import logging
import time
import numpy as np
from scipy.sparse.linalg import lsqr
from pylops import LinearOperator
from pylops.basicoperators import Diagonal
from pylops.optimization.leastsquares import NormalEquationsInversion, \
RegularizedInversion
try:
from spgl1 import spgl1
except ModuleNotFoundError:
spgl1 = None
spgl1_message = 'Spgl1 not installed. ' \
'Run "pip install spgl1".'
except Exception as e:
spgl1 = None
spgl1_message = 'Failed to import spgl1 (error:%s).' % e
def _hardthreshold(x, thresh):
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, 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
"""
x1 = x.copy()
x1[np.abs(x) <= np.sqrt(2*thresh)] = 0
return x1
def _softthreshold(x, thresh):
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, 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
"""
#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)
x1 = x - thresh * x / np.abs(x)
x1[np.abs(x) <= thresh] = 0
return x1
def _halfthreshold(x, thresh):
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, 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
"""
phi = 2. / 3. * np.arccos((thresh / 8.) * (np.abs(x) / 3.) ** (-1.5))
x1 = 2./3. * x * (1 + np.cos(2. * np.pi / 3. - phi))
#x1[np.abs(x) <= 1.5 * thresh ** (2. / 3.)] = 0
x1[np.abs(x) <= (54 ** (1. / 3.) / 4.) * thresh ** (2. / 3.)] = 0
return x1
def _hardthreshold_percentile(x, perc):
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 _halfthreshold(x, 0.5 * thresh ** 2)
def _softthreshold_percentile(x, perc):
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, perc):
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`
Tresholded vector
"""
thresh = np.percentile(np.abs(x), perc)
#return _halfthreshold(x, (2. / 3. * thresh) ** (1.5))
return _halfthreshold(x, (4. / 54 ** (1. / 3.) * thresh) ** 1.5)
def _shrinkage(x, thresh):
r"""Shrinkage.
Applies shrinkage to vector ``x``.
Parameters
----------
x : :obj:`numpy.ndarray`
Vector
thresh : :obj:`float`
Threshold
"""
xabs = np.abs(x)
return x/(xabs+1e-10) * np.maximum(xabs - thresh, 0)
[docs]def IRLS(Op, data, nouter, threshR=False, epsR=1e-10,
epsI=1e-10, x0=None, tolIRLS=1e-10,
returnhistory=False, **kwargs_cg):
r"""Iteratively reweighted least squares.
Solve an optimization problem with :math:`L1` cost function given the
operator ``Op`` and data ``y``. 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`.
The IRLS solver is robust to *outliers* since the L1 norm given less
weight to large residuals than L2 norm does.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
data : :obj:`numpy.ndarray`
Data
nouter : :obj:`int`
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
espI : :obj:`float`, optional
Tikhonov damping
x0 : :obj:`numpy.ndarray`, optional
Initial guess
tolIRLS : :obj:`float`, optional
Tolerance. Stop outer iterations if difference between inverted model
at subsequent iterations is smaller than ``tolIRLS``
returnhistory : :obj:`bool`, optional
Return history of inverted model for each outer iteration of IRLS
**kwargs_cg
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.cg` solver
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
nouter : :obj:`int`
Number of effective outer iterations
xinv_hist : :obj:`numpy.ndarray`, optional
History of inverted model
rw_hist : :obj:`numpy.ndarray`, optional
History of weights
Notes
-----
Solves the following optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_1
by a set of outer iterations which require to repeateadly solve a
weighted least squares problem of the form:
.. math::
\mathbf{x}^{(i+1)} = \operatorname*{arg\,min}_\mathbf{x} ||\mathbf{d} -
\mathbf{Op} \mathbf{x}||_{2, \mathbf{R}^{(i)}} +
\epsilon_I^2 ||\mathbf{x}||
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 j-th element of the diagonal of
:math:`\mathbf{R}^{(i)}` is
.. math::
R^{(i)}_{j,j} = \frac{1}{|r^{(i)}_j|+\epsilon_R}
or
.. math::
R^{(i)}_{j,j} = \frac{1}{max(|r^{(i)}_j|, \epsilon_R)}
depending on the choice ``threshR``. In either case,
:math:`\epsilon_R` is the user-defined stabilization/thresholding
factor [1]_.
