def par_y1step(i):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}_{1,G_i}`, one of the disjoint problems of
optimizing :math:`\mathbf{y}_1`.
Parameters
----------
i : int
Index of grouping to update
"""
global mp_Y1
grpind = slice(mp_grp[i], mp_grp[i+1])
XU1 = mp_X[grpind] + 1/mp_alpha*mp_U1[grpind]
if mp_wl1.shape[mp_axisM] is 1:
gamma = mp_lmbda/(mp_alpha**2*mp_rho)*mp_wl1
else:
gamma = mp_lmbda/(mp_alpha**2*mp_rho)*mp_wl1[grpind]
Y1 = sp.prox_l1(XU1, gamma)
if mp_NonNegCoef:
Y1[Y1 < 0.0] = 0.0
if mp_NoBndryCross:
for n in range(len(mp_Nv)):
Y1[(slice(None),) + (slice(None),)*n +
(slice(1-mp_Dshp[n], None),)] = 0.0
mp_Y1[mp_grp[i]:mp_grp[i+1]] = Y1
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y = np.asarray(sp.prox_l1(self.S - self.AX - self.U,
self.lmbda/self.rho), dtype=self.dtype)
def cbpdn_ystep(k):
"""Do the Y step of the cbpdn stage. The only parameter is the slice
index `k` and there are no return values; all inputs and outputs are
from and to global variables.
"""
AXU = mp_Z_X[k] + mp_Z_U[k]
mp_Z_Y[k] = sp.prox_l1(AXU, (mp_lmbda/mp_xrho))
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()
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
AXU = self.AX + self.U
self.Y[..., 0:-1] = sp.prox_l2(AXU[..., 0:-1], self.mu/self.rho)
self.Y[..., -1] = sp.prox_l1(AXU[..., -1],
(self.lmbda/self.rho) * self.Wl1)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
AXU = self.AX + self.U
self.block_sep0(self.Y)[:] = sp.prox_l1(
self.block_sep0(AXU), (self.lmbda/self.rho) * self.Wl1)
self.block_sep1(self.Y)[:] = sp.prox_l2(
self.block_sep1(AXU), self.mu/self.rho, axis=(self.cri.axisC, -1))
def resolvent_d(d, y, s, rho, N, M, d_size):
rootN = int(np.sqrt(N))
d = d.reshape(M, rootN, rootN)
d = d.transpose(1, 2, 0)
d_Pcn = cnvrep.Pcn(d.reshape(rootN, rootN, 1, 1, M), (d_size, d_size, M),
Nv=(rootN, rootN)).squeeze()
d_Pcn = d_Pcn.transpose(2, 0, 1)
d = d_Pcn.reshape(N * M)
y = y - (pr.prox_l1(y - s, 1 / rho) + s)
return np.concatenate([d, y], 0)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y[..., 0:-1] = prox_l2(self.AX[..., 0:-1] + self.U[..., 0:-1],
(self.lmbda / self.rho) * self.Wtvna,
axis=self.saxes)
self.Y[..., -1] = prox_l1(self.AX[..., -1] + self.U[..., -1] - self.S,
(1.0 / self.rho) * self.Wdf)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
AXU = self.AX + self.U
self.Y[..., 0:-1] = sp.prox_l2(AXU[..., 0:-1],
self.mu / self.rho,
axis=(self.cri.axisM, -1))
self.Y[..., -1] = sp.prox_l1(AXU[..., -1],
(self.lmbda / self.rho) * self.Wl1)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
AXU = self.AX + self.U
self.block_sep0(self.Y)[:] = sp.prox_l1(
self.block_sep0(AXU), (self.lmbda / self.rho) * self.Wl1)
self.block_sep1(self.Y)[:] = sp.prox_l2(self.block_sep1(AXU),
self.mu / self.rho,
axis=(self.cri.axisC, -1))
def prox_g(self, V):
"""Compute proximal operator of :math:`g`."""
