Huber.gradient¶

property Huber.gradient¶

Gradient operator of the functional.

The gradient of the Huber functional is given by

$\nabla f_{\gamma}(x) = \begin{cases} \frac{1}{\gamma} x & \text{if } \|x\|_2 \leq \gamma \\ \frac{1}{\|x\|_2} x & \text{else} \end{cases}.$

Examples

Check that the gradient norm is less than the norm of the one element:

>>> space = odl.uniform_discr(0, 1, 14)
>>> norm_one = space.one().norm()
>>> x = odl.phantom.white_noise(space)
>>> huber_norm = odl.solvers.Huber(space, gamma=0.1)
>>> grad = huber_norm.gradient(x)
>>> tol = 1e-5
>>> grad.norm() <=  norm_one + tol
True

Redo previous example for a product space in two dimensions:

>>> domain = odl.uniform_discr([0, 0], [1, 1], [5, 5])
>>> space = odl.ProductSpace(domain, 2)
>>> norm_one = space.one().norm()
>>> x = odl.phantom.white_noise(space)
>>> huber_norm = odl.solvers.Huber(space, gamma=0.2)
>>> grad = huber_norm.gradient(x)
>>> tol = 1e-5
>>> grad.norm() <=  norm_one + tol
True