# -*- coding: utf-8 -*-
# Copyright (C) 2015-2019 by Brendt Wohlberg <brendt@ieee.org>
# All rights reserved. BSD 3-clause License.
# This file is part of the SPORCO package. Details of the copyright
# and user license can be found in the 'LICENSE.txt' file distributed
# with the package.
"""Classes for ADMM algorithm for the BPDN problem"""
from __future__ import division, absolute_import
import copy
import numpy as np
from sporco.admm import admm
import sporco.linalg as sl
import sporco.prox as sp
from sporco.util import u
__author__ = """Brendt Wohlberg <brendt@ieee.org>"""
[docs]
class GenericBPDN(admm.ADMMEqual):
r"""
Base class for ADMM algorithm for solving variants of the
Basis Pursuit DeNoising (BPDN) :cite:`chen-1998-atomic` problem.
|
.. inheritance-diagram:: GenericBPDN
:parts: 2
|
The generic problem form is
.. math::
\mathrm{argmin}_\mathbf{x} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + g(\mathbf{x}) \;\;,
where :math:`g(\cdot)` is a penalty term or the indicator function
of a constraint, and is solved via the ADMM problem
.. math::
\mathrm{argmin}_{\mathbf{x}, \mathbf{y}} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + g(\mathbf{y})
\quad \text{such that} \quad \mathbf{x} = \mathbf{y} \;\;.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value
``DFid`` : Value of data fidelity term
:math:`(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2`
``Reg`` : Value of regularisation term
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
[docs]
class Options(admm.ADMMEqual.Options):
"""GenericBPDN algorithm options
Options include all of those defined in
:class:`.admm.ADMMEqual.Options`, together with
additional options:
``AuxVarObj`` : Flag indicating whether the objective
function should be evaluated using variable X (``False``) or
Y (``True``) as its argument. Setting this flag to ``True``
often gives a better estimate of the objective function.
``LinSolveCheck`` : Flag indicating whether to compute
relative residual of X step solver.
``NonNegCoef`` : If ``True``, force solution to be non-negative.
"""
defaults = copy.deepcopy(admm.ADMMEqual.Options.defaults)
# Warning: although __setitem__ below takes care of setting
# 'fEvalX' and 'gEvalY' from the value of 'AuxVarObj', this
# cannot be relied upon for initialisation since the order of
# initialisation of the dictionary keys is not deterministic;
# if 'AuxVarObj' is initialised first, the other two keys are
# correctly set, but this setting is overwritten when 'fEvalX'
# and 'gEvalY' are themselves initialised
defaults.update({'AuxVarObj': True, 'fEvalX': False,
'gEvalY': True, 'ReturnX': False,
'LinSolveCheck': False, 'RelaxParam': 1.8,
'NonNegCoef': False})
defaults['AutoRho'].update({'Enabled': True, 'Period': 10,
'AutoScaling': True, 'Scaling': 1000.0,
'RsdlRatio': 1.2})
def __init__(self, opt=None):
"""
Parameters
----------
opt : dict or None, optional (default None)
GenericBPDN algorithm options
"""
if opt is None:
opt = {}
admm.ADMMEqual.Options.__init__(self, opt)
def __setitem__(self, key, value):
"""Set options 'fEvalX' and 'gEvalY' appropriately when
option 'AuxVarObj' is set.
"""
admm.ADMMEqual.Options.__setitem__(self, key, value)
if key == 'AuxVarObj':
if value is True:
self['fEvalX'] = False
self['gEvalY'] = True
else:
self['fEvalX'] = True
self['gEvalY'] = False
itstat_fields_objfn = ('ObjFun', 'DFid', 'Reg')
itstat_fields_extra = ('XSlvRelRes',)
hdrtxt_objfn = ('Fnc', 'DFid', 'Reg')
hdrval_objfun = {'Fnc': 'ObjFun', 'DFid': 'DFid', 'Reg': 'Reg'}
def __init__(self, D, S, opt=None):
"""
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (N, K)
Signal vector or matrix
opt : :class:`BPDN.Options` object
Algorithm options
"""
Nc = D.shape[1]
Nm = S.shape[1]
if opt is None:
opt = GenericBPDN.Options()
super(GenericBPDN, self).__init__((Nc, Nm), S.dtype, opt)
self.S = np.asarray(S, dtype=self.dtype)
self.setdict(D)
[docs]
def setdict(self, D):
"""Set dictionary array."""
self.D = np.asarray(D, dtype=self.dtype)
self.DTS = self.D.T.dot(self.S)
# Factorise dictionary for efficient solves
self.lu, self.piv = sl.cho_factor(self.D, self.rho)
self.lu = np.asarray(self.lu, dtype=self.dtype)
[docs]
def getcoef(self):
"""Get final coefficient array."""
return self.Y
[docs]
def xstep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{x}`.