U = prox_l1(V, (self.lmbda / self.L) * self.wl1)
if self.opt['NonNegCoef']:
U[U < 0.0] = 0.0
if self.opt['NoBndryCross']:
for n in range(0, self.cri.dimN):
U[(slice(None), ) * n +
(slice(1 - self.D.shape[n], None), )] = 0.0
return U
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y[..., 0:-1] = sp.prox_l2(
self.AX[..., 0:-1] + self.U[..., 0:-1],
(self.lmbda/self.rho)*self.Wtvna, axis=self.saxes)
self.Y[..., -1] = sp.prox_l1(
self.AX[..., -1] + self.U[..., -1] - self.S,
(1.0/self.rho)*self.Wdf)
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 = sl.proj_l2ball(self.block_sep1(AXU), self.S, self.epsilon, axes=0)
self.Y = self.block_cat(Y0, Y1)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
scalar_factor = (self.lmbda / self.rho) * self.wl1
print('AX dtype %s \n' % (self.AX.dtype, ))
print('U dtype %s \n' % (self.U.dtype, ))
print('sc_factor shape %s \n' % (scalar_factor.shape, ))
self.Y = sp.prox_l1(self.AX + self.U,
(self.lmbda / self.rho) * self.wl1)
super(KConvBPDN, self).ystep()
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 = sl.proj_l2ball(self.block_sep1(AXU), self.S, self.epsilon, axes=0)
self.Y = self.block_cat(Y0, Y1)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
AXU = self.AX + self.U
Y0 = sp.prox_l1(self.block_sep0(AXU) - self.S, (1.0/self.rho)*self.W)
Y1 = sp.prox_l1l2(self.block_sep1(AXU), 0.0,
(self.lmbda/self.rho)*self.wl21,
axis=self.cri.axisC)
self.Y = self.block_cat(Y0, Y1)
cbpdn.ConvTwoBlockCnstrnt.ystep(self)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
AXU = self.AX + self.U
Y0 = sp.prox_l1(self.block_sep0(AXU) - self.S, (1.0/self.rho)*self.W)
Y1 = sp.prox_sl1l2(self.block_sep1(AXU), 0.0,
(self.lmbda/self.rho)*self.wl21,
axis=self.cri.axisC)
self.Y = self.block_cat(Y0, Y1)
cbpdn.ConvTwoBlockCnstrnt.ystep(self)
def cbpdnmd_ystep(k):
"""Do the Y step of the cbpdn stage. The only parameter is the slice
index `k` and there are no return values; all inputs and outputs are
from and to global variables.
"""
if mp_W.shape[0] > 1:
W = mp_W[k]
else:
W = mp_W
AXU0 = mp_DX[k] - mp_S[k] + mp_Z_U0[k]
AXU1 = mp_Z_X[k] + mp_Z_U1[k]
mp_Z_Y0[k] = mp_xrho*AXU0 / (W**2 + mp_xrho)
mp_Z_Y1[k] = sp.prox_l1(AXU1, (mp_lmbda/mp_xrho))
def cbpdnmd_ystep(k):
"""Do the Y step of the cbpdn stage. The only parameter is the slice
index `k` and there are no return values; all inputs and outputs are
from and to global variables.
"""
if mp_W.shape[0] > 1:
W = mp_W[k]
else:
W = mp_W
AXU0 = mp_DX[k] - mp_S[k] + mp_Z_U0[k]
AXU1 = mp_Z_X[k] + mp_Z_U1[k]
mp_Z_Y0[k] = mp_xrho*AXU0 / (W**2 + mp_xrho)
mp_Z_Y1[k] = sp.prox_l1(AXU1, (mp_lmbda/mp_xrho))
def dictionary_learning_by_L1_Dstep(Xf, S, G1, H1, G0, H0, parameter_rho_dic,
iteration, cri, dsz):
GH = con.convert_to_Df(G1 - H1, S, cri)
GSH = con.convert_to_Sf(G0 + S - H0, cri)
b = GH + sl.inner(np.conj(Xf), GSH, cri.axisK)
Df = sl.solvemdbi_ism(Xf, 1, b, cri.axisM, cri.axisK)
D = con.convert_to_D(Df, dsz, cri)
XfDf = np.sum(Xf * Df, axis=cri.axisM)
XD = con.convert_to_S(XfDf, cri)
G0 = sp.prox_l1(XD - S + H0, (1 / parameter_rho_dic))
H0 = H0 + XD - G0 - S
return D, G0, H0, XD
def test_03(self):
N = 64
lmbda = 0.1
s = np.random.randn(N, 1)
def proxf(x, rho):
return s / (1 + rho) + (rho / (1 + rho)) * x
def proxg(x, rho):
return sp.prox_l1(x, lmbda / rho)
opt = ppp.PPPConsensus.Options({'Verbose': False, 'RelStopTol': 1e-5,
'MaxMainIter': 100, 'rho': 8e-1})
b = ppp.PPPConsensus(s.shape, (proxf,), proxg=proxg, opt=opt)
xce = b.solve()
xdrct = sp.prox_l1(s, lmbda)
assert sm.mse(xdrct, xce) < 1e-9
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`."""
