(y\). Default None.

fn(x, y, physics, *args, **kwargs)

Computes the data fidelity term \(\datafid{x}{y} = \distance{\forw{x}}{y}\).

Parameters:

• x (torch.Tensor) – Variable \(x\) at which the data fidelity is computed.

• y (torch.Tensor) – Data \(y\).

• physics (deepinv.physics.Physics) – physics model.

Returns:

(torch.Tensor) data fidelity \(\datafid{x}{y}\).

grad(x, y, physics, *args, **kwargs)

Calculates the gradient of the data fidelity term \(\datafidname\) at \(x\).

The gradient is computed using the chain rule:

\[\nabla_x \distance{\forw{x}}{y} = \left. \frac{\partial A}{\partial x} \right|_x^\top \nabla_u \distance{u}{y},\]

where \(\left. \frac{\partial A}{\partial x} \right|_x\) is the Jacobian of \(A\) at \(x\), and \(\nabla_u \distance{u}{y}\) is computed using grad_d with \(u = \forw{x}\). The multiplication is computed using the A_vjp method of the physics.

Parameters:

• x (torch.Tensor) – Variable \(x\) at which the gradient is computed.

• y (torch.Tensor) – Data \(y\).

• physics (deepinv.physics.Physics) – physics model.

Returns:

(torch.Tensor) gradient \(\nabla_x \datafid{x}{y}\), computed in \(x\).

grad_d(u, y, *args, **kwargs)

Computes the gradient \(\nabla_u\distance{u}{y}\), computed in \(u\). Note that this is the gradient of \(\distancename\) and not \(\datafidname\). This function direclty calls deepinv.optim.Distance.grad() for the speficic distance function \(\distance\). :param torch.Tensor u: Variable \(u\) at which the gradient is computed. :param torch.Tensor y: Data \(y\) of the same dimension as \(u\). :return: (torch.Tensor) gradient of \(d\) in \(u\), i.e. \(\nabla_u\distance{u}{y}\).

prox_d(u, y, *args, **kwargs)

Computes the proximity operator \(\operatorname{prox}_{\gamma\distance{\cdot}{y}}(u)\), computed in \(u\). Note that this is the proximity operator of \(\distancename\) and not \(\datafidname\). This function direclty calls deepinv.optim.Distance.prox() for the speficic distance function \(\distance\). :param torch.Tensor u: Variable \(u\) at which the gradient is computed. :param torch.Tensor y: Data \(y\) of the same dimension as \(u\). :return: (torch.Tensor) gradient of \(d\) in \(u\), i.e. \(\nabla_u\distance{u}{y}\).

prox_d_conjugate(u, y, *args, **kwargs)

Computes the proximity operator of the convex conjugate of the distance function \(\distance{u}{y}\). This function direclty calls deepinv.optim.Distance.prox_conjugate() for the speficic distance function \(\distance\).