def Maximum_likelihood_estimation_of_an_increasing_nonnegative_signal():
'''
Setup
'''
N = 100
xtrue = np.zeros((N, 1))
xtrue[0:40] = 0.1
xtrue[49] = 2 # matlab is 1-indexed
xtrue[70:80] = 0.15
xtrue[79] = 1 # matlab is 1-indexed
xtrue = np.cumsum(xtrue, axis=0)
h = np.array([[1, -0.85, 0.7, -0.3]])
k = len(h)
yhat = np.array([np.convolve(h.flatten(), xtrue.flatten())]).T
y = yhat[:-3] + np.array(
[[-0.43], [-1.7], [0.13], [0.29],
[-1.1], [1.2], [1.2], [-0.038], [0.33], [0.17], [-0.19], [0.73],
[-0.59], [2.2], [-0.14], [0.11], [1.1], [0.059], [-0.096], [-0.83],
[0.29], [-1.3], [0.71], [1.6], [-0.69], [0.86], [1.3], [-1.6], [-1.4],
[0.57], [-0.4], [0.69], [0.82], [0.71], [1.3], [0.67], [1.2], [-1.2],
[-0.02], [-0.16], [-1.6], [0.26], [-1.1], [1.4], [-0.81], [0.53],
[0.22], [-0.92], [-2.2], [-0.059], [-1], [0.61], [0.51],
[1.7], [0.59], [-0.64], [0.38], [-1], [-0.02], [-0.048], [4.3e-05],
[-0.32], [1.1], [-1.9], [0.43], [0.9], [0.73], [0.58], [0.04], [0.68],
[0.57], [-0.26], [-0.38], [-0.3], [-1.5], [-0.23], [0.12], [0.31],
[1.4], [-0.35], [0.62], [0.8], [0.94], [-0.99], [0.21], [0.24], [-1],
[-0.74], [1.1], [-0.13], [0.39], [0.088], [-0.64], [-0.56], [0.44],
[-0.95], [0.78], [0.57], [-0.82], [-0.27]])
plt.plot(range(len(xtrue)), xtrue, label="xtrue")
plt.plot(range(len(y)), y, label="y")
x_ml = cp.Variable((N, 1))
constraints = [x_ml >= 0] + [x_ml[i + 1] >= x_ml[i] for i in range(N - 1)]
# obj = cp.sum(cp.log(y - cp.conv(h, x_ml)))
# prob = cp.Problem(cp.Maximize(obj), constraints)
obj = cp.norm2(y - cp.conv(h, x_ml))
prob = cp.Problem(cp.Minimize(obj), constraints)
prob.solve()
print("Status = " + str(prob.status))
plt.plot(range(len(x_ml.value)), x_ml.value, label="x_ml")
x_ml_free = cp.Variable((N, 1))
obj = cp.norm2(y - cp.conv(h, x_ml_free))
prob = cp.Problem(cp.Minimize(obj))
prob.solve()
print("Status = " + str(prob.status))
plt.plot(range(len(x_ml_free.value)), x_ml_free.value, label="x_ml_free")
plt.legend()
plt.show()
def main():
TARGET = 0
y = []
x = []
for line in open("features.train"):
vec = line.strip().split()
x.append([float(vec[1]),float(vec[2])])
if int(float(vec[0])) == TARGET:
y.append(1)
else:
y.append(-1)
w = cp.Variable(2)
b = cp.Variable()
Q = np.identity(2)
xi = cp.Variable(len(x))
C = 0.01
objective = cp.Minimize(0.5 * cp.quad_form(w,Q) + C * cp.sum_entries(xi))
constraints = []
for i in range(0,len(x)):
constraints.append(y[i]*(cp.conv(x[i],w) + b) >= 1 - xi[i])
constraints.append(xi[i]>=0)
prob = cp.Problem(objective, constraints)
prob.solve()
print "w=",sum([ item**2 for item in w.value])
#choose any support vector (z,y) to calculate the b
print "b=",y[1] - np.dot(w.value, x[1])
print "status:", prob.status
print "optimal value", prob.value
def test_0D_conv(self) -> None:
"""Convolution with 0D input.