.. [1] https://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares
"""
if x0 is not None:
data = data - Op * x0
if returnhistory:
xinv_hist = np.zeros((nouter+1, Op.shape[1]))
rw_hist = np.zeros((nouter+1, Op.shape[0]))
# first iteration (unweighted least-squares)
xinv = NormalEquationsInversion(Op, None, data, epsI=epsI,
returninfo=False,
**kwargs_cg)
r = data-Op*xinv
if returnhistory:
xinv_hist[0] = xinv
for iiter in range(nouter):
# other iterations (weighted least-squares)
xinvold = xinv.copy()
if threshR:
rw = 1./np.maximum(np.abs(r), epsR)
else:
rw = 1./(np.abs(r)+epsR)
rw = rw / rw.max()
R = Diagonal(rw)
xinv = NormalEquationsInversion(Op, [], data, Weight=R,
epsI=epsI,
returninfo=False,
**kwargs_cg)
r = data-Op*xinv
# save history
if returnhistory:
rw_hist[iiter] = rw
xinv_hist[iiter+1] = xinv
# check tolerance
if np.linalg.norm(xinv - xinvold) < tolIRLS:
nouter = iiter
break
# adding initial guess
if x0 is not None:
xinv = x0 + xinv
if returnhistory:
xinv_hist = x0 + xinv_hist
if returnhistory:
return xinv, nouter, xinv_hist[:nouter+1], rw_hist[:nouter+1]
else:
return xinv, nouter
[docs]def OMP(Op, data, niter_outer=10, niter_inner=40, sigma=1e-4,
normalizecols=False, show=False):
r"""Orthogonal Matching Pursuit (OMP).
Solve an optimization problem with :math:`L0` 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
data : :obj:`numpy.ndarray`
Data
niter_outer : :obj:`int`
Number of iterations of outer loop
niter_inner : :obj:`int`
Number of iterations of inner loop
sigma : :obj:`list`
Maximum L2 norm of residual. When smaller stop iterations.
normalizecols : :obj:`list`
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 iterations log
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
iiter : :obj:`int`
Number of effective outer iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
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{d}`:
.. math::
||\mathbf{x}||_0 \quad subj. to \quad
||\mathbf{Op}\mathbf{x}-\mathbf{b}||_2 <= \sigma,
using Orthogonal Matching Pursuit (OMP). This is a very
simple iterative algorithm which applies the following step:
.. math::
\Lambda_k = \Lambda_{k-1} \cup \{ arg max_j
|\mathbf{Op}_j^H \mathbf{r}_k| \} \\
\mathbf{x}_k = \{ arg min_{\mathbf{x}}
||\mathbf{Op}_{\Lambda_k} \mathbf{x} - \mathbf{b}||_2
"""
Op = LinearOperator(Op)
if show:
tstart = time.time()
print('OMP optimization\n'
'-----------------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'sigma = %.2e\tniter_outer = %d\tniter_inner = %d\n'
'normalization=%s' %
(Op.shape[0], Op.shape[1], sigma, niter_outer,
niter_inner, normalizecols))
# find normalization factor for each column
if normalizecols:
ncols = Op.shape[1]
norms = np.zeros(ncols)
for icol in range(ncols):
unit = np.zeros(ncols)
unit[icol] = 1
norms[icol] = np.linalg.norm(Op.matvec(unit))
if show:
print('-----------------------------------------------------------------')
head1 = ' Itn r2norm'
print(head1)
cols = []
res = data.copy()
cost = np.zeros(niter_outer + 1)
cost[0] = np.linalg.norm(data)
iiter = 0
while iiter < niter_outer and cost[iiter] > sigma:
cres = np.abs(Op.rmatvec(res))
if normalizecols:
cres = cres / norms
# exclude columns already chosen by putting them negative
if iiter > 0:
cres[cols] = -1
# choose column with max cres
imax = np.argwhere(cres == np.max(cres)).ravel()
nimax = len(imax)
if nimax > 0:
imax = imax[np.random.permutation(nimax)[0]]
else:
imax = imax[0]
cols.append(imax)
# estimate model for current set of columns
Opcol = Op.apply_columns(cols)
x = lsqr(Opcol, data, iter_lim=niter_inner)[0]
res = data - Opcol.matvec(x)
iiter += 1
cost[iiter] = np.linalg.norm(res)
if show:
