__all__ = ["IRLS"]
import logging
import time
from typing import Any, Callable, Dict, List, Optional, Sequence, Tuple
import numpy as np
from scipy.sparse.linalg import lsqr
from pylops import LinearOperator
from pylops.basicoperators import Diagonal, Identity, VStack
from pylops.optimization.basesolver import Solver
from pylops.optimization.basic import cgls
from pylops.optimization.eigs import power_iteration
from pylops.optimization.leastsquares import regularized_inversion
from pylops.utils import deps
from pylops.utils.backend import get_array_module, get_module_name
from pylops.utils.typing import InputDimsLike, NDArray, SamplingLike
spgl1_message = deps.spgl1_import("the spgl1 solver")
if spgl1_message is None:
from spgl1 import spgl1 as ext_spgl1
def _hardthreshold(x: NDArray, thresh: float) -> NDArray:
r"""Hard thresholding.
Applies hard thresholding to vector ``x`` (equal to the proximity
operator for :math:`\|\mathbf{x}\|_0`) as shown in [1]_.
.. [1] Chen, F., Shen, L., Suter, B.W., “Computing the proximity
operator of the ℓp norm with 0 < p < 1”,
IET Signal Processing, vol. 10. 2016.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
thresh : :obj:`float`
Threshold
Returns
-------
x1 : :obj:`numpy.ndarray`
Tresholded vector
"""
x1 = x.copy()
x1[np.abs(x) <= np.sqrt(2 * thresh)] = 0
return x1
def _softthreshold(x: NDArray, thresh: float) -> NDArray:
r"""Soft thresholding.
Applies soft thresholding to vector ``x`` (equal to the proximity
operator for :math:`\|\mathbf{x}\|_1`) as shown in [1]_.
.. [1] Chen, F., Shen, L., Suter, B.W., “Computing the proximity
operator of the ℓp norm with 0 < p < 1”,
IET Signal Processing, vol. 10. 2016.
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, 0.0) * np.exp(1j * np.angle(x))
else:
x1 = np.maximum(np.abs(x) - thresh, 0.0) * np.sign(x)
return x1
def _halfthreshold(x: NDArray, thresh: float) -> NDArray:
r"""Half thresholding.
Applies half thresholding to vector ``x`` (equal to the proximity
operator for :math:`\|\mathbf{x}\|_{1/2}^{1/2}`) as shown in [1]_.
.. [1] Chen, F., Shen, L., Suter, B.W., “Computing the proximity
operator of the ℓp norm with 0 < p < 1”,
IET Signal Processing, vol. 10. 2016.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
thresh : :obj:`float`
Threshold
Returns
-------
x1 : :obj:`numpy.ndarray`
Tresholded vector
.. warning::
Since version 1.17.0 does not produce ``np.nan`` on bad input.
"""
arg = np.ones_like(x)
arg[x != 0] = (thresh / 8.0) * (np.abs(x[x != 0]) / 3.0) ** (-1.5)
arg = np.clip(arg, -1, 1)
phi = 2.0 / 3.0 * np.arccos(arg)
x1 = 2.0 / 3.0 * x * (1 + np.cos(2.0 * np.pi / 3.0 - phi))
# x1[np.abs(x) <= 1.5 * thresh ** (2. / 3.)] = 0
x1[np.abs(x) <= (54 ** (1.0 / 3.0) / 4.0) * thresh ** (2.0 / 3.0)] = 0
return x1
def _hardthreshold_percentile(x: NDArray, perc: float) -> NDArray:
r"""Percentile Hard thresholding.
Applies hard thresholding to vector ``x`` using a percentile to define
the amount of values in the input vector to be preserved as shown in [1]_.
.. [1] Chen, Y., Chen, K., Shi, P., Wang, Y., “Irregular seismic
data reconstruction using a percentile-half-thresholding algorithm”,
Journal of Geophysics and Engineering, vol. 11. 2014.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
thresh : :obj:`float`
Threshold
Returns
-------
x1 : :obj:`numpy.ndarray`
Tresholded vector
"""
thresh = np.percentile(np.abs(x), perc)
return _hardthreshold(x, 0.5 * thresh**2)
def _softthreshold_percentile(x: NDArray, perc: float) -> NDArray:
r"""Percentile Soft thresholding.
Applies soft thresholding to vector ``x`` using a percentile to define
the amount of values in the input vector to be preserved as shown in [1]_.
.. [1] Chen, Y., Chen, K., Shi, P., Wang, Y., “Irregular seismic
data reconstruction using a percentile-half-thresholding algorithm”,
Journal of Geophysics and Engineering, vol. 11. 2014.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
perc : :obj:`float`
Percentile
Returns
-------
x : :obj:`numpy.ndarray`
Tresholded vector
"""
thresh = np.percentile(np.abs(x), perc)
return _softthreshold(x, thresh)
def _halfthreshold_percentile(x: NDArray, perc: float) -> NDArray:
r"""Percentile Half thresholding.
Applies half thresholding to vector ``x`` using a percentile to define
the amount of values in the input vector to be preserved as shown in [1]_.
.. [1] Xu, Z., Xiangyu, C., Xu, F. and Zhang, H., “L1/2 Regularization: A
Thresholding Representation Theory and a Fast Solver”, IEEE Transactions
on Neural Networks and Learning Systems, vol. 23. 2012.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
perc : :obj:`float`
Percentile
Returns
-------
x : :obj:`numpy.ndarray`
Thresholded vector
"""
thresh = np.percentile(np.abs(x), perc)
# return _halfthreshold(x, (2. / 3. * thresh) ** (1.5))
return _halfthreshold(x, (4.0 / 54 ** (1.0 / 3.0) * thresh) ** 1.5)
[docs]class IRLS(Solver):
r"""Iteratively reweighted least squares.
Solve an optimization problem with :math:`L_1` cost function (data IRLS)
or :math:`L_1` regularization term (model IRLS) given the operator ``Op``
and data ``y``.
In the *data IRLS*, the cost function is minimized by iteratively solving a
weighted least squares problem with the weight at iteration :math:`i`
being based on the data residual at iteration :math:`i-1`. This IRLS solver
is robust to *outliers* since the L1 norm given less weight to large
residuals than L2 norm does.
Similarly in the *model IRLS*, the weight at at iteration :math:`i`
is based on the model at iteration :math:`i-1`. This IRLS solver inverts
for a sparse model vector.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
Raises
------
NotImplementedError
If ``kind`` is different from model or data
Notes
-----
*Data IRLS* solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{y}`:
.. math::
J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_1
by a set of outer iterations which require to repeatedly solve a
weighted least squares problem of the form:
.. math::
\DeclareMathOperator*{\argmin}{arg\,min}
\mathbf{x}^{(i+1)} = \argmin_\mathbf{x} \|\mathbf{y} -
\mathbf{Op}\,\mathbf{x}\|_{2, \mathbf{R}^{(i)}}^2 +
\epsilon_\mathbf{I}^2 \|\mathbf{x}\|_2^2
where :math:`\mathbf{R}^{(i)}` is a diagonal weight matrix
whose diagonal elements at iteration :math:`i` are equal to the absolute
inverses of the residual vector :math:`\mathbf{r}^{(i)} =
\mathbf{y} - \mathbf{Op}\,\mathbf{x}^{(i)}` at iteration :math:`i`.
More specifically the :math:`j`-th element of the diagonal of
:math:`\mathbf{R}^{(i)}` is
.. math::
R^{(i)}_{j,j} = \frac{1}{\left| r^{(i)}_j \right| + \epsilon_\mathbf{R}}
or
.. math::
R^{(i)}_{j,j} = \frac{1}{\max\{\left|r^{(i)}_j\right|, \epsilon_\mathbf{R}\}}
depending on the choice ``threshR``. In either case,
:math:`\epsilon_\mathbf{R}` is the user-defined stabilization/thresholding
factor [1]_.
Similarly *model IRLS* solves the following optimization problem for the
operator :math:`\mathbf{Op}` and the data :math:`\mathbf{y}`:
.. math::
J = \|\mathbf{x}\|_1 \quad \text{subject to} \quad
\mathbf{y} = \mathbf{Op}\,\mathbf{x}
by a set of outer iterations which require to repeatedly solve a
weighted least squares problem of the form [2]_:
.. math::
\mathbf{x}^{(i+1)} = \operatorname*{arg\,min}_\mathbf{x}
\|\mathbf{x}\|_{2, \mathbf{R}^{(i)}}^2 \quad \text{subject to} \quad
\mathbf{y} = \mathbf{Op}\,\mathbf{x}
where :math:`\mathbf{R}^{(i)}` is a diagonal weight matrix
whose diagonal elements at iteration :math:`i` are equal to the absolutes
of the model vector :math:`\mathbf{x}^{(i)}` at iteration
:math:`i`. More specifically the :math:`j`-th element of the diagonal of
:math:`\mathbf{R}^{(i)}` is
.. math::
R^{(i)}_{j,j} = \left|x^{(i)}_j\right|.
