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
def FISTA(Op,
data,
niter,
eps=0.1,
alpha=None,
eigsiter=None,
eigstol=0,
tol=1e-10,
returninfo=False,
show=False):
r"""Fast Iterative Soft 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``, estimated to satisfy the
condition, otherwise the condition will not be checked)
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
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
See Also
--------
OMP: Orthogonal Matching Pursuit (OMP).
ISTA: Iterative Soft 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}||_1
using the Fast Iterative Soft Thresholding Algorithm (FISTA) [1]_. This is
a modified version of ISTA solver with improved convergence properties and
limitied additional computational cost.
.. [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 show:
tstart = time.time()
print('FISTA optimization\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' %
(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:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
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
xinv = _softthreshold(xinv_unthesh, thresh)
# 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
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
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
def ISTA(Op, data, niter, eps=0.1, alpha=None, eigsiter=None, eigstol=0,
tol=1e-10, monitorres=False, returninfo=False, show=False):
r"""Iterative Soft Thresholding Algorithm (ISTA).
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``, estimated to satisfy the
condition, otherwise the condition will not be checked)
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
Returns
-------
xinv : :obj:`numpy.ndarray`
Inverted model
niter : :obj:`int`
Number of effective iterations
cost : :obj:`numpy.ndarray`, optional
History of cost function
Raises
------
ValueError
If ``monitorres=True`` and residual increases
See Also
--------
FISTA: Fast Iterative Soft Thresholding Algorithm (FISTA).
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}||_1
using the Iterative Soft Thresholding Algorithm (ISTA) [1]_. This is a very
simple iterative algorithm which applies the following step:
.. math::
\mathbf{x}^{(i+1)} = soft (\mathbf{x}^{(i)} + \alpha \mathbf{Op}^H
(\mathbf{d} - \mathbf{Op} \mathbf{x}^{(i)})), \epsilon \alpha /2)
where :math:`\epsilon \alpha /2` is the
threshold and :math:`soft()` is the so-called soft-thresholding rule.
.. [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 show:
tstart = time.time()
print('ISTA optimization\n'
'-----------------------------------------------------------\n'
'The Operator Op has %d rows and %d cols\n'
'eps = %10e\ttol = %10e\tniter = %d' % (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:
print('alpha = %10e\tthresh = %10e' % (alpha, thresh))
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 modyfing '
'eps and/or alpha...' % iiter)
else:
normresold = normres
# compute gradient
grad = alpha*Op.rmatvec(res)
# update inverted model
xinv_unthesh = xinv + grad
xinv = _softthreshold(xinv_unthesh, thresh)
# 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
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