"""
x = cvx.Variable((1, )) # or cvx.Variable((1,1))
problem = cvx.Problem(
cvx.Minimize(cvx.max(cvx.conv([1.], cvx.multiply(1., x)))),
[x >= 0])
problem.solve(cvx.ECOS)
assert problem.status == cvx.OPTIMAL
def op_convolution(n):
x = cvx.Variable(n)
sigma = n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-4] = 0
k = 5
x.value = np.zeros(n)
x.value[np.random.choice(n,k),0] = np.random.rand(k)*sigma
return cvx.conv(c, x), x
def op_convolution(n):
x = cvx.Variable(n)
sigma = n / 10
c = np.exp(-np.arange(-n / 2., n / 2.)**2. /
(2 * sigma**2)) / np.sqrt(2 * sigma**2 * np.pi)
c[c < 1e-4] = 0
k = 5
x.value = np.zeros(n)
x.value[np.random.choice(n, k), 0] = np.random.rand(k) * sigma
return cvx.conv(c, x), x
def test_conv_prob(self):
"""Test a problem with convolution.
"""
import numpy as np
N = 5
y = np.asmatrix(np.random.randn(N, 1))
h = np.asmatrix(np.random.randn(2, 1))
x = cvx.Variable(N)
v = cvx.conv(h, x)
obj = cvx.Minimize(cvx.sum(cvx.multiply(y, v[0:N])))
print((cvx.Problem(obj, []).solve()))
def test_1D_conv(self):
"""Test 1D convolution.
"""
n = 3
x = cvx.Variable(n)
f = [1, 2, 3]
g = [0, 1, 0.5]
f_conv_g = [0., 1., 2.5, 4., 1.5]
expr = cvx.conv(f, g)
assert expr.is_constant()
self.assertEqual(expr.shape, (5, 1))
self.assertItemsAlmostEqual(expr.value, f_conv_g)
expr = cvx.conv(f, x)
assert expr.is_affine()
self.assertEqual(expr.shape, (5, 1))
# Matrix stuffing.
prob = cvx.Problem(cvx.Minimize(cvx.norm(expr, 1)), [x == g])
result = prob.solve()
self.assertAlmostEqual(result, sum(f_conv_g), places=3)
self.assertItemsAlmostEqual(expr.value, f_conv_g)
def test_conv_prob(self):
"""Test a problem with convolution.
"""
import cvxpy as cvx
import numpy as np
N = 5
y = np.asmatrix(np.random.randn(N, 1))
h = np.asmatrix(np.random.randn(2, 1))
x = cvx.Variable(N)
v = cvx.conv(h, x)
obj = cvx.Minimize(cvx.sum_entries(cvx.mul_elemwise(y, v[0:N])))
print(cvx.Problem(obj, []).solve())
def _direct_deconvolution(w, y, Nx, gamma_L2=0, gamma_L1=0):
w = np.asarray(w)
y = np.asarray(y)
cplx = np.issubdtype(w.dtype, complex) or np.issubdtype(y.dtype, complex)
if cplx:
wr, wi = w.real, w.imag
yr, yi = y.real, y.imag
xr = cvx.Variable(Nx)
xi = cvx.Variable(Nx)
error = (cvx.sum_squares(cvx.conv(wr, xr) - cvx.conv(wi, xi) - yr) +
cvx.sum_squares(cvx.conv(wi, xr) + cvx.conv(wr, xi) - yi))
u = cvx.norm(cvx.hstack(xr, xi), 2, axis=1)
else:
x = cvx.Variable(Nx)
error = cvx.sum_squares(cvx.conv(w, x) - y)
u = x
cost = error
if gamma_L1 != 0:
gamma_L1 = cvx.Parameter(value=gamma_L1, sign="positive")
cost = cost + gamma_L1 * cvx.norm(x, 1)
if gamma_L2 != 0:
gamma_L2 = cvx.Parameter(value=gamma_L2, sign="positive")
cost = cost + gamma_L2 * cvx.sum_squares(x)
objective = cvx.Minimize(cost)