.. [1] https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares
.. [2] Chartrand, R., and Yin, W. "Iteratively reweighted algorithms for
compressive sensing", IEEE. 2008.
"""
def _print_setup(self, xcomplex: bool = False) -> None:
self._print_solver(f" ({self.kind})")
strpar = f"threshR = {self.threshR}\tepsR = {self.epsR}\tepsI = {self.epsI}"
if self.nouter is not None:
strpar1 = f"tolIRL = {self.nouter}\tnouter = {self.nouter}"
else:
strpar1 = f"tolIRL = {self.nouter}"
print(strpar)
print(strpar1)
print("-" * 80)
if not xcomplex:
head1 = " Itn x[0] r2norm"
else:
head1 = " Itn x[0] r2norm"
print(head1)
def _print_step(self, x: NDArray) -> None:
strx = f"{x[0]:1.2e} " if np.iscomplexobj(x) else f"{x[0]:11.4e}"
str1 = f"{self.iiter:6g} " + strx
str2 = f" {self.rnorm:10.3e}"
print(str1 + str2)
def setup(
self,
y: NDArray,
nouter: Optional[int] = None,
threshR: bool = False,
epsR: float = 1e-10,
epsI: float = 1e-10,
tolIRLS: float = 1e-10,
warm: bool = False,
kind: str = "data",
show: bool = False,
) -> None:
r"""Setup solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
nouter : :obj:`int`, optional
Number of outer iterations
threshR : :obj:`bool`, optional
Apply thresholding in creation of weight (``True``)
or damping (``False``)
epsR : :obj:`float`, optional
Damping to be applied to residuals for weighting term
epsI : :obj:`float`, optional
Tikhonov damping (for ``kind="data"``) or L1 model damping
(for ``kind="datamodel"``)
tolIRLS : :obj:`float`, optional
Tolerance. Stop outer iterations if difference between inverted model
at subsequent iterations is smaller than ``tolIRLS``
warm : :obj:`bool`, optional
Warm start each inversion inner step with previous estimate (``True``) or not (``False``).
This only applies to ``kind="data"`` and ``kind="datamodel"``
kind : :obj:`str`, optional
Kind of solver (``model``, ``data`` or ``datamodel``)
show : :obj:`bool`, optional
Display setup log
"""
self.y = y
self.nouter = nouter
self.threshR = threshR
self.epsR = epsR
self.epsI = epsI
self.tolIRLS = tolIRLS
self.warm = warm
self.kind = kind
self.ncp = get_array_module(y)
self.iiter = 0
# choose step to use
if self.kind == "data":
self._step = self._step_data
elif self.kind == "model":
self._step = self._step_model
self.Iop = Identity(y.size, dtype=y.dtype)
elif self.kind == "datamodel":
self._step = self._step_data
# augment Op and y
self.Op = VStack([self.Op, epsI * Identity(self.Op.shape[1])])
self.epsI = 0.0 # as epsI is added to the augmented system already
self.y = np.hstack([self.y, np.zeros(self.Op.shape[1])])
else:
raise NotImplementedError("kind must be model, data or datamodel")
# print setup
if show:
self._print_setup()
def _step_data(self, x: NDArray, **kwargs_solver) -> NDArray:
r"""Run one step of solver with L1 data term"""
if self.iiter == 0:
x = regularized_inversion(
self.Op,
self.y,
None,
x0=x if self.warm else None,
damp=self.epsI,
**kwargs_solver,
)[0]
else:
# other iterations (weighted least-squares)
if self.threshR:
self.rw = 1.0 / self.ncp.maximum(self.ncp.abs(self.r), self.epsR)
else:
self.rw = 1.0 / (self.ncp.abs(self.r) + self.epsR)
self.rw = self.rw / self.rw.max()
R = Diagonal(np.sqrt(self.rw))
x = regularized_inversion(
self.Op,
self.y,
None,
Weight=R,
x0=x if self.warm else None,
damp=self.epsI,
**kwargs_solver,
)[0]
return x
def _step_model(self, x: NDArray, **kwargs_solver) -> NDArray:
r"""Run one step of solver with L1 model term"""
if self.iiter == 0:
# first iteration (unweighted least-squares)
if self.ncp == np:
x = self.Op.rmatvec(
lsqr(
self.Op @ self.Op.H + (self.epsI**2) * self.Iop,
self.y,
**kwargs_solver,
)[0]
)
else:
x = self.Op.rmatvec(
cgls(
self.Op @ self.Op.H + (self.epsI**2) * self.Iop,
self.y,
self.ncp.zeros(int(self.Op.shape[0]), dtype=self.Op.dtype),
**kwargs_solver,
)[0]
)
else:
# other iterations (weighted least-squares)
self.rw = np.abs(x)
self.rw = self.rw / self.rw.max()
R = Diagonal(self.rw, dtype=self.rw.dtype)
if self.ncp == np:
x = R.matvec(
self.Op.rmatvec(
lsqr(
self.Op @ R @ self.Op.H + self.epsI**2 * self.Iop,
self.y,
**kwargs_solver,
)[0]
)
)
else:
x = R.matvec(
self.Op.rmatvec(
cgls(
self.Op @ R @ self.Op.H + self.epsI**2 * self.Iop,
self.y,
self.ncp.zeros(int(self.Op.shape[0]), dtype=self.Op.dtype),
**kwargs_solver,
)[0]
)
)
return x
def step(self, x: NDArray, show: bool = False, **kwargs_solver) -> NDArray:
r"""Run one step of solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by a step of ISTA
show : :obj:`bool`, optional
Display iteration log
**kwargs_solver
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.cg` solver for data IRLS and
:py:func:`scipy.sparse.linalg.lsqr` solver for model IRLS when using
numpy data(or :py:func:`pylops.optimization.solver.cg` and
:py:func:`pylops.optimization.solver.cgls` when using cupy data)
Returns
-------
x : :obj:`np.ndarray`
Updated model vector
"""
# update model
x = self._step(x, **kwargs_solver)
# compute residual
self.r: NDArray = self.y - self.Op.matvec(x)
self.rnorm = self.ncp.linalg.norm(self.r)
self.iiter += 1
if show:
self._print_step(x)
return x
def run(
self,
x: NDArray,
nouter: int = 10,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
**kwargs_solver,
) -> NDArray:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of IRLS
nouter : :obj:`int`, optional
Number of outer iterations.
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
**kwargs_solver
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.cg` solver for data IRLS and
:py:func:`scipy.sparse.linalg.lsqr` solver for model IRLS when using
numpy data(or :py:func:`pylops.optimization.solver.cg` and
:py:func:`pylops.optimization.solver.cgls` when using cupy data)
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
nouter = nouter if self.nouter is None else self.nouter
if x is not None:
self.x0 = x.copy()
self.y = self.y - self.Op.matvec(x)
# choose xold to ensure tolerance test is passed initially
xold = x.copy() + np.inf
while self.iiter < nouter and self.ncp.linalg.norm(x - xold) >= self.tolIRLS:
showstep = (
True
if show
and (
self.iiter < itershow[0]
or nouter - self.iiter < itershow[1]
or self.iiter % itershow[2] == 0
)
else False
)
xold = x.copy()
x = self.step(x, showstep, **kwargs_solver)
self.callback(x)
# adding initial guess
if hasattr(self, "x0"):
x = self.x0 + x
return x
def finalize(self, show: bool = False) -> None:
r"""Finalize solver
Parameters
----------
show : :obj:`bool`, optional
Display finalize log
"""
self.tend = time.time()
self.telapsed = self.tend - self.tstart
self.nouter = self.iiter
if show:
self._print_finalize()
def solve(
self,
y: NDArray,
x0: Optional[NDArray] = None,
nouter: int = 10,
threshR: bool = False,
epsR: float = 1e-10,
epsI: float = 1e-10,
tolIRLS: float = 1e-10,
kind: str = "data",
warm: bool = False,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
**kwargs_solver,
) -> NDArray:
r"""Run entire solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
x0 : :obj:`np.ndarray`, optional
Initial guess of size :math:`[N \times 1]`. If ``None``, initialize
internally as zero vector
nouter : :obj:`int`, optional
Number of outer iterations
threshR : :obj:`bool`, optional
Apply thresholding in creation of weight (``True``)
or damping (``False``)
epsR : :obj:`float`, optional
Damping to be applied to residuals for weighting term
epsI : :obj:`float`, optional
Tikhonov damping
tolIRLS : :obj:`float`, optional
Tolerance. Stop outer iterations if difference between inverted model
at subsequent iterations is smaller than ``tolIRLS``
warm : :obj:`bool`, optional
Warm start each inversion inner step with previous estimate (``True``) or not (``False``).