"""
self.X = np.asarray(sl.cho_solve_ATAI(
self.D, self.rho, self.DTS + self.rho * (self.Y - self.U),
self.lu, self.piv), dtype=self.dtype)
if self.opt['LinSolveCheck']:
b = self.DTS + self.rho * (self.Y - self.U)
ax = self.D.T.dot(self.D.dot(self.X)) + self.rho*self.X
self.xrrs = sl.rrs(ax, b)
else:
self.xrrs = None
[docs]
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`. If this method is not overridden, the
problem is solved without any regularisation other than the
option enforcement of non-negativity of the solution. When it
is overridden, it should be explicitly called at the end of
the overriding method.
"""
if self.opt['NonNegCoef']:
self.Y[self.Y < 0.0] = 0.0
[docs]
def eval_objfn(self):
"""Compute components of objective function as well as total
contribution to objective function.
"""
dfd = self.obfn_dfd()
reg = self.obfn_reg()
obj = dfd + reg[0]
return (obj, dfd) + reg[1:]
[docs]
def obfn_dfd(self):
r"""Compute data fidelity term :math:`(1/2) \| D \mathbf{x} -
\mathbf{s} \|_2^2`.
"""
return 0.5*np.linalg.norm((self.D.dot(self.obfn_fvar()) - self.S))**2
[docs]
def obfn_reg(self):
"""Compute regularisation term(s) and contribution to objective
function.
"""
raise NotImplementedError()
[docs]
def rhochange(self):
"""Re-factorise matrix when rho changes."""
self.lu, self.piv = sl.cho_factor(self.D, self.rho)
self.lu = np.asarray(self.lu, dtype=self.dtype)
[docs]
class BPDN(GenericBPDN):
r"""
ADMM algorithm for the Basis Pursuit DeNoising (BPDN)
:cite:`chen-1998-atomic` problem.
|
.. inheritance-diagram:: BPDN
:parts: 2
|
Solve the Single Measurement Vector (SMV) BPDN problem
.. math::
\mathrm{argmin}_\mathbf{x} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + \lambda \| \mathbf{x}
\|_1
via the ADMM problem
.. math::
\mathrm{argmin}_{\mathbf{x}, \mathbf{y}} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + \lambda \| \mathbf{y}
\|_1 \quad \text{such that} \quad \mathbf{x} = \mathbf{y} \;\;.
The Multiple Measurement Vector (MMV) BPDN problem
.. math::
\mathrm{argmin}_X \;
(1/2) \| D X - S \|_F^2 + \lambda \| X \|_1
is also supported.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value
``DFid`` : Value of data fidelity term :math:`(1/2) \| D
\mathbf{x} - \mathbf{s} \|_2^2`
``RegL1`` : Value of regularisation term :math:`\| \mathbf{x}
\|_1`
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
[docs]
class Options(GenericBPDN.Options):
r"""BPDN algorithm options
Options include all of those defined in
:class:`.GenericBPDN.Options`, together with additional
options:
``L1Weight`` : An array of weights for the :math:`\ell_1`
norm. The array shape must be such that the array is
compatible for multiplication with the X/Y variables. If this
option is defined, the regularization term is :math:`\lambda
\| \mathbf{w} \odot \mathbf{x} \|_1` where :math:`\mathbf{w}`
denotes the weighting array.