if iiter < 10 or niter_outer - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e' % (iiter + 1, cost[iiter])
print(msg)
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
xinv[cols] = x
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (iiter, time.time() - tstart))
print('-----------------------------------------------------------------\n')
return xinv, iiter, cost
[docs]def ISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, monitorres=False, returninfo=False, show=False,
threshkind='soft', perc=None, callback=None):
r"""Iterative Shrinkage-Thresholding Algorithm (ISTA).
Solve an optimization problem with :math:`Lp, \quad p=0, 1/2, 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
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`float`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
monitorres : :obj:`bool`, optional
Monitor that residual is decreasing
returninfo : :obj:`bool`, optional
Return info of CG solver
show : :obj:`bool`, optional
Display iterations log
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)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
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 optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Iterative Shrinkage-Thresholding Algorithms (ISTA) [1]_, where
:math:`p=0, 1, 1/2`. This is a very simple iterative algorithm which
applies the following step:
.. math::
\mathbf{x}^{(i+1)} = T_{(\epsilon \alpha /2, p)} (\mathbf{x}^{(i)} +
\alpha \mathbf{Op}^H (\mathbf{d} - \mathbf{Op} \mathbf{x}^{(i)}))
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 `p=0` or a half-thresholding
rule is used in the case of `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.
"""
if not threshkind 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')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
if show:
tstart = time.time()
print('ISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (threshkind,
Op.shape[0],
Op.shape[1],
eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(Op1.eigs(neigs=1, symmetric=True, niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1./maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
if monitorres:
normresold = np.inf
if returninfo:
cost = np.zeros(niter+1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
res = data - Op.matvec(xinv)
if monitorres:
normres = np.linalg.norm(res)
if normres > normresold:
raise ValueError('ISTA stopped at iteration %d due to '
'residual increasing, consider modifying '
'eps and/or alpha...' % iiter)
else:
normresold = normres
# compute gradient
grad = alpha * Op.rmatvec(res)
# update inverted model
xinv_unthesh = xinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5 * np.linalg.norm(res) ** 2
costreg = eps * np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter - iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
logging.warning('update smaller that tolerance for '
'iteration %d' % iiter)
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
[docs]def FISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, returninfo=False, show=False, threshkind='soft',
perc=None, callback=None):
r"""Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).
Solve an optimization problem with :math:`L1` 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
data : :obj:`numpy.ndarray`
Data
niter : :obj:`int`
Number of iterations
eps : :obj:`float`, optional
Sparsity damping
alpha : :obj:`float`, optional
Step size (:math:`\alpha \le 1/\lambda_{max}(\mathbf{Op}^H\mathbf{Op})`
guarantees convergence. If ``None``, the maximum eigenvalue is
estimated and the optimal step size is chosen. If provided, the
condition will not be checked internally).
