IndicatorL2

class deepinv.optim.data_fidelity.IndicatorL2(radius=None)

Bases: DataFidelity

Data-fidelity as the indicator of \(\ell_2\) ball with radius \(r\).

\[\begin{split}\iota_{\mathcal{B}_2(y,r)}(u)= \left. \begin{cases} 0, & \text{if } \|u-y\|_2\leq r \\ +\infty & \text{else.} \end{cases} \right.\end{split}\]

Parameters:

radius (float) – radius of the ball. Default: None.

prox(x, y, physics, *args, radius=None, stepsize=None, crit_conv=1e-05, max_iter=100, **kwargs)

Proximal operator of the indicator of \(\ell_2\) ball with radius radius, i.e.

\[\operatorname{prox}_{\gamma \iota_{\mathcal{B}_2(y, r)}(A\cdot)}(x) = \underset{u}{\text{argmin}} \,\, \iota_{\mathcal{B}_2(y, r)}(Au)+\frac{1}{2}\|u-x\|_2^2\]

Since no closed form is available for general measurement operators, we use a dual forward-backward algorithm, as suggested in Proximal Splitting Methods in Signal Processing.

Parameters:

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

• y (torch.Tensor) – Data \(y\) of the same dimension as \(\forw{x}\).

• radius (torch.Tensor) – radius of the \(\ell_2\) ball.

• stepsize (float) – step-size of the dual-forward-backward algorithm.

• crit_conv (float) – convergence criterion of the dual-forward-backward algorithm.

• max_iter (int) – maximum number of iterations of the dual-forward-backward algorithm.

• gamma (float) – factor in front of the indicator function. Notice that this does not affect the proximity operator since the indicator is scale invariant. Default: None.

Returns:

(torch.Tensor) projection on the \(\ell_2\) ball of radius radius and centered in y.