"""
defaults = copy.deepcopy(GenericBPDN.Options.defaults)
defaults.update({'L1Weight': 1.0})
def __init__(self, opt=None):
"""
Parameters
----------
opt : dict or None, optional (default None)
BPDN algorithm options
"""
if opt is None:
opt = {}
GenericBPDN.Options.__init__(self, opt)
itstat_fields_objfn = ('ObjFun', 'DFid', 'RegL1')
hdrtxt_objfn = ('Fnc', 'DFid', u('Regℓ1'))
hdrval_objfun = {'Fnc': 'ObjFun', 'DFid': 'DFid', u('Regℓ1'): 'RegL1'}
def __init__(self, D, S, lmbda=None, opt=None):
"""
|
**Call graph**
.. image:: ../_static/jonga/bpdn_init.svg
:width: 20%
:target: ../_static/jonga/bpdn_init.svg
|
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (N, K)
Signal vector or matrix
lmbda : float
Regularisation parameter
opt : :class:`BPDN.Options` object
Algorithm options
"""
# Set default options if necessary
if opt is None:
opt = BPDN.Options()
# Set dtype attribute based on S.dtype and opt['DataType']
self.set_dtype(opt, S.dtype)
# Set default lambda value if not specified
if lmbda is None:
DTS = D.T.dot(S)
lmbda = 0.1 * abs(DTS).max()
# Set l1 term scaling and weight array
self.lmbda = self.dtype.type(lmbda)
self.wl1 = np.asarray(opt['L1Weight'], dtype=self.dtype)
# Set penalty parameter
self.set_attr('rho', opt['rho'], dval=(50.0*self.lmbda + 1.0),
dtype=self.dtype)
# Set rho_xi attribute (see Sec. VI.C of wohlberg-2015-adaptive)
if self.lmbda != 0.0:
rho_xi = float((1.0 + (18.3)**(np.log10(self.lmbda) + 1.0)))
else:
rho_xi = 1.0
self.set_attr('rho_xi', opt['AutoRho', 'RsdlTarget'], dval=rho_xi,
dtype=self.dtype)
super(BPDN, self).__init__(D, S, opt)
[docs]
def uinit(self, ushape):
"""Return initialiser for working variable U"""
if self.opt['Y0'] is None:
return np.zeros(ushape, dtype=self.dtype)
else:
# If initial Y is non-zero, initial U is chosen so that
# the relevant dual optimality criterion (see (3.10) in
# boyd-2010-distributed) is satisfied.
return (self.lmbda / self.rho) * np.sign(self.Y)
[docs]
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
self.Y = np.asarray(sp.prox_l1(self.AX + self.U,
(self.lmbda / self.rho) * self.wl1),
dtype=self.dtype)
super(BPDN, self).ystep()
[docs]
def obfn_reg(self):
"""Compute regularisation term and contribution to objective
function.
"""
rl1 = np.linalg.norm((self.wl1 * self.obfn_gvar()).ravel(), 1)
return (self.lmbda*rl1, rl1)
[docs]
class BPDNJoint(BPDN):
r"""
ADMM algorithm for BPDN with joint sparsity via an :math:`\ell_{2,1}`
norm term.
|
.. inheritance-diagram:: BPDNJoint
:parts: 2
|
Solve the optimisation problem
.. math::
\mathrm{argmin}_X \; (1/2) \| D X - S \|_2^2 + \lambda \| X \|_1
+ \mu \| X \|_{2,1}
via the ADMM problem
.. math::
\mathrm{argmin}_{X, Y} \; (1/2) \| D X - S \|_2^2 +
\lambda \| Y \|_1 + \mu \| Y \|_{2,1} \quad \text{such that} \quad
X = Y \;\;.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value
``DFid`` : Value of data fidelity term :math:`(1/2) \| D X - S
\|_2^2`
``RegL1`` : Value of regularisation term :math:`\| X \|_1`
``RegL21`` : Value of regularisation term :math:`\| X \|_{2,1}`
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual Residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
itstat_fields_objfn = ('ObjFun', 'DFid', 'RegL1', 'RegL21')
hdrtxt_objfn = ('Fnc', 'DFid', u('Regℓ1'), u('Regℓ2,1'))
hdrval_objfun = {'Fnc': 'ObjFun', 'DFid': 'DFid',
u('Regℓ1'): 'RegL1', u('Regℓ2,1'): 'RegL21'}
def __init__(self, D, S, lmbda=None, mu=0.0, opt=None):
"""
|
**Call graph**
.. image:: ../_static/jonga/bpdnjnt_init.svg
:width: 20%
:target: ../_static/jonga/bpdnjnt_init.svg
|
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (M, K)
Signal vector or matrix
lmbda : float
Regularisation parameter (l1)
mu : float
Regularisation parameter (l2,1)
opt : :class:`BPDN.Options` object
Algorithm options
"""
if opt is None:
opt = BPDN.Options()
super(BPDNJoint, self).__init__(D, S, lmbda, opt)
self.mu = self.dtype.type(mu)
[docs]
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
self.Y = np.asarray(sp.prox_sl1l2(
self.AX + self.U, (self.lmbda / self.rho) * self.wl1,
self.mu / self.rho, axis=-1), dtype=self.dtype)
GenericBPDN.ystep(self)
[docs]
def obfn_reg(self):
r"""Compute regularisation terms and contribution to objective
function. Regularisation terms are :math:`\| Y \|_1` and
:math:`\| Y \|_{2,1}`.