eigsiter : :obj:`int`, optional
Number of iterations for eigenvalue estimation if ``alpha=None``
eigstol : :obj:`float`, optional
Tolerance for eigenvalue estimation if ``alpha=None``
tol : :obj:`float`, optional
Tolerance. Stop iterations if difference between inverted model
at subsequent iterations is smaller than ``tol``
returninfo : :obj:`bool`, optional
Return info of FISTA solver
show : :obj:`bool`, optional
Display iterations log
threshkind : :obj:`str`, optional
Kind of thresholding ('hard', 'soft', 'half', '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)
callback : :obj:`callable`, optional
Function with signature (``callback(x)``) to call after each iteration
where ``x`` is the current model vector
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
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 optimization problem for the operator
:math:`\mathbf{Op}` and the data :math:`\mathbf{d}`:
.. math::
J = ||\mathbf{d} - \mathbf{Op} \mathbf{x}||_2^2 +
\epsilon ||\mathbf{x}||_p
using the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [1]_,
where :math:`p=0, 1, 1/2`. 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.
"""
if not threshkind 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')
# choose thresholding function
if threshkind == 'soft':
threshf = _softthreshold
elif threshkind == 'hard':
threshf = _hardthreshold
elif threshkind == 'half':
threshf = _halfthreshold
elif threshkind == 'hard-percentile':
threshf = _hardthreshold_percentile
elif threshkind == 'soft-percentile':
threshf = _softthreshold_percentile
else:
threshf = _halfthreshold_percentile
if show:
tstart = time.time()
print('FISTA optimization (%s thresholding)\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (threshkind,
Op.shape[0],
Op.shape[1],
eps, tol, niter))
# step size
if alpha is None:
if not isinstance(Op, LinearOperator):
Op = LinearOperator(Op, explicit=False)
# compute largest eigenvalues of Op^H * Op
Op1 = LinearOperator(Op.H * Op, explicit=False)
maxeig = np.abs(Op1.eigs(neigs=1, symmetric=True, niter=eigsiter,
**dict(tol=eigstol, which='LM')))[0]
alpha = 1./maxeig
# define threshold
thresh = eps * alpha * 0.5
if show:
if perc is None:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
else:
print('alpha = %10e\tperc = %.1f' % (alpha, perc))
print('-----------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm xupdate'
print(head1)
# initialize model and cost function
xinv = np.zeros(Op.shape[1], dtype=Op.dtype)
zinv = xinv.copy()
t = 1
if returninfo:
cost = np.zeros(niter+1)
# iterate
for iiter in range(niter):
xinvold = xinv.copy()
# compute residual
resz = data - Op.matvec(zinv)
# compute gradient
grad = alpha * Op.rmatvec(resz)
# update inverted model
xinv_unthesh = zinv + grad
if perc is None:
xinv = threshf(xinv_unthesh, thresh)
else:
xinv = threshf(xinv_unthesh, 100 - perc)
# update auxiliary coefficients
told = t
t = (1. + np.sqrt(1. + 4. * t ** 2)) / 2.
zinv = xinv + ((told - 1.) / t) * (xinv - xinvold)
# model update
xupdate = np.linalg.norm(xinv - xinvold)
if returninfo or show:
costdata = 0.5*np.linalg.norm(data - Op.matvec(xinv))**2
costreg = eps*np.linalg.norm(xinv, ord=1)
if returninfo:
cost[iiter] = costdata + costreg
# run callback
if callback is not None:
callback(xinv)
if show:
if iiter < 10 or niter-iiter < 10 or iiter % 10 == 0:
msg = '%6g %12.5e %10.3e %9.3e %10.3e' % \
(iiter+1, xinv[0], costdata, costdata+costreg, xupdate)
print(msg)
# check tolerance
if xupdate < tol:
niter = iiter
break
# get values pre-threshold at locations where xinv is different from zero
# xinv = np.where(xinv != 0, xinv_unthesh, xinv)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (niter, time.time() - tstart))
print('---------------------------------------------------------\n')
if returninfo:
return xinv, niter, cost[:niter]
else:
return xinv, niter
[docs]def SPGL1(Op, data, SOp=None, tau=0, sigma=0, x0=None, **kwargs_spgl1):
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 sparsyfing 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
data : :obj:`numpy.ndarray`
Data
SOp : :obj:`pylops.LinearOperator`
Sparsyfing transform
tau : :obj:`float`
Non-negative LASSO scalar. If different from ``0``,
SPGL1 will solve LASSO problem
sigma : :obj:`list`
BPDN scalar. If different from ``0``,
SPGL1 will solve BPDN problem
x0 : :obj:`numpy.ndarray`
Initial guess
**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``,
``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
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 subj. to \quad
||\mathbf{Op}\mathbf{S}^H\mathbf{x}-\mathbf{b}||_2 <= \sigma,
while the second problem is the *l1-regularized least-squares or LASSO*
problem and its cost function is
.. math::
||\mathbf{Op}\mathbf{S}^H\mathbf{x}-\mathbf{b}||_2 \quad subj.