prob = cvx.Problem(objective)
prob.solve()
print("Problem Status: {0}".format(prob.status))
if cplx:
result = np.array(xr.value).ravel() + 1j * np.array(xi.value).ravel()
else:
result = np.asarray(x.value).ravel()
return result
def lasso_conv(n):
sigma = n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-4] = 0
x0 = np.array(sp.rand(n, 1, 0.1).todense()).ravel()
b = np.convolve(c, x0) + 1e-2*np.random.randn(2*n-1)
lam = 0.2*np.max(np.abs(np.convolve(b, c, "valid")))
print lam
x = cvx.Variable(n)
f = cvx.sum_squares(cvx.conv(c, x) - b) + lam*cvx.norm1(x)
return cvx.Problem(cvx.Minimize(f))
def nn_deconv(n):
sigma = 0.1*n
k = 5
x0 = np.zeros(n)
x0[np.random.choice(n,k)] = np.random.randn(k)*n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-6] = 0
b = np.convolve(c, x0) + 0.1*np.random.randn(2*n-1)
x = cvx.Variable(n)
f = cvx.norm(cvx.conv(c, x) - b)
return cvx.Problem(cvx.Minimize(f), [x >= 0])
def nn_deconv(n):
sigma = 0.1*n
k = 5
x0 = np.zeros(n)
x0[np.random.choice(n,k)] = np.random.randn(k)*n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-6] = 0
b = np.convolve(c, x0) + 0.1*np.random.randn(2*n-1)
x = cvx.Variable(n)
f = cvx.norm(cvx.conv(c, x) - b)
return cvx.Problem(cvx.Minimize(f), [x >= 0])
def lasso_conv(n):
sigma = n/10
c = np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/np.sqrt(2*sigma**2*np.pi)
c[c < 1e-4] = 0
x0 = np.array(sp.rand(n, 1, 0.1).todense()).ravel()
b = np.convolve(c, x0) + 1e-2*np.random.randn(2*n-1)
lam = 0.2*np.max(np.abs(np.convolve(b, c, "valid")))
print lam
x = cvx.Variable(n)
# f = cvx.sum_squares(cvx.conv(c, x) - b) + lam*cvx.norm1(x)
f = cvx.norm1(cvx.conv(c, x) - b) + lam*cvx.norm1(x)
return cvx.Problem(cvx.Minimize(f))
def nonnegative_deconvolution():
np.random.seed(0)
n = 10
k = 3
sigma = n/10.
c = (np.exp(-np.arange(-n/2., n/2.)**2./(2*sigma**2))/
np.sqrt(2*sigma**2*np.pi))
x0 = np.zeros(n)
x0[np.random.choice(n,k)] = np.random.randn(k)*sigma
b = np.convolve(c, x0)
b += np.random.randn(2*n-1)*np.linalg.norm(b)/np.sqrt(2*n-1)/20
x = cvx.Variable(n)
f = cvx.sum_squares(cvx.conv(c, x) - b)
return cvx.Problem(cvx.Minimize(f), [x >= 0])
def test_conv(self):
"""Test the conv atom.
"""
a = np.ones((3, 1))
b = Parameter(2, nonneg=True)
expr = cp.conv(a, b)
assert expr.is_nonneg()
self.assertEqual(expr.shape, (4, 1))
b = Parameter(2, nonpos=True)
expr = cp.conv(a, b)
assert expr.is_nonpos()
with self.assertRaises(Exception) as cm:
cp.conv(self.x, -1)
self.assertEqual(str(cm.exception),
"The first argument to conv must be constant.")
with self.assertRaises(Exception) as cm:
cp.conv([[0, 1], [0, 1]], self.x)
self.assertEqual(str(cm.exception),
"The arguments to conv must resolve to vectors.")