if (self.Wg is None or self.mu == 0) and self.gamma == 0:
# Skip unnecessary inhibition steps and run standard CBPDN
super(ConvBPDNInhib, self).ystep()
else:
# Perform soft-thresholding step of Y subproblem using l1 weights
self.Y = sp.prox_l1(self.AX + self.U,
(self.lmbda * self.wl1 + self.mu * self.wml +
self.gamma * self.wms) / self.rho)
# Compute the frequency domain representation of the
# magnitude of X
Xaf = rfftn(np.abs(self.X), self.cri.Nv, self.cri.axisN)
if self.mu > 0 and self.Wg is not None:
# Update previous lateral inhibition term
self.wml_prev = self.wml
# Convolve the lateral spatial weighting matrix with the
# magnitude of X
WhXal = irfftn(self.Whfl * Xaf, self.cri.Nv, self.cri.axisN)
# Sum the weights across in-group members for each element
self.wml = np.dot(np.dot(WhXal, self.Wg.T),
self.Wg) - np.sum(self.Wg, axis=0) * WhXal
# Smooth lateral inhibition term
self.wml = self.smooth * self.wml_prev + \
(1 - self.smooth) * self.wml
if self.gamma > 0:
# Update previous self inhibition term
self.wms_prev = self.wms
# Convolve the self spatial weighting matrix with the
# magnitude of X
self.wms = irfftn(
self.Whfs * Xaf, self.cri.Nv, self.cri.axisN)
# Smooth self inhibition term
self.wms = self.smooth * self.wms_prev + \
(1 - self.smooth) * self.wms
# Handle negative coefficients and boundary crossings
super(cbpdn.ConvBPDN, self).ystep()
def test_01(self):
N = 64
lmbda = 0.1
s = np.random.randn(N, 1)
def f(x):
return 0.5 * np.linalg.norm((x - s).ravel())**2
def gradf(x):
return x - s
def proxg(x, L):
return sp.prox_l1(x, lmbda / L)
opt = ppp.PPP.Options({'Verbose': False, 'RelStopTol': 1e-5,
'MaxMainIter': 100, 'L': 9e-1})
b = ppp.PPP(s.shape, f, gradf, proxg, opt=opt)
xppp = b.solve()
xdrct = sp.prox_l1(s, lmbda)
assert sm.mse(xdrct, xppp) < 1e-9
def projection_to_solution_space_by_L1(D, X, S, Y, U, parameter_rho_coef,
parameter_gamma, iteration, thr, cri,
dsz):
Df = con.convert_to_Df(D, S, cri)
Xf = con.convert_to_Xf(X, S, cri)
for i in range(iteration):
YSUf = con.convert_to_Sf(Y + S - U, cri)
b = (1 / parameter_rho_coef) * Xf + np.conj(Df) * YSUf
Xf = sl.solvedbi_sm(Df, (1 / parameter_rho_coef), b)
X = con.convert_to_X(Xf, cri)
#X = np.where(X <= 0, 0, X)
Xf = con.convert_to_Xf(X, S, cri)
DfXf = np.sum(Df * Xf, axis=cri.axisM)
DX = con.convert_to_S(DfXf, cri)
Y = sp.prox_l1(DX - S + U, (parameter_gamma / parameter_rho_coef))
U = U + DX - Y - S
return X, Y, U
def proxg(x, rho):
return sp.prox_l1(x, lmbda / rho)
def eval_proxop(self, V):
"""Compute proximal operator of :math:`g`."""
return sp.prox_l1(V, (self.lmbda / self.L) * self.wl1)
def eval_proxop(self, V):
"""Compute proximal operator of :math:`g`."""
return np.asarray(prox_l1(V, (self.lmbda / self.L) * self.wl1),
dtype=self.dtype)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y = sp.prox_l1(self.AX - self.S + self.U, self.Wdf / self.rho)
def proxg(x, L):
return sp.prox_l1(x, lmbda / L)
def ystep(self):
r"""Minimise Augmented Lagrangian with respect to
:math:`\mathbf{y}`.
"""
self.Y = sp.prox_l1(self.AX - self.S + self.U, self.Wdf / self.rho)
def prox_g(self, X, rho):
r"""
Proximal operator of :math:`(\lambda/\rho) \|\cdot\|_1`.
"""
return sp.prox_l1(X, (self.lmbda / rho))
def eval_proxop(self, V):
"""Compute proximal operator of :math:`g`."""
return np.asarray(sp.prox_l1(V, (self.lmbda / self.L) * self.wl1),
dtype=self.dtype)
def eval_proxop(self, V):
"""Compute proximal operator of :math:`g`."""
return sp.prox_l1(V, (self.lmbda / self.L) * self.wl1)