This only applies to ``kind="data"`` and ``kind="datamodel"``
kind : :obj:`str`, optional
Kind of solver (``data`` or ``model``)
show : :obj:`bool`, optional
Display setup log
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
**kwargs_solver
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.cg` solver for data IRLS and
:py:func:`scipy.sparse.linalg.lsqr` solver for model IRLS when using
numpy data(or :py:func:`pylops.optimization.solver.cg` and
:py:func:`pylops.optimization.solver.cgls` when using cupy data)
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[N \times 1]`
"""
self.setup(
y=y,
threshR=threshR,
epsR=epsR,
epsI=epsI,
tolIRLS=tolIRLS,
warm=warm,
kind=kind,
show=show,
)
if x0 is None:
x0 = self.ncp.zeros(self.Op.shape[1], dtype=self.y.dtype)
x = self.run(x0, nouter=nouter, show=show, itershow=itershow, **kwargs_solver)
self.finalize(show)
return x, self.nouter
[docs]class OMP(Solver):
r"""Orthogonal Matching Pursuit (OMP).
Solve an optimization problem with :math:`L_0` regularization function given
the operator ``Op`` and data ``y``. The operator can be real or complex,
and should ideally be either square :math:`N=M` or underdetermined
:math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
See Also
--------
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{y}`:
.. math::
\|\mathbf{x}\|_0 \quad \text{subject to} \quad
\|\mathbf{Op}\,\mathbf{x}-\mathbf{y}\|_2^2 \leq \sigma^2,
using Orthogonal Matching Pursuit (OMP). This is a very
simple iterative algorithm which applies the following step:
.. math::
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\Lambda_k = \Lambda_{k-1} \cup \left\{\argmax_j
\left|\mathbf{Op}_j^H\,\mathbf{r}_k\right| \right\} \\
\mathbf{x}_k = \argmin_{\mathbf{x}}
\left\|\mathbf{Op}_{\Lambda_k}\,\mathbf{x} - \mathbf{y}\right\|_2^2
Note that by choosing ``niter_inner=0`` the basic Matching Pursuit (MP)
algorithm is implemented instead. In other words, instead of solving an
optimization at each iteration to find the best :math:`\mathbf{x}` for the
currently selected basis functions, the vector :math:`\mathbf{x}` is just
updated at the new basis function by taking directly the value from
the inner product :math:`\mathbf{Op}_j^H\,\mathbf{r}_k`.
In this case it is highly recommended to provide a normalized basis
function. If different basis have different norms, the solver is likely
to diverge. Similar observations apply to OMP, even though mild unbalancing
between the basis is generally properly handled.
"""
def _print_setup(self, xcomplex: bool = False) -> None:
self._print_solver("(Only MP)" if self.niter_inner == 0 else "", nbar=55)
strpar = (
f"sigma = {self.sigma:.2e}\tniter_outer = {self.niter_outer}\n"
f"niter_inner = {self.niter_inner}\tnormalization={self.normalizecols}"
)
print(strpar)
print("-" * 55)
if not xcomplex:
head1 = " Itn x[0] r2norm"
else:
head1 = " Itn x[0] r2norm"
print(head1)
def _print_step(self, x: NDArray) -> None:
strx = f"{x[0]:1.2e} " if np.iscomplexobj(x) else f"{x[0]:11.4e}"
str1 = f"{self.iiter:6g} " + strx
str2 = f" {self.cost[-1]:10.3e}"
print(str1 + str2)
def setup(
self,
y: NDArray,
niter_outer: int = 10,
niter_inner: int = 40,
sigma: float = 1e-4,
normalizecols: bool = False,
show: bool = False,
) -> None:
r"""Setup solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
niter_outer : :obj:`int`, optional
Number of iterations of outer loop
niter_inner : :obj:`int`, optional
Number of iterations of inner loop. By choosing ``niter_inner=0``, the
Matching Pursuit (MP) algorithm is implemented.
sigma : :obj:`list`
Maximum :math:`L^2` norm of residual. When smaller stop iterations.
normalizecols : :obj:`list`, optional
Normalize columns (``True``) or not (``False``). Note that this can be
expensive as it requires applying the forward operator
:math:`n_{cols}` times to unit vectors (i.e., containing 1 at
position j and zero otherwise); use only when the columns of the
operator are expected to have highly varying norms.
show : :obj:`bool`, optional
Display setup log
"""
self.y = y
self.niter_outer = niter_outer
self.niter_inner = niter_inner
self.sigma = sigma
self.normalizecols = normalizecols
self.ncp = get_array_module(y)
# find normalization factor for each column
if self.normalizecols:
ncols = self.Op.shape[1]
self.norms = self.ncp.zeros(ncols)
for icol in range(ncols):
unit = self.ncp.zeros(ncols, dtype=self.Op.dtype)
unit[icol] = 1
self.norms[icol] = np.linalg.norm(self.Op.matvec(unit))
# create variables to track the residual norm and iterations
self.res = self.y.copy()
self.cost = [
float(np.linalg.norm(self.y)),
]
self.iiter = 0
if show:
self._print_setup()
def step(
self,
x: NDArray,
cols: InputDimsLike,
show: bool = False,
) -> NDArray:
r"""Run one step of solver
Parameters
----------
x : :obj:`list` or :obj:`np.ndarray`
Current model vector to be updated by a step of OMP
cols : :obj:`list`
Current list of chosen elements of vector x to be updated by a step of OMP
show : :obj:`bool`, optional
Display iteration log
Returns
-------
x : :obj:`np.ndarray`
Updated model vector
cols : :obj:`list`
Current list of chosen elements
"""
# compute inner products
cres = self.Op.rmatvec(self.res)
cres_abs = np.abs(cres)
if self.normalizecols:
cres_abs = cres_abs / self.norms
# choose column with max cres
cres_max = np.max(cres_abs)
imax = np.argwhere(cres_abs == cres_max).ravel()
nimax = len(imax)
if nimax > 0:
imax = imax[np.random.permutation(nimax)[0]]
else:
imax = imax[0]
# update active set
if imax not in cols:
addnew = True
cols.append(int(imax))
else:
addnew = False
imax_in_cols = cols.index(imax)
# estimate model for current set of columns
if self.niter_inner == 0:
# MP update
Opcol = self.Op.apply_columns(
[
int(imax),
]
)
self.res -= Opcol.matvec(cres[imax] * self.ncp.ones(1))
if addnew:
x.append(cres[imax])
else:
x[imax_in_cols] += cres[imax]
else:
# OMP update
Opcol = self.Op.apply_columns(cols)
if self.ncp == np:
x = lsqr(Opcol, self.y, iter_lim=self.niter_inner)[0]
else:
x = cgls(
Opcol,
self.y,
self.ncp.zeros(int(Opcol.shape[1]), dtype=Opcol.dtype),
niter=self.niter_inner,
)[0]
self.res = self.y - Opcol.matvec(x)
self.iiter += 1
self.cost.append(float(np.linalg.norm(self.res)))
if show:
self._print_step(x)
return x, cols
def run(
self,
x: NDArray,
cols: InputDimsLike,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
) -> Tuple[NDArray, InputDimsLike]:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of IRLS
cols : :obj:`list`
Current list of chosen elements of vector x to be updated by a step of OMP
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
cols : :obj:`list`
Current list of chosen elements
"""
while self.iiter < self.niter_outer and self.cost[self.iiter] > self.sigma:
showstep = (
True
if show
and (
self.iiter < itershow[0]
or self.niter_outer - self.iiter < itershow[1]
or self.iiter % itershow[2] == 0
)
else False
)
x, cols = self.step(x, cols, showstep)
self.callback(x)
return x, cols
def finalize(
self,
x: NDArray,
cols: InputDimsLike,
show: bool = False,
) -> NDArray:
r"""Finalize solver
Parameters
----------
x : :obj:`list` or :obj:`np.ndarray`
Current model vector to be updated by a step of OMP
cols : :obj:`list`
Current list of chosen elements of vector x to be updated by a step of OMP
show : :obj:`bool`, optional
Display finalize log
Returns
-------
xfin : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
self.tend = time.time()
self.telapsed = self.tend - self.tstart
self.cost = np.array(self.cost)
self.nouter = self.iiter
xfin = self.ncp.zeros(int(self.Op.shape[1]), dtype=self.Op.dtype)
xfin[cols] = self.ncp.array(x)
if show:
self._print_finalize(nbar=55)
return xfin
def solve(
self,
y: NDArray,
niter_outer: int = 10,
niter_inner: int = 40,
sigma: float = 1e-4,
normalizecols: bool = False,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
) -> Tuple[NDArray, int, NDArray]:
r"""Run entire solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
niter_outer : :obj:`int`, optional
Number of iterations of outer loop
niter_inner : :obj:`int`, optional
Number of iterations of inner loop. By choosing ``niter_inner=0``, the
Matching Pursuit (MP) algorithm is implemented.
