Source code for pyproximal.proximal.Huber

import numpy as np

from pyproximal.ProxOperator import _check_tau
from pyproximal.projection import BoxProj
from pyproximal import ProxOperator


class Huber(ProxOperator):
    r"""Huber norm proximal operator.

    Proximal operator of the Huber norm defined as:

    .. math::

        H_\alpha(\mathbf{x}) =
        \begin{cases}
        \frac{\|\mathbf{x}\|_2^2}{2 \alpha}, & \|\mathbf{x}\|_2 \leq \alpha \\
        \|\mathbf{x}\|_2 - \frac{\alpha}{2}, & \|\mathbf{x}\|_2 > \alpha \\
        \end{cases}

    which behaves like a :math:`\ell_2` norm for :math:`|x_i| \leq \alpha` and a
    :math:`\ell_1` norm for :math:`|x_i| < \alpha`.

    Parameters
    ----------
    alpha : :obj:`float`
        Huber parameter

    Notes
    -----
    The Huber proximal operator is defined as:

    .. math::

        \prox^*_{\tau H_\alpha(\cdot)}(\mathbf{x}) =
        \left( 1 - \frac{\tau}{\max\{\|\mathbf{x}\|_2, \tau\} + \alpha} \right) \mathbf{x}

    """
    def __init__(self, alpha):
        super().__init__(None, False)
        self.alpha = alpha

    def __call__(self, x):
        l2 = np.linalg.norm(x)
        if l2 <= self.alpha:
            h = l2 ** 2 / (2 * self.alpha)
        else:
            h = l2 - self.alpha / 2.
        return h

    @_check_tau
    def prox(self, x, tau):
        x = (1. - tau / max(np.linalg.norm(x), tau + self.alpha)) * x
        return x