import numpy as np
from pyproximal.ProxOperator import _check_tau
from pyproximal.projection import BoxProj, L1BallProj
from pyproximal import ProxOperator
def _softthreshold(x, thresh):
r"""Soft thresholding.
Applies soft thresholding to vector ``x - g``.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
thresh : :obj:`float`
Threshold
Returns
-------
x1 : :obj:`numpy.ndarray`
Tresholded vector
"""
if np.iscomplexobj(x):
# https://stats.stackexchange.com/questions/357339/soft-thresholding-
# for-the-lasso-with-complex-valued-data
x1 = np.maximum(np.abs(x) - thresh, 0.) * np.exp(1j * np.angle(x))
else:
x1 = np.maximum(np.abs(x) - thresh, 0.) * np.sign(x)
return x1
def _current_sigma(sigma, count):
if not callable(sigma):
return sigma
else:
return sigma(count)
[docs]class L1(ProxOperator):
r"""L1 norm proximal operator.
Proximal operator of the :math:`\ell_1` norm:
:math:`\sigma\|\mathbf{x} - \mathbf{g}\|_1 = \sigma \sum |x_i - g_i|`.
Parameters
----------
sigma : :obj:`float` or :obj:`list` or :obj:`np.ndarray` or :obj:`func`, optional
Multiplicative coefficient of L1 norm. This can be a constant number, a list
of values (for multidimensional inputs, acting on the second dimension) or
a function that is called passing a counter which keeps track of how many
times the ``prox`` method has been invoked before and returns a scalar (or a list of)
``sigma`` to be used.
g : :obj:`np.ndarray`, optional
Vector to be subtracted
Notes
-----
The :math:`\ell_1` proximal operator is defined as [1]_:
.. math::
\prox_{\tau \sigma \|\cdot\|_1}(\mathbf{x}) =
\operatorname{soft}(\mathbf{x}, \tau \sigma) =
\begin{cases}
x_i + \tau \sigma, & x_i - g_i < -\tau \sigma \\
g_i, & -\tau\sigma \leq x_i - g_i \leq \tau\sigma \\
x_i - \tau\sigma, & x_i - g_i > \tau\sigma\\
\end{cases}
where :math:`\operatorname{soft}` is the so-called called *soft thresholding*.
Moreover, as the conjugate of the :math:`\ell_1` norm is the orthogonal projection of
its dual norm (i.e., :math:`\ell_\inf` norm) onto a unit ball, its dual
operator (when :math:`\mathbf{g}=\mathbf{0}`) is defined as:
.. math::
\prox^*_{\tau \sigma \|\cdot\|_1}(\mathbf{x}) = P_{\|\cdot\|_{\infty} <=\sigma}(\mathbf{x}) =
\begin{cases}
-\sigma, & x_i < -\sigma \\
x_i,& -\sigma \leq x_i \leq \sigma \\
\sigma, & x_i > \sigma\\
\end{cases}
.. [1] Chambolle, and A., Pock, "A first-order primal-dual algorithm for
convex problems with applications to imaging", Journal of Mathematical
Imaging and Vision, 40, 8pp. 120–145. 2011.
"""
def __init__(self, sigma=1., g=None):
super().__init__(None, False)
self.sigma = sigma
self.g = g
self.gdual = 0 if g is None else g
if not callable(sigma):
self.box = BoxProj(-sigma, sigma)
else:
self.box = BoxProj(-sigma(0), sigma(0))
self.count = 0
def __call__(self, x):
sigma = _current_sigma(self.sigma, self.count)
return sigma * np.sum(np.abs(x))
def _increment_count(func):
"""Increment counter
"""
def wrapped(self, *args, **kwargs):
self.count += 1
return func(self, *args, **kwargs)
return wrapped
@_increment_count
@_check_tau
def prox(self, x, tau):
sigma = _current_sigma(self.sigma, self.count)
if self.g is None:
x = _softthreshold(x, tau * sigma)
else:
# use precomposition property
x = _softthreshold(x - self.g, tau * sigma) + self.g
return x
@_check_tau
def proxdual(self, x, tau):
if not isinstance(self.gdual, np.ndarray):
x = self.box(x)
else:
x = self._proxdual_moreau(x, tau)
return x
[docs]class L1Ball(ProxOperator):
r"""L1 ball proximal operator.
Proximal operator of the :math:`\ell_1` ball: :math:`L1_{r} =
\{ \mathbf{x}: \|\mathbf{x}\|_1 \leq r \}`.
Parameters
----------
n : :obj:`int`
Number of elements of input vector
radius : :obj:`float`
Radius
maxiter : :obj:`int`, optional
Maximum number of iterations used by :func:`scipy.optimize.bisect`
xtol : :obj:`float`, optional
Absolute tolerance of :func:`scipy.optimize.bisect`
Notes
-----
As the L1 ball is an indicator function, the proximal operator
corresponds to its orthogonal projection
(see :class:`pyproximal.projection.L1BallProj` for details.
"""
def __init__(self, n, radius, maxiter=100, xtol=1e-5):
super().__init__(None, False)
self.n = n
self.radius = radius
self.maxiter = maxiter
self.xtol = xtol
self.ball = L1BallProj(self.n, self.radius, self.maxiter, self.xtol)
def __call__(self, x, tol=1e-4):
return np.sum(np.abs(x)) - self.radius < tol
@_check_tau
def prox(self, x, tau):
return self.ball(x)