sigma : :obj:`list`
Maximum :math:`L^2` norm of residual. When smaller stop iterations.
normalizecols : :obj:`list`, optional
Normalize columns (``True``) or not (``False``). Note that this can be
expensive as it requires applying the forward operator
:math:`n_{cols}` times to unit vectors (i.e., containing 1 at
position j and zero otherwise); use only when the columns of the
operator are expected to have highly varying norms.
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
niter_outer : :obj:`int`
Number of effective outer iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
"""
self.setup(
y,
niter_outer=niter_outer,
niter_inner=niter_inner,
sigma=sigma,
normalizecols=normalizecols,
show=show,
)
x: List[NDArray] = []
cols: List[InputDimsLike] = []
x, cols = self.run(x, cols, show=show, itershow=itershow)
x = self.finalize(x, cols, show)
return x, self.nouter, self.cost
[docs]class ISTA(Solver):
r"""Iterative Shrinkage-Thresholding Algorithm (ISTA).
Solve an optimization problem with :math:`L_p, \; p=0, 0.5, 1`
regularization, given the operator ``Op`` and data ``y``. The operator
can be real or complex, and should ideally be either square :math:`N=M`
or underdetermined :math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
ValueError
If ``monitorres=True`` and residual increases
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
FISTA: Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following synthesis problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{y}`:
.. math::
J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 +
\epsilon \|\mathbf{x}\|_p
or the analysis problem:
.. math::
J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 +
\epsilon \|\mathbf{SOp}^H\,\mathbf{x}\|_p
if ``SOp`` is provided. Note that in the first case, ``SOp`` should be
assimilated in the modelling operator (i.e., ``Op=GOp * SOp``).
The Iterative Shrinkage-Thresholding Algorithms (ISTA) [1]_ is used, where
:math:`p=0, 0.5, 1`. This is a very simple iterative algorithm which
applies the following step:
.. math::
\mathbf{x}^{(i+1)} = T_{(\epsilon \alpha /2, p)} \left(\mathbf{x}^{(i)} +
\alpha\,\mathbf{Op}^H \left(\mathbf{y} - \mathbf{Op}\,\mathbf{x}^{(i)}\right)\right)
or
.. math::
\mathbf{x}^{(i+1)} = \mathbf{SOp}\,\left\{T_{(\epsilon \alpha /2, p)}
\mathbf{SOp}^H\,\left(\mathbf{x}^{(i)} + \alpha\,\mathbf{Op}^H \left(\mathbf{y} -
\mathbf{Op} \,\mathbf{x}^{(i)}\right)\right)\right\}
where :math:`\epsilon \alpha /2` is the threshold and :math:`T_{(\tau, p)}`
is the thresholding rule. The most common variant of ISTA uses the
so-called soft-thresholding rule :math:`T(\tau, p=1)`. Alternatively an
hard-thresholding rule is used in the case of :math:`p=0` or a half-thresholding
rule is used in the case of :math:`p=1/2`. Finally, percentile bases thresholds
are also implemented: the damping factor is not used anymore an the
threshold changes at every iteration based on the computed percentile.
.. [1] Daubechies, I., Defrise, M., and De Mol, C., “An iterative
thresholding algorithm for linear inverse problems with a sparsity
constraint”, Communications on pure and applied mathematics, vol. 57,
pp. 1413-1457. 2004.
"""
def _print_setup(self) -> None:
self._print_solver(f" ({self.threshkind} thresholding)")
if self.niter is not None:
strpar = f"eps = {self.eps:10e}\ttol = {self.tol:10e}\tniter = {self.niter}"
else:
strpar = f"eps = {self.eps:10e}\ttol = {self.tol:10e}"
if self.perc is None:
strpar1 = f"alpha = {self.alpha:10e}\tthresh = {self.thresh:10e}"
else:
strpar1 = f"alpha = {self.alpha:10e}\tperc = {self.perc:.1f}"
head1 = " Itn x[0] r2norm r12norm xupdate"
print(strpar)
print(strpar1)
print("-" * 80)
print(head1)
def _print_step(
self,
x: NDArray,
costdata: float,
costreg: float,
xupdate: float,
) -> None:
strx = (
f" {x[0]:1.2e} " if np.iscomplexobj(x) else f" {x[0]:11.4e} "
)
msg = (
f"{self.iiter:6g} "
+ strx
+ f"{costdata:10.3e} {costdata + costreg:9.3e} {xupdate:10.3e}"
)
print(msg)
def setup(
self,
y: NDArray,
x0: Optional[NDArray] = None,
niter: Optional[int] = None,
SOp: Optional[LinearOperator] = None,
eps: float = 0.1,
alpha: Optional[float] = None,
eigsdict: Optional[Dict[str, Any]] = None,
tol: float = 1e-10,
threshkind: str = "soft",
perc: Optional[float] = None,
decay: Optional[NDArray] = None,
monitorres: bool = False,
show: bool = False,
) -> NDArray:
r"""Setup solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]` or :math:`[N \times R]` where
a solution for multiple right-hand-side is found when ``R>1``.
x0: :obj:`numpy.ndarray`, optional
Initial guess
niter : :obj:`int`
Number of iterations
SOp : :obj:`pylops.LinearOperator`, optional
Regularization operator (use when solving the analysis problem)
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size. To guarantee convergence, ensure
:math:`\alpha \le 1/\lambda_\text{max}`, where :math:`\lambda_\text{max}`
is the largest eigenvalue of :math:`\mathbf{Op}^H\mathbf{Op}`.
If ``None``, the maximum eigenvalue is estimated and the optimal step size
is chosen as :math:`1/\lambda_\text{max}`. If provided, the
convergence criterion will not be checked internally.
