import numpy as np
from pylops.utils.typing import NDArray
[docs]
class HalfSpaceProj:
r"""Half space projection.
Parameters
----------
w : :obj:`numpy.ndarray`
Coefficients of the half space
b : :obj:`float`
bias of the half space
Notes
-----
Given an half space defined as:
.. math::
\operatorname{H}_{[w, b]}
= \{ \mathbf{x}: \mathbf{w}^T \mathbf{x} \leq b \}
its orthogonal projection is:
.. math::
P_{\operatorname{H}_{[w, b]}} (\mathbf{x}) =
\begin{cases}
\mathbf{x},
& \mathbf{w}^T \mathbf{x} \leq b \\
\mathbf{x} - \frac{\mathbf{w}^T \mathbf{x} - b}{||\mathbf{w}||^2} \mathbf{w},
& \text{otherwise}
\end{cases}
Note the this is the proximal operator of the corresponding
indicator function :math:`\mathcal{I}_{\operatorname{H}_{[w, b]}}`.
"""
def __init__(self, w: NDArray, b: float) -> None:
self.w = w
self.b = b
self.w_norm_sq = np.dot(w, w)
def __call__(self, x: NDArray) -> NDArray:
w_x = np.dot(self.w, x)
if w_x <= self.b:
return x
factor = (w_x - self.b) / self.w_norm_sq
x_proj = x - factor * self.w
return x_proj
PyProximal