def test_bp(par):
"""Basis pursuit (BP) problem:
minimize ||x||_1 subject to Ax = b
"""
np.random.seed(0)
# Create random m-by-n encoding matrix
n = par['n']
m = par['m']
k = par['k']
A, A1 = np.linalg.qr(np.random.randn(n, m), 'reduced')
if m > n:
A = A1.copy()
A = A.astype(par['dtype'])
# Create sparse vector
p = np.random.permutation(m)
p = p[0:k]
x = np.zeros(m, dtype=par['dtype'])
x[p] = np.random.randn(k)
# Set up vector b, and run solver
b = A.dot(x)
xinv, _, _, _ = spg_bp(A, b, verbosity=0)
assert xinv.dtype == par['dtype']
assert_array_almost_equal(x, xinv, decimal=3)
# Run solver with subspace minimization
xinv, _, _, _ = spg_bp(A, b, subspace_min=True, verbosity=0)
assert xinv.dtype == par['dtype']
assert_array_almost_equal(x, xinv, decimal=3)
def test_weighted_bp(par):
"""Weighted Basis pursuit (WBP) problem:
minimize ||y||_1 subject to AW^{-1}y = b
or
minimize ||Wx||_1 subject to Ax = b
"""
# Create random m-by-n encoding matrix
n = par['n']
m = par['m']
k = par['k']
A, A1 = np.linalg.qr(np.random.randn(n, m), 'reduced')
if m > n:
A = A1.copy()
# Create sparse vector
p = np.random.permutation(m)
p = p[0:k]
x = np.zeros(m)
x[p] = np.random.randn(k)
# Set up weights w and vector b
w = 0.1*np.random.rand(m) + 0.1 # Weights
b = A.dot(x / w) # Signal
xinv, _, _, _ = spg_bp(A, b, iter_lim=1000, weights=w, verbosity=0)
# Reconstructed solution, with weighting
xinv *= w
assert_array_almost_equal(x, xinv, decimal=3)
# % -----------------------------------------------------------
# % Solve the basis pursuit (BP) problem:
# %
# % minimize ||x||_1 subject to Ax = b
# %
# % -----------------------------------------------------------
print('%s ' % ('-'*78))
print('Solve the basis pursuit (BP) problem: ')
print(' ')
print(' minimize ||x||_1 subject to Ax = b')
print(' ')
print('%s%s ' % ('-'*78, '\n'))
# % Set up vector b, and run solver
b = A.dot(x0) # signal
x,resid,grad,info = spg_bp(A, b)
figure()
plot(x,'b')
hold(True)
plot(x0,'ro')
legend(('Recovered coefficients','Original coefficients'))
title('(a) Basis Pursuit')
print('%s%s%s' % ('-'*35,' Solution ','-'*35))
print('See figure 1(a)')
print('%s%s ' % ('-'*78, '\n'))
# % -----------------------------------------------------------
# % Solve the basis pursuit denoise (BPDN) problem:
def basis_pursuit(t, y, fmin=None, fmax=None, nfreqs=5000, polyorder=2, method="basis", tau=0.1,
noise=True):
# preprocess
ndata = np.size(y)
tmin = t.min()
t -= tmin
yscale = y.max()-y.min()
ymin = y.min()
y = (y-ymin)/(yscale)-0.5
trange = np.nanmax(t)-np.nanmin(t)
dt = np.abs(np.nanmedian(t-np.roll(t,-1)))
nt = np.size(t)
if fmin is None:
fmin = 1./trange
if fmax is None:
fmax = 2./dt
freqs = np.linspace(fmin,fmax,nfreqs)
df = np.abs(np.nanmedian(freqs-np.roll(freqs,-1)))
if noise is True:
ndirac = ndata
else:
ndirac = 0
X = np.zeros((nt,nfreqs*2+polyorder+1+ndirac))
# set up matrix of sines and cosines
for j in range(nfreqs):
X[:,j] = np.sin(t*freqs[j])
X[:,nfreqs+j] = np.cos(t*freqs[j])
# now do polynomial bits
for j in range(polyorder+1):
pp = t**(polyorder-j)
X[:,nfreqs*2 +j] = pp/np.abs(pp.max())
# now do the dirac delta functions
for j in range(ndirac):
X[j,-ndirac+j] = 1.
if method == "basis":
x,resid,grad,info = spg_bp(X, y)
elif method == "lasso":
tau = 0.1
x, resid, grad, info = spg_lasso(X, y, tau)
else:
print("Did not select a method")
return 0
sines = x[:nfreqs]
cosines = x[nfreqs:2*nfreqs]
power = (sines**2 + cosines**2)
if noise:
diracs = x[-ndirac:]
else:
diracs = None
output = {'freqs':freqs,
'sines': sines,
'cosines': cosines,
'power':power,
'polys':x[2*nfreqs:2*nfreqs+polyorder+1],
'diracs':diracs,
'resid':resid,
'grad':grad,
'info':info,
'coeffs':x,
'matrix':X,
'model':(np.dot(X,x)+0.5)*yscale+ymin
}
return output
t = np.linspace(-np.pi * frac, np.pi * frac, n)
x = np.fft.ifft(np.fft.fft(2.5 * np.sin(t)))
# Create dictionary, D
m = int(n / 2)
D = np.fft.fft(np.eye(n, n))
# Create A
np.random.seed(0)
idx = np.random.permutation(n)
idx = idx[0:m]
A = D[idx, :]
# Sense b in the time domain
b = x[idx]
# Run spgl1 solver to get coefficients
c, resid, grad, info = spg_bp(A, b)
# Reconstruct estimate of x
x_hat = D.dot(c)
# Get estimate from b
x_b = np.zeros(x.shape, dtype='complex')
x_b[idx] = b
plt.plot(np.abs(x))
plt.plot(np.abs(x_b))
plt.plot(np.abs(x_hat))
plt.show()
x, resid, grad, info = spgl1.spg_lasso(A, b, tau, verbosity=1)
print('%s%s%s' % ('-' * 35, ' Solution ', '-' * 35))
print('nonzeros(x) = %i, ||x||_1 = %12.6e, ||x||_1 - pi = %13.6e' %
(np.sum(np.abs(x) > 1e-5), np.linalg.norm(
x, 1), np.linalg.norm(x, 1) - np.pi))
print('%s' % ('-' * 80))
###############################################################################
# Solve the basis pursuit (BP) problem:
#
# .. math::
# min. ||\mathbf{x}||_1 \quad subject \quad to \quad
# \mathbf{Ax} = \mathbf{b}
b = A.dot(x0)
x, resid, grad, info = spgl1.spg_bp(A, b, verbosity=2)
plt.figure()
plt.plot(x, 'b')
plt.plot(x0, 'ro')
plt.legend(('Recovered coefficients', 'Original coefficients'))
plt.title('Basis Pursuit')
plt.figure()
plt.plot(info['xnorm1'], info['rnorm2'], '.-k')
plt.xlabel(r'$||x||_1$')
plt.ylabel(r'$||r||_2$')
plt.title('Pareto curve')
plt.figure()
plt.plot(np.arange(info['niters']),