"""
rl1 = np.linalg.norm((self.wl1 * self.obfn_gvar()).ravel(), 1)
rl21 = np.sum(np.sqrt(np.sum(self.obfn_gvar()**2, axis=1)))
return (self.lmbda*rl1 + self.mu*rl21, rl1, rl21)
[docs]
class ElasticNet(BPDN):
r"""
ADMM algorithm for the elastic net :cite:`zou-2005-regularization`
problem.
|
.. inheritance-diagram:: ElasticNet
:parts: 2
|
Solve the optimisation problem
.. math::
\mathrm{argmin}_\mathbf{x} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + \lambda \| \mathbf{x}
\|_1 + (\mu/2) \| \mathbf{x} \|_2^2
via the ADMM problem
.. math::
\mathrm{argmin}_{\mathbf{x}, \mathbf{y}} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + \lambda \| \mathbf{y}
\|_1 + (\mu/2) \| \mathbf{x} \|_2^2 \quad \text{such that} \quad
\mathbf{x} = \mathbf{y} \;\;.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value
``DFid`` : Value of data fidelity term :math:`(1/2) \| D
\mathbf{x} - \mathbf{s} \|_2^2`
``RegL1`` : Value of regularisation term :math:`\| \mathbf{x}
\|_1`
``RegL2`` : Value of regularisation term :math:`(1/2) \|
\mathbf{x} \|_2^2`
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual Residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
itstat_fields_objfn = ('ObjFun', 'DFid', 'RegL1', 'RegL2')
hdrtxt_objfn = ('Fnc', 'DFid', u('Regℓ1'), u('Regℓ2'))
hdrval_objfun = {'Fnc': 'ObjFun', 'DFid': 'DFid',
u('Regℓ1'): 'RegL1', u('Regℓ2'): 'RegL2'}
def __init__(self, D, S, lmbda=None, mu=0.0, opt=None):
"""
|
**Call graph**
.. image:: ../_static/jonga/elnet_init.svg
:width: 20%
:target: ../_static/jonga/elnet_init.svg
|
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (M, K)
Signal vector or matrix
lmbda : float
Regularisation parameter (l1)
mu : float
Regularisation parameter (l2)
opt : :class:`BPDN.Options` object
Algorithm options
"""
if opt is None:
opt = BPDN.Options()
# Set dtype attribute based on S.dtype and opt['DataType']
self.set_dtype(opt, S.dtype)
self.mu = self.dtype.type(mu)
super(ElasticNet, self).__init__(D, S, lmbda, opt)
[docs]
def setdict(self, D):
"""Set dictionary array."""
self.D = np.asarray(D)
self.DTS = self.D.T.dot(self.S)
# Factorise dictionary for efficient solves
self.lu, self.piv = sl.cho_factor(self.D, self.mu + self.rho)
self.lu = np.asarray(self.lu, dtype=self.dtype)
[docs]
def xstep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{x}`.
"""
self.X = np.asarray(sl.cho_solve_ATAI(
self.D, self.mu + self.rho, self.DTS +
self.rho * (self.Y - self.U),
self.lu, self.piv), dtype=self.dtype)
if self.opt['LinSolveCheck']:
b = self.DTS + self.rho * (self.Y - self.U)
ax = self.D.T.dot(self.D.dot(self.X)) + (self.mu+self.rho)*self.X
self.xrrs = sl.rrs(ax, b)
else:
self.xrrs = None
[docs]
def obfn_reg(self):
"""Compute regularisation term and contribution to objective
function.