eigsdict : :obj:`dict`, optional
Dictionary of parameters to be passed to :func:`pylops.LinearOperator.eigs` method
when computing the maximum eigenvalue
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', or 'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
decay : :obj:`numpy.ndarray`, optional
Decay factor to be applied to thresholding during iterations
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
show : :obj:`bool`, optional
Display setup log
Returns
-------
x : :obj:`np.ndarray`
Initial model vector
"""
self.y = y
self.SOp = SOp
self.niter = niter
self.eps = eps
self.eigsdict = {} if eigsdict is None else eigsdict
self.tol = tol
self.threshkind = threshkind
self.perc = perc
self.decay = decay
self.monitorres = monitorres
self.ncp = get_array_module(y)
# choose matvec/rmatvec or matmat/rmatmat based on R
if y.ndim > 1 and y.shape[1] > 1:
self.Opmatvec = self.Op.matmat
self.Oprmatvec = self.Op.rmatmat
if self.SOp is not None:
self.SOpmatvec = self.SOp.matmat
self.SOprmatvec = self.SOp.rmatmat
else:
self.Opmatvec = self.Op.matvec
self.Oprmatvec = self.Op.rmatvec
if self.SOp is not None:
self.SOpmatvec = self.SOp.matvec
self.SOprmatvec = self.SOp.rmatvec
# choose thresholding function
if threshkind not in [
"hard",
"soft",
"half",
"hard-percentile",
"soft-percentile",
"half-percentile",
]:
raise NotImplementedError(
"threshkind should be hard, soft, half,"
"hard-percentile, soft-percentile, "
"or half-percentile"
)
if (
threshkind in ["hard-percentile", "soft-percentile", "half-percentile"]
and perc is None
):
raise ValueError(
"Provide a percentile when choosing hard-percentile,"
"soft-percentile, or half-percentile thresholding"
)
self.threshf: Callable[[NDArray, float], NDArray]
if threshkind == "soft":
self.threshf = _softthreshold
elif threshkind == "hard":
self.threshf = _hardthreshold
elif threshkind == "half":
self.threshf = _halfthreshold
elif threshkind == "hard-percentile":
self.threshf = _hardthreshold_percentile
elif threshkind == "soft-percentile":
self.threshf = _softthreshold_percentile
else:
self.threshf = _halfthreshold_percentile
# prepare decay (if not passed)
if perc is None and decay is None:
self.decay = self.ncp.ones(niter)
# step size
if alpha is not None:
self.alpha = alpha
elif not hasattr(self, "alpha"):
# compute largest eigenvalues of Op^H * Op
Op1 = self.Op.H @ self.Op
if get_module_name(self.ncp) == "numpy":
maxeig: float = np.abs(
Op1.eigs(
neigs=1,
symmetric=True,
**self.eigsdict,
)[0]
)
else:
maxeig = np.abs(
power_iteration(
Op1,
dtype=Op1.dtype,
backend="cupy",
**self.eigsdict,
)[0]
)
self.alpha = 1.0 / maxeig
# define threshold
self.thresh = eps * self.alpha * 0.5
# initialize model and cost function
if x0 is None:
if y.ndim == 1:
x = self.ncp.zeros(int(self.Op.shape[1]), dtype=self.Op.dtype)
else:
x = self.ncp.zeros(
(int(self.Op.shape[1]), y.shape[1]), dtype=self.Op.dtype
)
else:
if y.ndim != x0.ndim:
# error for wrong dimensions
raise ValueError("Number of columns of x0 and data are not the same")
elif x0.shape[0] != self.Op.shape[1]:
# error for wrong dimensions
raise ValueError("Operator and input vector have different dimensions")
else:
x = x0.copy()
# create variable to track residual
if monitorres:
self.normresold = np.inf
# for fista
self.t = 1.0
# create variables to track the residual norm and iterations
self.cost: List[float] = []
self.iiter = 0
# print setup
if show:
self._print_setup()
return x
def step(self, x: NDArray, show: bool = False) -> Tuple[NDArray, float]:
r"""Run one step of solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by a step of ISTA
show : :obj:`bool`, optional
Display iteration log
Returns
-------
x : :obj:`np.ndarray`
Updated model vector
xupdate : :obj:`float`
Norm of the update
"""
# store old vector
xold = x.copy()
# compute residual
res: NDArray = self.y - self.Opmatvec(x)
if self.monitorres:
self.normres = np.linalg.norm(res)
if self.normres > self.normresold:
raise ValueError(
f"ISTA stopped at iteration {self.iiter} due to "
"residual increasing, consider modifying "
"eps and/or alpha..."
)
else:
self.normresold = self.normres
# compute gradient
grad: NDArray = self.alpha * (self.Oprmatvec(res))
# update inverted model
x_unthesh: NDArray = x + grad
if self.SOp is not None:
x_unthesh = self.SOprmatvec(x_unthesh)
if self.perc is None and self.decay is not None:
x = self.threshf(x_unthesh, self.decay[self.iiter] * self.thresh)
elif self.perc is not None:
x = self.threshf(x_unthesh, 100 - self.perc)
if self.SOp is not None:
x = self.SOpmatvec(x)
# model update
xupdate = np.linalg.norm(x - xold)
costdata = 0.5 * np.linalg.norm(res) ** 2
costreg = self.eps * np.linalg.norm(x, ord=1)
self.cost.append(float(costdata + costreg))
self.iiter += 1
if show:
self._print_step(x, costdata, costreg, xupdate)
return x, xupdate
def run(
self,
x: NDArray,
niter: Optional[int] = None,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
) -> NDArray:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of CG
niter : :obj:`int`, optional
Number of iterations. Can be set to ``None`` if already
provided in the setup call
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
xupdate = np.inf
niter = self.niter if niter is None else niter
if niter is None:
raise ValueError("niter must not be None")
while self.iiter < niter and xupdate > self.tol:
showstep = (
True
if show
and (
self.iiter < itershow[0]
or niter - self.iiter < itershow[1]
or self.iiter % itershow[2] == 0
)
else False
)
x, xupdate = self.step(x, showstep)
self.callback(x)
if xupdate <= self.tol:
logging.warning(
"update smaller that tolerance for " "iteration %d", self.iiter
)
return x
def finalize(self, show: bool = False) -> None:
r"""Finalize solver
Parameters
----------
show : :obj:`bool`, optional
Display finalize log
"""
self.tend = time.time()
self.telapsed = self.tend - self.tstart
self.cost = np.array(self.cost)
if show:
self._print_finalize()
def solve(
self,
y: NDArray,
x0: Optional[NDArray] = None,
niter: Optional[int] = None,
SOp: Optional[LinearOperator] = None,
eps: float = 0.1,
alpha: Optional[float] = None,
eigsdict: Optional[Dict[str, Any]] = None,
tol: float = 1e-10,
threshkind: str = "soft",
perc: Optional[float] = None,
decay: Optional[NDArray] = None,
monitorres: bool = False,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
) -> Tuple[NDArray, int, NDArray]:
r"""Run entire solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
x0: :obj:`numpy.ndarray`, optional
Initial guess
niter : :obj:`int`
Number of iterations
SOp : :obj:`pylops.LinearOperator`, optional
Regularization operator (use when solving the analysis problem)
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size. To guarantee convergence, ensure
:math:`\alpha \le 1/\lambda_\text{max}`, where :math:`\lambda_\text{max}`
is the largest eigenvalue of :math:`\mathbf{Op}^H\mathbf{Op}`.
If ``None``, the maximum eigenvalue is estimated and the optimal step size
is chosen as :math:`1/\lambda_\text{max}`. If provided, the
convergence criterion will not be checked internally.
eigsdict : :obj:`dict`, optional
Dictionary of parameters to be passed to :func:`pylops.LinearOperator.eigs` method
when computing the maximum eigenvalue
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', 'hard-percentile',
'soft-percentile', or 'half-percentile' - 'soft' used as default)
perc : :obj:`float`, optional
Percentile, as percentage of values to be kept by thresholding (to be
provided when thresholding is soft-percentile or half-percentile)
decay : :obj:`numpy.ndarray`, optional
Decay factor to be applied to thresholding during iterations
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
"""
x = self.setup(
y=y,
x0=x0,
niter=niter,
SOp=SOp,
eps=eps,
alpha=alpha,
eigsdict=eigsdict,
tol=tol,
threshkind=threshkind,
perc=perc,
decay=decay,
monitorres=monitorres,
show=show,
)
x = self.run(x, niter, show=show, itershow=itershow)
self.finalize(show)
return x, self.iiter, self.cost
[docs]class FISTA(ISTA):
r"""Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
Solve an optimization problem with :math:`L_p, \; p=0, 0.5, 1`
regularization, given the operator ``Op`` and data ``y``.
The operator can be real or complex, and should ideally be either square
:math:`N=M` or underdetermined :math:`N<M`.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
Raises
------
NotImplementedError
If ``threshkind`` is different from hard, soft, half, soft-percentile,
or half-percentile
ValueError
If ``perc=None`` when ``threshkind`` is soft-percentile or
half-percentile
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
ISTA: Iterative Shrinkage-Thresholding Algorithm (ISTA).
SPGL1: Spectral Projected-Gradient for L1 norm (SPGL1).
SplitBregman: Split Bregman for mixed L2-L1 norms.
Notes
-----
Solves the following synthesis problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{y}`:
.. math::
J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 +
\epsilon \|\mathbf{x}\|_p
or the analysis problem:
.. math::
J = \|\mathbf{y} - \mathbf{Op}\,\mathbf{x}\|_2^2 +
\epsilon \|\mathbf{SOp}^H\,\mathbf{x}\|_p
if ``SOp`` is provided.
The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [1]_ is used,
where :math:`p=0, 0.5, 1`. This is a modified version of ISTA solver with
improved convergence properties and limited additional computational cost.
Similarly to the ISTA solver, the choice of the thresholding algorithm to
apply at every iteration is based on the choice of :math:`p`.
.. [1] Beck, A., and Teboulle, M., “A Fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems”, SIAM Journal on
Imaging Sciences, vol. 2, pp. 183-202. 2009.
"""
def step(self, x: NDArray, z: NDArray, show: bool = False) -> NDArray:
r"""Run one step of solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by a step of ISTA
z : :obj:`np.ndarray`
Current auxiliary model vector to be updated by a step of ISTA
show : :obj:`bool`, optional
Display iteration log
Returns
-------
x : :obj:`np.ndarray`
Updated model vector
z : :obj:`np.ndarray`
Updated auxiliary model vector
xupdate : :obj:`float`
Norm of the update
"""
# store old vector
xold = x.copy()
# compute residual
resz: NDArray = self.y - self.Opmatvec(z)
if self.monitorres:
self.normres = np.linalg.norm(resz)
if self.normres > self.normresold:
raise ValueError(
f"ISTA stopped at iteration {self.iiter} due to "
"residual increasing, consider modifying "
"eps and/or alpha..."
