# -*- coding: utf-8 -*-
# Copyright (C) 2020 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.
"""Dictionary learning based on weighted BPDN sparse coding"""
from __future__ import print_function, absolute_import
import copy
import numpy as np
from sporco.util import u
from sporco.array import atleast_nd
from sporco.pgm import bpdn
from sporco.pgm import cmod
from sporco.dictlrn import dictlrn
__author__ = """Brendt Wohlberg <brendt@ieee.org>"""
[docs]class WeightedBPDNDictLearn(dictlrn.DictLearn):
r"""
Dictionary learning based on weighted BPDN and CnstrMOD
|
.. inheritance-diagram:: WeightedBPDNDictLearn
:parts: 2
|
Solve the optimisation problem
.. math::
\mathrm{argmin}_{D, X} \; (1/2) \| D X - S \|_W^2 + \lambda \|
X \|_1 \quad \text{such that} \quad \|\mathbf{d}_m\|_2 = 1
via interleaved alternation between the PGM steps of the
:class:`.pgm.bpdn.WeightedBPDN` and
:class:`.pgm.cmod.WeightedCnstrMOD` problems. Note that
:math:`\| \cdot \|_W` denotes a non-standard definition of the
weighted Frobenius norm defined as
.. math::
\| X \|_W^2 = \| W^{1/2} \odot X \|_F
so that
.. math::
\| X \|_W^2 = \sum_i \| W_i^{1/2} \mathbf{x}_i \|_2^2 =
\sum_i \| \mathbf{x}_i \|_{W_i}^2 \;,
where :math:`\mathbf{x}_i` and :math:`\mathbf{w}_i` are the
:math:`i^{\text{th}}` columns of :math:`X` and :math:`W`
respectively, and :math:`W_i = \mathrm{diag}(\mathbf{w}_i)`.
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 \|_W^2`
``RegL1`` : Value of regularisation term :math:`\| X \|_1`
``Cnstr`` : Constraint violation measure
``XRsdl`` : Norm of X residual
``XL`` : X inverse of gradient step parameter
``DRsdl`` : Norm of D residual
``DL`` : D inverse of gradient step parameter
``Time`` : Cumulative run time
"""
[docs] class Options(dictlrn.DictLearn.Options):
"""BPDN dictionary learning algorithm options.
Options include all of those defined in
:class:`sporco.dictlrn.dictlrn.DictLearn.Options`, together with
additional options:
``AccurateDFid`` : Flag determining whether data fidelity term is
estimated from the value computed in the X update (``False``) or
is computed after every outer iteration over an X update and a D
update (``True``), which is slower but more accurate.
``BPDN`` : Options :class:`sporco.pgm.bpdn.WeightedBPDN.Options`
``CMOD`` : Options :class:`sporco.pgm.cmod.WeightedCnstrMOD.Options`
"""
defaults = copy.deepcopy(dictlrn.DictLearn.Options.defaults)
defaults.update(
{'AccurateDFid': False,
'BPDN': copy.deepcopy(bpdn.WeightedBPDN.Options.defaults),
'CMOD': copy.deepcopy(cmod.WeightedCnstrMOD.Options.defaults)})
def __init__(self, opt=None):
"""
Parameters
----------
opt : dict or None, optional (default None)
WeightedBPDNDictLearn algorithm options
"""
dictlrn.DictLearn.Options.__init__(
self, {'BPDN': bpdn.WeightedBPDN.Options(
{'MaxMainIter': 1}),
'CMOD': cmod.WeightedCnstrMOD.Options(
{'MaxMainIter': 1})
})
if opt is None:
opt = {}
self.update(opt)
def __init__(self, D0, S, lmbda=None, W=None, opt=None):
"""
Parameters
----------
D0 : array_like, shape (N, M)
Initial dictionary matrix
S : array_like, shape (N, K)
Signal vector or matrix
lmbda : float
Regularisation parameter
W : array_like, shape (N, K)
Weight matrix
opt : :class:`WeightedBPDNDictLearn.Options` object
Algorithm options
"""
if opt is None:
opt = WeightedBPDNDictLearn.Options()
self.opt = opt
# Normalise dictionary according to D update options
D0 = cmod.getPcn(opt['CMOD', 'ZeroMean'],
opt['CMOD', 'NonNegCoef'])(D0)
# Modify D update options to include initial values for Y and U
Nc = D0.shape[1]
opt['CMOD'].update({'X0': D0})
# Create X update object
xstep = bpdn.WeightedBPDN(D0, S, lmbda, W=W, opt=opt['BPDN'])
# Create D update object
Nm = S.shape[1]
dstep = cmod.WeightedCnstrMOD(xstep.Y, S, W=W, dsz=(Nc, Nm),
opt=opt['CMOD'])
if W is None:
W = np.array([1.0], dtype=xstep.dtype)
if W.ndim > 0:
W = atleast_nd(2, W)
self.W = np.asarray(W, dtype=xstep.dtype)
# Configure iteration statistics reporting
if self.opt['AccurateDFid']:
isxmap = {'XRsdl': 'Rsdl', 'XL': 'L'}
evlmap = {'ObjFun': 'ObjFun', 'DFid': 'DFid', 'RegL1': 'RegL1'}
else:
isxmap = {'ObjFun': 'ObjFun', 'DFid': 'DFid', 'RegL1': 'RegL1',
'XRsdl': 'Rsdl', 'XL': 'L'}
evlmap = {}
isc = dictlrn.IterStatsConfig(
isfld=['Iter', 'ObjFun', 'DFid', 'RegL1', 'Cnstr', 'XRsdl',
'XL', 'DRsdl', 'DL', 'Time'],
isxmap=isxmap,
isdmap={'Cnstr': 'Cnstr', 'DRsdl': 'Rsdl', 'DL': 'L'},
evlmap=evlmap,
hdrtxt=['Itn', 'Fnc', 'DFid', u('ℓ1'), 'Cnstr', 'X_Rsdl',
'X_L', 'D_Rsdl', 'D_L'],
hdrmap={'Itn': 'Iter', 'Fnc': 'ObjFun', 'DFid': 'DFid',
u('ℓ1'): 'RegL1', 'Cnstr': 'Cnstr', 'X_Rsdl': 'XRsdl',
'X_L': 'XL', 'D_Rsdl': 'DRsdl', 'D_L': 'DL'}
)
# Call parent constructor
super(WeightedBPDNDictLearn, self).__init__(xstep, dstep, opt, isc)
[docs] def evaluate(self):
"""Evaluate functional value of previous iteration"""
if self.opt['AccurateDFid']:
D = self.dstep.var_x()
X = self.xstep.var_x()
S = self.xstep.S
dfd = 0.5*np.linalg.norm(np.sqrt(self.W) * (D.dot(X) - S))**2
rl1 = np.sum(np.abs(X))
return dict(DFid=dfd, RegL1=rl1, ObjFun=dfd+self.xstep.lmbda*rl1)
else:
return None