0.17, -0.19, 0.73, -0.59, 2.2, -0.14, 0.11, 1.1, 0.059,
-0.096, -0.83, 0.29, -1.3, 0.71, 1.6, -0.69, 0.86, 1.3,
-1.6, -1.4, 0.57, -0.4, 0.69, 0.82, 0.71, 1.3, 0.67, 1.2,
-1.2, -0.02, -0.16, -1.6, 0.26, -1.1, 1.4, -0.81, 0.53,
0.22, -0.92, -2.2, -0.059, -1, 0.61, 0.51, 1.7, 0.59,
-0.64, 0.38, -1, -0.02, -0.048, 4.3e-05, -0.32, 1.1,
-1.9, 0.43, 0.9, 0.73, 0.58, 0.04, 0.68, 0.57, -0.26,
-0.38, -0.3, -1.5, -0.23, 0.12, 0.31, 1.4, -0.35, 0.62,
0.8, 0.94, -0.99, 0.21, 0.24, -1, -0.74, 1.1, -0.13, 0.39,
0.088, -0.64, -0.56, 0.44, -0.95, 0.78, 0.57, -0.82, -0.27
])
s = y + noise
# Solving the ML estimation
x_ = cp.Variable(N)
y_ = cp.conv(h, x_)[:-k+1].flatten()
constraints = [0 <= x_[0]] + [
x_[i] <= x_[i+1] for i in range(0, N-1)
]
# With constraints
cp.Problem(cp.Minimize(cp.norm(y_ - s)), constraints).solve()
x_hat = x_.value
y_hat = y_.value
# Without constraints
cp.Problem(cp.Minimize(cp.norm(y_ - s))).solve()
x_hat_free = x_.value
y_hat_free = y_.value
fig, axs = plt.subplots(nrows=2, ncols=1, figsize=(14, 8))
# Plotting the results
axs[0].plot(x, label="Origial")
import numpy as np
import cvxpy as cp
N = 100
xtrue = np.zeros((N, 1))
xtrue[0:39, ] = 0.1
xtrue[49, ] = 2
xtrue[69:79, ] = 0.15
xtrue[79, ] = 1
xtrue = xtrue.cumsum()
h = np.array([1, -0.85, 0.7, -0.3])
k = len(h)
yhat = cp.conv(h, xtrue).value
noise = np.array([
-0.43, -1.7, 0.13, 0.29, -1.1, 1.2, 1.2, -0.038, 0.33, 0.17, -0.19, 0.73,
-0.59, 2.2, -0.14, 0.11, 1.1, 0.059, -0.096, -0.83, 0.29, -1.3, 0.71, 1.6,
-0.69, 0.86, 1.3, -1.6, -1.4, 0.57, -0.4, 0.69, 0.82, 0.71, 1.3, 0.67, 1.2,
-1.2, -0.02, -0.16, -1.6, 0.26, -1.1, 1.4, -0.81, 0.53, 0.22, -0.92, -2.2,
-0.059, -1, 0.61, 0.51, 1.7, 0.59, -0.64, 0.38, -1, -0.02, -0.048, 4.3e-05,
-0.32, 1.1, -1.9, 0.43, 0.9, 0.73, 0.58, 0.04, 0.68, 0.57, -0.26, -0.38,
-0.3, -1.5, -0.23, 0.12, 0.31, 1.4, -0.35, 0.62, 0.8, 0.94, -0.99, 0.21,
0.24, -1, -0.74, 1.1, -0.13, 0.39, 0.088, -0.64, -0.56, 0.44, -0.95, 0.78,
0.57, -0.82, -0.27
])
y = yhat[0:N] + noise
A = np.zeros(shape=(N, N))
for i in range(n):
for j in range(len(h)):
# create problem data
N = 100;
# create an increasing input signal
xtrue = np.zeros((N,1))
xtrue[1:40] = 0.1
xtrue[50] = 2
xtrue[70:80] = 0.15
xtrue[80] = 1
xtrue = np.cumsum(xtrue)
# pass the increasing input through a moving-average filter
# and add Gaussian noise
h = np.array([1, -0.85 ,0.7 ,-0.3])
k = h.shape[0]
yhat = np.convolve(h,xtrue)
y = yhat[:-3] + np.random.randn(N)
x = cp.Variable((100,),nonneg = True)
z = y[:,None] - cp.conv(h,x)[:-3]
objective = cp.Minimize(cp.sum_squares(z))
constraints = [cp.diff(x) >= 0]
prob=cp.Problem(objective,constraints=constraints)
prob.solve()
#plot
t = list(range(0,xtrue.size))
plt.plot(t,list(xtrue), color='red',label='x_true')
plt.plot(t,list(x.value), color='blue',label='x_hat')
plt.legend(loc="upper left")
plt.savefig('prob_66.png')
plt.show()
from numpy.testing import assert_allclose
import cvxpy as cvx
import numpy as np
import scipy.sparse as sp
import tensorflow as tf
from cvxflow import cvxpy_expr
np.random.seed(0)
A_sparse = sp.rand(5, 3, 0.5)
x_var = cvx.Variable(3)
EXPRESSIONS = [
("conv", cvx.conv([1, 2, 3], x_var), [1, 2, 4], [1, 2, 4, 6, 8]),
("sparse", A_sparse * x_var, [1, 2, 4], [1, 2, 4, 6, 8]),
]
class TensorTest(tf.test.TestCase):
pass
def get_tensor_test(f_expr, x, y):
f = f_expr.canonicalize()[0]
x = np.array(x).reshape(-1, 1)
y = np.array(y).reshape(-1, 1)
def test(self):
prob = cvx.Problem(cvx.Minimize(0), [f_expr == 0])
A = prob.get_problem_data(cvx.SCS)["A"]
xt = tf.constant(x, dtype=tf.float32)
n = 100
m = 40
y = cvx.Variable(m)
x = cvx.Variable(n)
x0 = np.zeros((n, 1))
x0[n / 2 + 3:n / 2 + 4] = 1
x0[n / 2 + 23:n / 2 + 24] = 0.8
x0[n / 2 + 43:n / 2 + 44] = 0.6
t = np.linspace(-2, 2, m)
y0 = np.exp(-np.square(t) * 2)
d = np.convolve(np.array(x0).flatten(), np.array(y0).flatten())
cost = cvx.norm(cvx.conv(y, x) - d) + 0.15 * cvx.norm(
x, 1) # cvx.conv does not yet support variable as first argument.