"""
rl1 = np.linalg.norm((self.wl1 * self.obfn_gvar()).ravel(), 1)
rl2 = 0.5 * np.linalg.norm(self.obfn_gvar())**2
return (self.lmbda*rl1 + self.mu*rl2, rl1, rl2)
[docs]
def rhochange(self):
"""Re-factorise matrix when rho changes."""
self.lu, self.piv = sl.cho_factor(self.D, self.mu + self.rho)
self.lu = np.asarray(self.lu, dtype=self.dtype)
[docs]
class BPDNProjL1(GenericBPDN):
r"""
ADMM algorithm for a BPDN variant with projection onto the
:math:`\ell_1` ball instead of an :math:`\ell_1` penalty.
|
.. inheritance-diagram:: BPDNProjL1
:parts: 2
|
This variant of the BPDN problem was originally referred to as the
lasso :cite:`tibshirani-1996-regression`, but that name is now also
frequently applied to the penalised form that is referred to here as
the BPDN problem.
Solve the problem
.. math::
\mathrm{argmin}_\mathbf{x} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 \; \text{such that} \;
\| \mathbf{x} \|_1 \leq \gamma
via the ADMM problem
.. math::
\mathrm{argmin}_{\mathbf{x}, \mathbf{y}} \;
(1/2) \| D \mathbf{x} - \mathbf{s} \|_2^2 + \iota_{C(\gamma)}
(\mathbf{y}) \quad \text{such that} \quad \mathbf{x} = \mathbf{y}
\;\;,
where :math:`\iota_{C(\gamma)}(\cdot)` is the indicator function
of the :math:`\ell_1` ball of radius :math:`\gamma` about the origin.
The algorithm is very similar to that for the BPDN problem (see
:class:`BPDN`), the only difference being in the replacement in the
:math:`\mathbf{y}` step of the proximal operator of the :math:`\ell_1`
norm with the projection operator of the :math:`\ell_1` norm.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value :math:`(1/2) \| D
\mathbf{x} - \mathbf{s} \|_2^2`
``Cnstr`` : Constraint violation measure
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
[docs]
class Options(GenericBPDN.Options):
"""BPDNProjL1 algorithm options
Options are the same as those defined in
:class:`.GenericBPDN.Options`.
"""
defaults = copy.deepcopy(GenericBPDN.Options.defaults)
defaults['AutoRho'].update({'RsdlTarget': 1.0})
def __init__(self, opt=None):
"""
Parameters
----------
opt : dict or None, optional (default None)
BPDNProjL1 algorithm options
"""
if opt is None:
opt = {}
GenericBPDN.Options.__init__(self, opt)
itstat_fields_objfn = ('ObjFun', 'Cnstr')
hdrtxt_objfn = ('Fnc', 'Cnstr')
hdrval_objfun = {'Fnc': 'ObjFun', 'Cnstr': 'Cnstr'}
def __init__(self, D, S, gamma, opt=None):
"""
|
**Call graph**
.. image:: ../_static/jonga/bpdnprjl1_init.svg
:width: 20%
:target: ../_static/jonga/bpdnprjl1_init.svg
|
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (N, K)
Signal vector or matrix
gamma : float
Constraint parameter
opt : :class:`BPDNProjL1.Options` object
Algorithm options
"""
# Set default options if necessary
if opt is None:
opt = BPDNProjL1.Options()
super(BPDNProjL1, self).__init__(D, S, opt)
self.gamma = self.dtype.type(gamma)
[docs]
def uinit(self, ushape):
"""Return initialiser for working variable U."""
if self.opt['Y0'] is None:
return np.zeros(ushape, dtype=self.dtype)
else:
# If initial Y is non-zero, initial U is chosen so that
# the relevant dual optimality criterion (see (3.10) in
# boyd-2010-distributed) is satisfied.
# NB: still needs to be worked out.
return np.zeros(ushape, dtype=self.dtype)
[docs]
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y = np.asarray(sp.proj_l1(self.AX + self.U, self.gamma, axis=0),
dtype=self.dtype)
super(BPDNProjL1, self).ystep()
[docs]
def eval_objfn(self):
"""Compute components of regularisation function as well as total
contribution to objective function.