)
else:
self.normresold = self.normres
# compute gradient
grad: NDArray = self.alpha * (self.Oprmatvec(resz))
# update inverted model
x_unthesh: NDArray = z + grad
if self.SOp is not None:
x_unthesh = self.SOprmatvec(x_unthesh)
if self.perc is None and self.decay is not None:
x = self.threshf(x_unthesh, self.decay[self.iiter] * self.thresh)
elif self.perc is not None:
x = self.threshf(x_unthesh, 100 - self.perc)
if self.SOp is not None:
x = self.SOpmatvec(x)
# update auxiliary coefficients
told = self.t
self.t = (1.0 + np.sqrt(1.0 + 4.0 * self.t**2)) / 2.0
z = x + ((told - 1.0) / self.t) * (x - xold)
# model update
xupdate = np.linalg.norm(x - xold)
costdata = 0.5 * np.linalg.norm(self.y - self.Op @ x) ** 2
costreg = self.eps * np.linalg.norm(x, ord=1)
self.cost.append(float(costdata + costreg))
self.iiter += 1
if show:
self._print_step(x, costdata, costreg, xupdate)
return x, z, xupdate
def run(
self,
x: NDArray,
niter: Optional[int] = None,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
) -> NDArray:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of CG
niter : :obj:`int`, optional
Number of iterations. Can be set to ``None`` if already
provided in the setup call
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
z = x.copy()
xupdate = np.inf
niter = self.niter if niter is None else niter
if niter is None:
raise ValueError("niter must not be None")
while self.iiter < niter and xupdate > self.tol:
showstep = (
True
if show
and (
self.iiter < itershow[0]
or niter - self.iiter < itershow[1]
or self.iiter % itershow[2] == 0
)
else False
)
x, z, xupdate = self.step(x, z, showstep)
self.callback(x)
if xupdate <= self.tol:
logging.warning(
"update smaller that tolerance for " "iteration %d", self.iiter
)
return x
[docs]class SPGL1(Solver):
r"""Spectral Projected-Gradient for L1 norm.
Solve a constrained system of equations given the operator ``Op``
and a sparsyfing transform ``SOp`` aiming to retrive a model that
is sparse in the sparsifying domain.
This is a simple wrapper to :py:func:`spgl1.spgl1`
which is a porting of the well-known
`SPGL1 <https://www.cs.ubc.ca/~mpf/spgl1/>`_ MATLAB solver into Python.
In order to be able to use this solver you need to have installed the
``spgl1`` library.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert of size :math:`[N \times M]`.
Raises
------
ModuleNotFoundError
If the ``spgl1`` library is not installed
Notes
-----
Solve different variations of sparsity-promoting inverse problem by
imposing sparsity in the retrieved model [1]_.
The first problem is called *basis pursuit denoise (BPDN)* and
its cost function is
.. math::
\|\mathbf{x}\|_1 \quad \text{subject to} \quad
\left\|\mathbf{Op}\,\mathbf{S}^H\mathbf{x}-\mathbf{y}\right\|_2^2
\leq \sigma,
while the second problem is the *ℓ₁-regularized least-squares or LASSO*
problem and its cost function is
.. math::
\left\|\mathbf{Op}\,\mathbf{S}^H\mathbf{x}-\mathbf{y}\right\|_2^2
\quad \text{subject to} \quad \|\mathbf{x}\|_1 \leq \tau
.. [1] van den Berg E., Friedlander M.P., "Probing the Pareto frontier
for basis pursuit solutions", SIAM J. on Scientific Computing,
vol. 31(2), pp. 890-912. 2008.
"""
def _print_setup(self, xcomplex: bool = False) -> None:
self._print_solver()
strprec = f"SOp={self.SOp}"
strreg = f"tau={self.tau} sigma={self.sigma}"
print(strprec)
print(strreg)
print("-" * 80)
def _print_finalize(self) -> None:
print(f"\nTotal time (s) = {self.telapsed:.2f}")
print("-" * 80 + "\n")
def setup(
self,
y: NDArray,
SOp: Optional[LinearOperator] = None,
tau: int = 0,
sigma: int = 0,
show: bool = False,
) -> None:
r"""Setup solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
SOp : :obj:`pylops.LinearOperator`, optional
Sparsifying transform
tau : :obj:`float`, optional
Non-negative LASSO scalar. If different from ``0``,
SPGL1 will solve LASSO problem
sigma : :obj:`list`, optional
BPDN scalar. If different from ``0``,
SPGL1 will solve BPDN problem
show : :obj:`bool`, optional
Display setup log
"""
if spgl1_message is not None:
raise ModuleNotFoundError(spgl1_message)
self.y = y
self.SOp = SOp
self.tau = tau
self.sigma = sigma
self.ncp = get_array_module(y)
# print setup
if show:
self._print_setup()
def step(self) -> None:
raise NotImplementedError(
"SPGL1 uses as default the"
"spgl1.spgl1 solver, therefore the "
"step method is not implemented. Use directly run or solve."
)
def run(
self,
x: NDArray,
show: bool = False,
**kwargs_spgl1,
) -> Tuple[NDArray, NDArray, Dict[str, Any]]:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of the solver.
If ``None``, x is assumed to be a zero vector
show : :obj:`bool`, optional
Display iterations log
**kwargs_spgl1
Arbitrary keyword arguments for
:py:func:`spgl1.spgl1` solver
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model in original domain.
pinv : :obj:`numpy.ndarray`
Inverted model in sparse domain.
info : :obj:`dict`
Dictionary with the following information:
- ``tau``, final value of tau (see sigma above)
- ``rnorm``, two-norm of the optimal residual
- ``rgap``, relative duality gap (an optimality measure)
- ``gnorm``, Lagrange multiplier of (LASSO)
- ``stat``, final status of solver
* ``1``: found a BPDN solution,
* ``2``: found a BP solution; exit based on small gradient,
* ``3``: found a BP solution; exit based on small residual,
* ``4``: found a LASSO solution,
* ``5``: error, too many iterations,
* ``6``: error, linesearch failed,
* ``7``: error, found suboptimal BP solution,
* ``8``: error, too many matrix-vector products.
- ``niters``, number of iterations
- ``nProdA``, number of multiplications with A
- ``nProdAt``, number of multiplications with A'
- ``n_newton``, number of Newton steps
- ``time_project``, projection time (seconds)
- ``time_matprod``, matrix-vector multiplications time (seconds)
- ``time_total``, total solution time (seconds)
- ``niters_lsqr``, number of lsqr iterations (if ``subspace_min=True``)
- ``xnorm1``, L1-norm model solution history through iterations
- ``rnorm2``, L2-norm residual history through iterations
- ``lambdaa``, Lagrange multiplier history through iterations
"""
pinv, _, _, info = ext_spgl1(
self.Op if self.SOp is None else self.Op @ self.SOp.H,
self.y,
tau=self.tau,
sigma=self.sigma,
x0=x,
**kwargs_spgl1,
)
xinv = pinv.copy() if self.SOp is None else self.SOp.rmatvec(pinv)
return xinv, pinv, info
def solve(
self,
y: NDArray,
x0: Optional[NDArray] = None,
SOp: Optional[LinearOperator] = None,
tau: float = 0.0,
sigma: float = 0,
show: bool = False,
**kwargs_spgl1,
) -> Tuple[NDArray, NDArray, Dict[str, Any]]:
r"""Run entire solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
x0 : :obj:`numpy.ndarray`, optional
Initial guess
SOp : :obj:`pylops.LinearOperator`, optional
Sparsifying transform
tau : :obj:`float`, optional
Non-negative LASSO scalar. If different from ``0``,
SPGL1 will solve LASSO problem
sigma : :obj:`list`, optional
BPDN scalar. If different from ``0``,
SPGL1 will solve BPDN problem
show : :obj:`bool`, optional
Display log
**kwargs_spgl1
Arbitrary keyword arguments for
:py:func:`spgl1.spgl1` solver
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model in original domain.
pinv : :obj:`numpy.ndarray`
Inverted model in sparse domain.
info : :obj:`dict`
Dictionary with the following information:
- ``tau``, final value of tau (see sigma above)
- ``rnorm``, two-norm of the optimal residual
- ``rgap``, relative duality gap (an optimality measure)
- ``gnorm``, Lagrange multiplier of (LASSO)
- ``stat``, final status of solver
* ``1``: found a BPDN solution,
* ``2``: found a BP solution; exit based on small gradient,
* ``3``: found a BP solution; exit based on small residual,
* ``4``: found a LASSO solution,
* ``5``: error, too many iterations,
* ``6``: error, linesearch failed,
* ``7``: error, found suboptimal BP solution,
* ``8``: error, too many matrix-vector products.