to \quad ||\mathbf{x}||_1 <= \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.
"""
if spgl1 is None:
raise ModuleNotFoundError(spgl1_message)
pinv, _, _, info = \
spgl1(Op if SOp is None else Op*SOp.H, data,
tau=tau, sigma=sigma, x0=x0, **kwargs_spgl1)
xinv = pinv.copy() if SOp is None else SOp.H * pinv
return xinv, pinv, info
[docs]def SplitBregman(Op, RegsL1, data, niter_outer=3, niter_inner=5, RegsL2=None,
dataregsL2=None, mu=1., epsRL1s=None, epsRL2s=None,
tol=1e-10, tau=1., x0=None, restart=False,
show=False, **kwargs_lsqr):
r"""Split Bregman for mixed L2-L1 norms.
Solve an unconstrained system of equations with mixed L2-L1 regularization
terms given the operator ``Op``, a list of L1 regularization terms
``RegsL1``, and an optional list of L2 regularization terms ``RegsL2``.
Parameters
----------
Op : :obj:`pylops.LinearOperator`
Operator to invert
RegsL1 : :obj:`list`
L1 regularization operators
data : :obj:`numpy.ndarray`
Data
niter_outer : :obj:`int`
Number of iterations of outer loop
niter_inner : :obj:`int`
Number of iterations of inner loop
RegsL2 : :obj:`list`
Additional L2 regularization operators
(if ``None``, L2 regularization is not added to the problem)
dataregsL2 : :obj:`list`, optional
L2 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`
L1 Regularization dampings (must have the same number of elements
as ``RegsL1``)
epsRL2s : :obj:`list`
L2 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)
x0 : :obj:`numpy.ndarray`, optional
Initial guess
restart : :obj:`bool`, optional
The unconstrained inverse problem in inner loop is initialized with
the initial guess (``True``) or with the last estimate (``False``)
show : :obj:`bool`, optional
Display iterations log
**kwargs_lsqr
Arbitrary keyword arguments for
:py:func:`scipy.sparse.linalg.lsqr` solver
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
itn_out : :obj:`int`
Iteration number of outer loop upon termination
Notes
-----
Solve the following system of unconstrained, regularized equations
given the operator :math:`\mathbf{Op}` and a set of mixed norm (L2 and L1)
regularization terms :math:`\mathbf{R_{L2,i}}` and
:math:`\mathbf{R_{L1,i}}`, respectively:
.. math::
J = \mu/2 ||\textbf{d} - \textbf{Op} \textbf{x} |||_2 +
\sum_i \epsilon_{{R}_{L2,i}} ||\mathbf{d_{{R}_{L2,i}}} -
\mathbf{R_{L2,i}} \textbf{x} |||_2 +
\sum_i || \mathbf{R_{L1,i}} \textbf{x} |||_1
where :math:`\mu` and :math:`\epsilon_{{R}_{L2,i}}` are the damping factors
used to weight the different terms of the cost function.