"""
dfd = self.obfn_dfd()
prj = sp.proj_l1(self.obfn_gvar(), self.gamma, axis=0)
cns = np.linalg.norm(prj - self.obfn_gvar())
return (dfd, cns)
[docs]
class MinL1InL2Ball(admm.ADMMTwoBlockCnstrnt):
r"""
ADMM algorithm for the problem with an :math:`\ell_1` objective and
an :math:`\ell_2` constraint.
|
.. inheritance-diagram:: MinL1InL2Ball
:parts: 2
|
The solution is computed following the approach proposed in
:cite:`afonso-2011-augmented`.
Solve the Single Measurement Vector (SMV) problem
.. math::
\mathrm{argmin}_\mathbf{x} \| \mathbf{x} \|_1 \; \text{such that}
\; \| D \mathbf{x} - \mathbf{s} \|_2 \leq \epsilon
via the ADMM problem
.. math::
\mathrm{argmin}_{\mathbf{x},\mathbf{y}_0,\mathbf{y}_1} \;
\| \mathbf{y}_0 \|_1 + \iota_{C(\mathbf{s}, \epsilon)}
(\mathbf{y}_1) \;\text{such that}\;
\left( \begin{array}{c} I \\ D \end{array} \right) \mathbf{x}
- \left( \begin{array}{c} \mathbf{y}_0 \\ \mathbf{y}_1
\end{array} \right) = \left( \begin{array}{c} \mathbf{0} \\
\mathbf{0} \end{array} \right) \;\;,
where :math:`\iota_{C(\mathbf{s}, \epsilon)}(\cdot)` is the indicator
function of the :math:`\ell_2` ball of radius :math:`\epsilon` about
:math:`\mathbf{s}`. The Multiple Measurement Vector (MMV) problem
.. math::
\mathrm{argmin}_X \| X \|_1 \; \text{such that} \;
\| [D X - S]_k \|_2 \leq \epsilon \;\;\; \forall k \;\;,
where :math:`[X]_k` denotes column :math:`k` of matrix :math:`X`,
is also supported.
After termination of the :meth:`solve` method, attribute
:attr:`itstat` is a list of tuples representing statistics of each
iteration. The fields of the named tuple ``IterationStats`` are:
``Iter`` : Iteration number
``ObjFun`` : Objective function value :math:`\| \mathbf{x} \|_1`
``Cnstr`` : Constraint violation measure
``PrimalRsdl`` : Norm of primal residual
``DualRsdl`` : Norm of dual residual
``EpsPrimal`` : Primal residual stopping tolerance
:math:`\epsilon_{\mathrm{pri}}`
``EpsDual`` : Dual residual stopping tolerance
:math:`\epsilon_{\mathrm{dua}}`
``Rho`` : Penalty parameter
``Time`` : Cumulative run time
"""
[docs]
class Options(admm.ADMMTwoBlockCnstrnt.Options):
r"""MinL1InL2Ball algorithm options
Options include all of those defined in
:class:`.admm.ADMMTwoBlockCnstrnt.Options`, together
with additional options:
``L1Weight`` : An array of weights for the :math:`\ell_1`
norm. The array shape must be such that the array is
compatible for multiplication with the X/Y variables. If this
option is defined, the objective function is :math:`\lambda \|
\mathbf{w} \odot \mathbf{x} \|_1` where :math:`\mathbf{w}`
denotes the weighting array.
``NonNegCoef`` : If ``True``, force solution to be non-negative.