- ``niters``, number of iterations
- ``nProdA``, number of multiplications with A
- ``nProdAt``, number of multiplications with A'
- ``n_newton``, number of Newton steps
- ``time_project``, projection time (seconds)
- ``time_matprod``, matrix-vector multiplications time (seconds)
- ``time_total``, total solution time (seconds)
- ``niters_lsqr``, number of lsqr iterations (if ``subspace_min=True``)
- ``xnorm1``, L1-norm model solution history through iterations
- ``rnorm2``, L2-norm residual history through iterations
- ``lambdaa``, Lagrange multiplier history through iterations
"""
self.setup(y=y, SOp=SOp, tau=tau, sigma=sigma, show=show)
xinv, pinv, info = self.run(x0, show=show, **kwargs_spgl1)
self.finalize(show)
return xinv, pinv, info
[docs]class SplitBregman(Solver):
r"""Split Bregman for mixed L2-L1 norms.
Solve an unconstrained system of equations with mixed :math:`L_2` and :math:`L_1`
regularization terms given the operator ``Op``, a list of :math:`L_1`
regularization terms ``RegsL1``, and an optional list of :math:`L_2`
regularization terms ``RegsL2``.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
Notes
-----
Solve the following system of unconstrained, regularized equations
given the operator :math:`\mathbf{Op}` and a set of mixed norm
(:math:`L^2` and :math:`L_1`)
regularization terms :math:`\mathbf{R}_{2,i}` and
:math:`\mathbf{R}_{1,i}`, respectively:
.. math::
J = \frac{\mu}{2} \|\textbf{y} - \textbf{Op}\,\textbf{x} \|_2^2 +
\frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} -
\mathbf{R}_{2,i} \textbf{x} \|_2^2 +
\sum_i \epsilon_{\mathbf{R}_{1,i}} \| \mathbf{R}_{1,i} \textbf{x} \|_1
where :math:`\mu` is the reconstruction damping, :math:`\epsilon_{\mathbf{R}_{2,i}}`
are the damping factors used to weight the different :math:`L^2` regularization
terms of the cost function and :math:`\epsilon_{\mathbf{R}_{1,i}}`
are the damping factors used to weight the different :math:`L_1` regularization
terms of the cost function.
The generalized Split-Bergman algorithm [1]_ is used to solve such cost
function: the algorithm is composed of a sequence of unconstrained
inverse problems and Bregman updates.
The original system of equations is initially converted into a constrained
problem:
.. math::
J = \frac{\mu}{2} \|\textbf{y} - \textbf{Op}\,\textbf{x}\|_2^2 +
\frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} -
\mathbf{R}_{2,i} \textbf{x}\|_2^2 +
\sum_i \| \textbf{y}_i \|_1 \quad \text{subject to} \quad
\textbf{y}_i = \mathbf{R}_{1,i} \textbf{x} \quad \forall i
and solved as follows:
.. math::
\DeclareMathOperator*{\argmin}{arg\,min}
\begin{align}
(\textbf{x}^{k+1}, \textbf{y}_i^{k+1}) =
\argmin_{\mathbf{x}, \mathbf{y}_i}
\|\textbf{y} - \textbf{Op}\,\textbf{x}\|_2^2
&+ \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{2,i}} \|\mathbf{y}_{\mathbf{R}_{2,i}} -
\mathbf{R}_{2,i} \textbf{x}\|_2^2 \\
&+ \frac{1}{2}\sum_i \epsilon_{\mathbf{R}_{1,i}} \|\textbf{y}_i -
\mathbf{R}_{1,i} \textbf{x} - \textbf{b}_i^k\|_2^2 \\
&+ \sum_i \| \textbf{y}_i \|_1
\end{align}
.. math::
\textbf{b}_i^{k+1}=\textbf{b}_i^k +
(\mathbf{R}_{1,i} \textbf{x}^{k+1} - \textbf{y}^{k+1})
The :py:func:`scipy.sparse.linalg.lsqr` solver and a fast shrinkage
algorithm are used within a inner loop to solve the first step. The entire
procedure is repeated ``niter_outer`` times until convergence.
.. [1] Goldstein T. and Osher S., "The Split Bregman Method for
L1-Regularized Problems", SIAM J. on Scientific Computing, vol. 2(2),
pp. 323-343. 2008.
"""
def _print_setup(self, xcomplex: bool = False) -> None:
self._print_solver(nbar=65)
strpar = (
f"niter_outer = {self.niter_outer:3d} niter_inner = {self.niter_inner:3d} tol = {self.tol:2.2e}\n"
f"mu = {self.mu:2.2e} epsL1 = {self.epsRL1s}\t epsL2 = {self.epsRL2s}"
)
print(strpar)
print("-" * 65)
if not xcomplex:
head1 = " Itn x[0] r2norm r12norm"
else:
head1 = " Itn x[0] r2norm r12norm"
print(head1)
def _print_step(self, x: NDArray) -> None:
strx = f"{x[0]:1.2e} " if np.iscomplexobj(x) else f"{x[0]:11.4e} "
str1 = f"{self.iiter:6g} " + strx
str2 = f"{self.costdata:10.3e} {self.costtot:9.3e}"
print(str1 + str2)
def setup(
self,
y: NDArray,
RegsL1: List[LinearOperator],
x0: Optional[NDArray] = None,
niter_outer: int = 3,
niter_inner: int = 5,
RegsL2: Optional[List[LinearOperator]] = None,
dataregsL2: Optional[Sequence[NDArray]] = None,
mu: float = 1.0,
epsRL1s: Optional[SamplingLike] = None,
epsRL2s: Optional[SamplingLike] = None,
tol: float = 1e-10,
tau: float = 1.0,
restart: bool = False,
show: bool = False,
) -> NDArray:
r"""Setup solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
RegsL1 : :obj:`list`
:math:`L_1` regularization operators
x0 : :obj:`np.ndarray`, optional
Initial guess of size :math:`[M \times 1]`. If ``None``, initialize
internally as zero vector
niter_outer : :obj:`int`, optional
Number of iterations of outer loop
niter_inner : :obj:`int`, optional
Number of iterations of inner loop of first step of the Split Bregman
algorithm. A small number of iterations is generally sufficient and
for many applications optimal efficiency is obtained when only one
iteration is performed.
RegsL2 : :obj:`list`, optional
Additional :math:`L^2` regularization operators
(if ``None``, :math:`L^2` regularization is not added to the problem)
dataregsL2 : :obj:`list`, optional
:math:`L^2` Regularization data (must have the same number of elements
of ``RegsL2`` or equal to ``None`` to use a zero data for every
regularization operator in ``RegsL2``)
mu : :obj:`float`, optional
Data term damping
epsRL1s : :obj:`list`
:math:`L_1` Regularization dampings (must have the same number of elements
as ``RegsL1``)
epsRL2s : :obj:`list`
:math:`L^2` Regularization dampings (must have the same number of elements
as ``RegsL2``)
tol : :obj:`float`, optional
Tolerance. Stop outer iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
tau : :obj:`float`, optional
Scaling factor in the Bregman update (must be close to 1)
restart : :obj:`bool`, optional
Initialize the unconstrained inverse problem in inner loop with
the initial guess (``True``) or with the last estimate (``False``).