The generalized Split Bergman algorithm is used to solve such cost
function: the algorithm is composed of a sequence of unconstrained
inverse problems and Bregman updates. Note that the L1 terms are not
weighted in the original cost function but are first converted into
constraints and then re-inserted in the cost function with Lagrange
multipliers :math:`\epsilon_{{R}_{L1,i}}`, which effectively act as
damping factors for those terms. See [1]_ for detailed derivation.
The :py:func:`scipy.sparse.linalg.lsqr` solver and a fast shrinkage
algorithm are used within the inner loop to solve the unconstrained
inverse problem, and the same 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.
"""
if show:
tstart = time.time()
print('Split-Bregman optimization\n'
'---------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'niter_outer = %3d niter_inner = %3d tol = %2.2e\n'
'mu = %2.2e epsL1 = %s\t epsL2 = %s '
% (Op.shape[0], Op.shape[1],
niter_outer, niter_inner, tol,
mu, str(epsRL1s), str(epsRL2s)))
print('---------------------------------------------------------\n')
head1 = ' Itn x[0] r2norm r12norm'
print(head1)
# L1 regularizations
nregsL1 = len(RegsL1)
b = [np.zeros(RegL1.shape[0]) for RegL1 in RegsL1]
d = b.copy()
# L2 regularizations
nregsL2 = 0 if RegsL2 is None else len(RegsL2)
if nregsL2 > 0:
Regs = RegsL2 + RegsL1
if dataregsL2 is None:
dataregsL2 = [np.zeros(Op.shape[1])] * nregsL2
else:
Regs = RegsL1
dataregsL2 = []
# Rescale dampings
epsRs = [np.sqrt(epsRL2s[ireg] / 2) / np.sqrt(mu / 2) for ireg in
range(nregsL2)] + \
[np.sqrt(epsRL1s[ireg] / 2) / np.sqrt(mu / 2) for ireg in
range(nregsL1)]
xinv = np.zeros_like(np.zeros(Op.shape[1])) if x0 is None else x0
xold = np.inf * np.ones_like(np.zeros(Op.shape[1]))
itn_out = 0
while np.linalg.norm(xinv - xold) > tol and itn_out < niter_outer:
xold = xinv
for _ in range(niter_inner):
# Regularized problem
dataregs = \
dataregsL2 + [d[ireg] - b[ireg] for ireg in range(nregsL1)]
xinv = RegularizedInversion(Op, Regs, data,
dataregs=dataregs,
epsRs=epsRs,
x0=x0 if restart else xinv,
**kwargs_lsqr)
# Shrinkage
d = [_shrinkage(RegsL1[ireg] * xinv + b[ireg], epsRL1s[ireg])
for ireg in range(nregsL1)]
# Bregman update
b = [b[ireg] + tau * (RegsL1[ireg] * xinv - d[ireg]) for ireg in
range(nregsL1)]
itn_out += 1
if show:
costdata = mu/2. * np.linalg.norm(data - Op.matvec(xinv)) ** 2
costregL2 = 0 if RegsL2 is None else \
[epsRL2 * np.linalg.norm(dataregL2 - RegL2.matvec(xinv)) ** 2
for epsRL2, RegL2, dataregL2 in zip(epsRL2s, RegsL2, dataregsL2)]
costregL1 = [np.linalg.norm(RegL1.matvec(xinv), ord=1)
for epsRL1, RegL1 in zip(epsRL1s, RegsL1)]
cost = costdata + np.sum(np.array(costregL2)) + \
np.sum(np.array(costregL1))
msg = '%6g %12.5e %10.3e %9.3e' % \
(np.abs(itn_out), xinv[0], costdata, cost)
print(msg)
if show:
print('\nIterations = %d Total time (s) = %.2f'
% (itn_out, time.time() - tstart))
print('---------------------------------------------------------\n')
return xinv, itn_out