"""
defaults = copy.deepcopy(admm.ADMMTwoBlockCnstrnt.Options.defaults)
defaults.update({'AuxVarObj': False, 'fEvalX': True,
'gEvalY': False, 'RelaxParam': 1.8,
'L1Weight': 1.0, 'NonNegCoef': False,
'ReturnVar': 'X'})
defaults['AutoRho'].update({'Enabled': True, 'Period': 10,
'AutoScaling': True, 'Scaling': 1000.0,
'RsdlRatio': 1.2, 'RsdlTarget': 1.0})
def __init__(self, opt=None):
"""
Parameters
----------
opt : dict or None, optional (default None)
MinL1InL2Ball algorithm options
"""
if opt is None:
opt = {}
admm.ADMMTwoBlockCnstrnt.Options.__init__(self, opt)
itstat_fields_objfn = ('ObjFun', 'Cnstr')
hdrtxt_objfn = ('Fnc', 'Cnstr')
hdrval_objfun = {'Fnc': 'ObjFun', 'Cnstr': 'Cnstr'}
def __init__(self, D, S, epsilon, opt=None):
r"""
|
**Call graph**
.. image:: ../_static/jonga/bpdnml1l2_init.svg
:width: 20%
:target: ../_static/jonga/bpdnml1l2_init.svg
|
Parameters
----------
D : array_like, shape (N, M)
Dictionary matrix
S : array_like, shape (N, K)
Signal vector or matrix
epsilon : float
:math:`\ell_2` ball radius
opt : :class:`MinL1InL2Ball.Options` object
Algorithm options
"""
Nr, Nc = D.shape
Nm = S.shape[1]
if opt is None:
opt = MinL1InL2Ball.Options()
super(MinL1InL2Ball, self).__init__(Nc * Nm, (Nc + Nr, Nm), 0, Nc,
S.dtype, opt)
# Record epsilon value and l1 weight array
self.epsilon = self.dtype.type(epsilon)
self.wl1 = np.asarray(opt['L1Weight'], dtype=self.dtype)
self.S = np.asarray(S, dtype=self.dtype)
self.setdict(D)
[docs]
def uinit(self, ushape):
"""Return initialiser for working variable U."""
if self.opt['Y0'] is None:
return np.zeros(ushape, dtype=self.dtype)
else:
# If initial Y is non-zero, initial U is chosen so that
# the relevant dual optimality criterion (see (3.10) in
# boyd-2010-distributed) is satisfied.
U0 = np.sign(self.block_sep0(self.Y)) / self.rho
U1 = self.block_sep1(self.Y) - self.S
return self.block_cat(U0, U1)
[docs]
def setdict(self, D):
"""Set dictionary array."""
self.D = np.asarray(D, dtype=self.dtype)
# Factorise dictionary for efficient solves
self.lu, self.piv = sl.cho_factor(self.D, 1.0)
self.lu = np.asarray(self.lu, dtype=self.dtype)
[docs]
def getcoef(self):
"""Get final coefficient array."""
return self.X
[docs]
def xstep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{x}`.
"""
YU = self.Y - self.U
self.X = np.asarray(sl.cho_solve_ATAI(
self.D, 1.0, self.block_sep0(YU) +
self.D.T.dot(self.block_sep1(YU)), self.lu, self.piv),
dtype=self.dtype)
[docs]
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
AXU = self.AX + self.U
Y0 = np.asarray(sp.prox_l1(self.block_sep0(AXU), self.wl1 / self.rho),
dtype=self.dtype)
if self.opt['NonNegCoef']:
Y0[Y0 < 0.0] = 0.0
Y1 = self.S + sp.proj_l2(self.block_sep1(AXU) - self.S, self.epsilon,
axis=0)
self.Y = self.block_cat(Y0, Y1)
[docs]
def cnst_A1(self, X):
r"""Compute :math:`A_1 \mathbf{x}` component of ADMM problem
constraint."""
return self.D.dot(X)
[docs]
def cnst_A1T(self, X):
r"""Compute :math:`A_1^T \mathbf{x}` where :math:`A_1 \mathbf{x}`
is a component of ADMM problem constraint."""
return self.D.T.dot(X)
[docs]
def eval_objfn(self):
r"""Compute components of objective function as well as total
contribution to objective function. The objective function is
:math:`\| \mathbf{x} \|_1` and the constraint violation
measure is :math:`P(\mathbf{x}) - \mathbf{x}` where
:math:`P(\mathbf{x})` is the projection into the constraint
set.
"""
obj = np.linalg.norm((self.wl1 * self.obfn_g0var()).ravel(), 1)
cns = np.linalg.norm(self.S + sp.proj_l2(self.obfn_g1var() - self.S,
self.epsilon, axis=0) -
self.obfn_g1var())
return (obj, cns)
[docs]
def rsdl_s(self, Yprev, Y):
"""Compute dual residual vector."""
return self.rho * np.linalg.norm(self.cnst_AT(self.U))
[docs]
def rsdl_sn(self, U):
"""Compute dual residual normalisation term."""
return self.rho * np.linalg.norm(U)