Note that when this is set to ``True``, the ``x0`` provided in the setup will
be used in all iterations.
show : :obj:`bool`, optional
Display setup log
Returns
-------
x : :obj:`np.ndarray`
Initial guess of size :math:`[N \times 1]`
"""
self.y = y
self.RegsL1 = RegsL1
self.niter_outer = niter_outer
self.niter_inner = niter_inner
self.RegsL2 = RegsL2
self.dataregsL2 = list(dataregsL2) if dataregsL2 is not None else []
self.mu = mu
self.epsRL1s = list(epsRL1s) if epsRL1s is not None else []
self.epsRL2s = list(epsRL2s) if epsRL2s is not None else []
self.tol = tol
self.tau = tau
self.restart = restart
self.ncp = get_array_module(y)
# L1 regularizations
self.nregsL1 = len(RegsL1)
self.b = [
self.ncp.zeros(RegL1.shape[0], dtype=self.Op.dtype) for RegL1 in RegsL1
]
self.d = self.b.copy()
# L2 regularizations
self.nregsL2 = 0 if RegsL2 is None else len(RegsL2)
if self.nregsL2 > 0 and RegsL2 is not None:
self.Regs = RegsL2 + RegsL1
if dataregsL2 is None:
self.dataregsL2 = [
self.ncp.zeros(Reg.shape[0], dtype=self.Op.dtype) for Reg in RegsL2
]
else:
self.Regs = RegsL1
self.dataregsL2 = []
# Rescale dampings
self.epsRs: List[float] = []
if epsRL2s is not None:
self.epsRs += [
np.sqrt(epsRL2s[ireg] / 2) / np.sqrt(mu / 2)
for ireg in range(self.nregsL2)
]
if epsRL1s is not None:
self.epsRs += [
np.sqrt(epsRL1s[ireg] / 2) / np.sqrt(mu / 2)
for ireg in range(self.nregsL1)
]
self.x0 = x0
x = self.ncp.zeros(self.Op.shape[1], dtype=self.Op.dtype) if x0 is None else x0
# create variables to track the residual norm and iterations
self.cost: List[float] = []
self.iiter = 0
if show:
self._print_setup(np.iscomplexobj(x))
return x
def step(
self,
x: NDArray,
show: bool = False,
show_inner: bool = False,
**kwargs_lsqr,
) -> NDArray:
r"""Run one step of solver
Parameters
----------
x : :obj:`list` or :obj:`np.ndarray`
Current model vector to be updated by a step of OMP
show_inner : :obj:`bool`, optional
Display inner iteration logs of lsqr
show : :obj:`bool`, optional
Display iteration log
**kwargs_lsqr
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.lsqr` solver used to solve the first
subproblem in the first step of the Split Bregman algorithm.
Returns
-------
x : :obj:`np.ndarray`
Updated model vector
"""
for _ in range(self.niter_inner):
# regularized problem
dataregs = self.dataregsL2 + [
self.d[ireg] - self.b[ireg] for ireg in range(self.nregsL1)
]
x = regularized_inversion(
self.Op,
self.y,
self.Regs,
dataregs=dataregs,
epsRs=self.epsRs,
x0=self.x0 if self.restart else x,
show=show_inner,
**kwargs_lsqr,
)[0]
# Shrinkage
self.d = [
_softthreshold(
self.RegsL1[ireg].matvec(x) + self.b[ireg], self.epsRL1s[ireg]
)
for ireg in range(self.nregsL1)
]
# Bregman update
self.b = [
self.b[ireg] + self.tau * (self.RegsL1[ireg].matvec(x) - self.d[ireg])
for ireg in range(self.nregsL1)
]
# compute residual norms
self.costdata = (
self.mu / 2.0 * self.ncp.linalg.norm(self.y - self.Op.matvec(x)) ** 2
)
self.costregL2 = (
0
if self.RegsL2 is None
else [
epsRL2 * self.ncp.linalg.norm(dataregL2 - RegL2.matvec(x)) ** 2
for epsRL2, RegL2, dataregL2 in zip(
self.epsRL2s, self.RegsL2, self.dataregsL2
)
]
)
self.costregL1 = [
self.ncp.linalg.norm(RegL1.matvec(x), ord=1)
for epsRL1, RegL1 in zip(self.epsRL1s, self.RegsL1)
]
self.costtot = (
self.costdata
+ self.ncp.sum(self.ncp.array(self.costregL2))
+ self.ncp.sum(self.ncp.array(self.costregL1))
)
# update history parameters
self.iiter += 1
self.cost.append(float(self.costtot))
if show:
self._print_step(x)
return x
def run(
self,
x: NDArray,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
show_inner: bool = False,
**kwargs_lsqr,
) -> NDArray:
r"""Run solver
Parameters
----------
x : :obj:`np.ndarray`
Current model vector to be updated by multiple steps of IRLS
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
show_inner : :obj:`bool`, optional
Display inner iteration logs of lsqr
**kwargs_lsqr
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.lsqr` solver used to solve the first
subproblem in the first step of the Split Bregman algorithm.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
xold = x.copy() + 1.1 * self.tol
while (
self.ncp.linalg.norm(x - xold) > self.tol and self.iiter < self.niter_outer
):
xold = x.copy()
showstep = (
True
if show
and (
self.iiter < itershow[0]
or self.niter_outer - self.iiter < itershow[1]
or self.iiter % itershow[2] == 0
)
else False
)
x = self.step(x, showstep, show_inner, **kwargs_lsqr)
self.callback(x)
return x
def finalize(self, show: bool = False) -> NDArray:
r"""Finalize solver
Parameters
----------
show : :obj:`bool`, optional
Display finalize log
Returns
-------
xfin : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
"""
self.tend = time.time()
self.telapsed = self.tend - self.tstart
self.cost = np.array(self.cost)
if show:
self._print_finalize(nbar=65)
def solve(
self,
y: NDArray,
RegsL1: List[LinearOperator],
x0: Optional[NDArray] = None,
niter_outer: int = 3,
niter_inner: int = 5,
RegsL2: Optional[List[LinearOperator]] = None,
dataregsL2: Optional[List[NDArray]] = None,
mu: float = 1.0,
epsRL1s: Optional[SamplingLike] = None,
epsRL2s: Optional[SamplingLike] = None,
tol: float = 1e-10,
tau: float = 1.0,
restart: bool = False,
show: bool = False,
itershow: Tuple[int, int, int] = (10, 10, 10),
show_inner: bool = False,
**kwargs_lsqr,
) -> Tuple[NDArray, int, NDArray]:
r"""Run entire solver
Parameters
----------
y : :obj:`np.ndarray`
Data of size :math:`[N \times 1]`
RegsL1 : :obj:`list`
:math:`L_1` regularization operators
x0 : :obj:`np.ndarray`, optional
Initial guess of size :math:`[M \times 1]`. If ``None``, initialize
internally as zero vector
niter_outer : :obj:`int`, optional
Number of iterations of outer loop
niter_inner : :obj:`int`, optional
Number of iterations of inner loop of first step of the Split Bregman
algorithm. A small number of iterations is generally sufficient and
for many applications optimal efficiency is obtained when only one
iteration is performed.
RegsL2 : :obj:`list`, optional
Additional :math:`L^2` regularization operators
(if ``None``, :math:`L^2` regularization is not added to the problem)
dataregsL2 : :obj:`list`, optional
:math:`L^2` Regularization data (must have the same number of elements
of ``RegsL2`` or equal to ``None`` to use a zero data for every
regularization operator in ``RegsL2``)
mu : :obj:`float`, optional
Data term damping
epsRL1s : :obj:`list`
:math:`L_1` Regularization dampings (must have the same number of elements
as ``RegsL1``)
epsRL2s : :obj:`list`
:math:`L^2` Regularization dampings (must have the same number of elements
as ``RegsL2``)
tol : :obj:`float`, optional
Tolerance. Stop outer iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
tau : :obj:`float`, optional
Scaling factor in the Bregman update (must be close to 1)
restart : :obj:`bool`, optional
Initialize the unconstrained inverse problem in inner loop with
the initial guess (``True``) or with the last estimate (``False``).
Note that when this is set to ``True``, the ``x0`` provided in the setup will
be used in all iterations.
show : :obj:`bool`, optional
Display logs
itershow : :obj:`tuple`, optional
Display set log for the first N1 steps, last N2 steps,
and every N3 steps in between where N1, N2, N3 are the
three element of the list.
show_inner : :obj:`bool`, optional
Display inner iteration logs of lsqr
**kwargs_lsqr
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.lsqr` solver used to solve the first
subproblem in the first step of the Split Bregman algorithm.
Returns
-------
x : :obj:`np.ndarray`
Estimated model of size :math:`[M \times 1]`
iiter : :obj:`int`
Iteration number of outer loop upon termination
cost : :obj:`numpy.ndarray`
History of cost function through iterations
"""
x = self.setup(
y,
RegsL1,
x0=x0,
niter_outer=niter_outer,
niter_inner=niter_inner,
RegsL2=RegsL2,
dataregsL2=dataregsL2,
mu=mu,
epsRL1s=epsRL1s,
epsRL2s=epsRL2s,
tol=tol,
tau=tau,
restart=restart,
show=show,
)
x = self.run(
x, show=show, itershow=itershow, show_inner=show_inner, **kwargs_lsqr
)
self.finalize(show)
return x, self.iiter, self.cost