prob = cvx.Problem(cvx.Minimize(cost), [cvx.norm(y, "inf") <= 1])
x.value = np.ones((n, 1))
y.value = np.ones((m, 1))
prob.solve(method='bcd')
plt.plot(np.array(abs(x).value).flatten(), 'b-o')
plt.plot(np.array(abs(y).value).flatten(), 'c-s')
plt.plot(d, 'g-')
plt.plot(x0, 'r--', linewidth=2)
plt.plot(y0, 'm-.', linewidth=2)
print cvx.norm(cvx.conv(x, y) - d).value
plt.legend(["$x$", "$y$", "$d$", "ground truth $x_0$", "ground truth $y_0$"])
plt.show()
from numpy.testing import assert_allclose
import cvxpy as cvx
import numpy as np
import scipy.sparse as sp
import tensorflow as tf
from cvxflow import cvxpy_expr
np.random.seed(0)
A_sparse = sp.rand(5, 3, 0.5)
x_var = cvx.Variable(3)
EXPRESSIONS = [
("conv", cvx.conv([1,2,3], x_var), [1,2,4], [1,2,4,6,8]),
("sparse", A_sparse*x_var, [1,2,4], [1,2,4,6,8]),
]
class TensorTest(tf.test.TestCase):
pass
def get_tensor_test(f_expr, x, y):
f = f_expr.canonicalize()[0]
x = np.array(x).reshape(-1,1)
y = np.array(y).reshape(-1,1)
def test(self):
prob = cvx.Problem(cvx.Minimize(0), [f_expr == 0])
A = prob.get_problem_data(cvx.SCS)["A"]
xt = tf.constant(x, dtype=tf.float32)
yt = tf.constant(y, dtype=tf.float32)
yhat = np.convolve(h, ytrue)
y = yhat[:-3] + np.array([
-0.43, -1.7, 0.13, 0.29, -1.1, 1.2, 1.2, -0.038, 0.33, 0.17, -0.19, 0.73,
-0.59, 2.2, -0.14, 0.11, 1.1, 0.059, -0.096, -0.83, 0.29, -1.3, 0.71, 1.6,
-0.69, 0.86, 1.3, -1.6, -1.4, 0.57, -0.4, 0.69, 0.82, 0.71, 1.3, 0.67, 1.2,
-1.2, -0.02, -0.16, -1.6, 0.26, -1.1, 1.4, -0.81, 0.53, 0.22, -0.92, -2.2,
-0.059, -1, 0.61, 0.51, 1.7, 0.59, -0.64, 0.38, -1, -0.02, -0.048, 4.3e-05,
-0.32, 1.1, -1.9, 0.43, 0.9, 0.73, 0.58, 0.04, 0.68, 0.57, -0.26, -0.38,
-0.3, -1.5, -0.23, 0.12, 0.31, 1.4, -0.35, 0.62, 0.8, 0.94, -0.99, 0.21,
0.24, -1, -0.74, 1.1, -0.13, 0.39, 0.088, -0.64, -0.56, 0.44, -0.95, 0.78,
0.57, -0.82, -0.27
])
#maximum likelihood estimation with no monotonicity taken in to account. can be solved analytically
y_ls = cp.Variable(N)
yhat = cp.conv(h, y_ls)[:-3]
y = y.reshape(-1, 1)
objective = cp.Minimize(cp.sum_squares(yhat - y))
prob = cp.Problem(objective)
_ = prob.solve()
error = np.sum((yhat.value - ytrue)**2)
print('error for y_ml,free is ', error)
#monotonic and non-negative signal estimation
y_mono = cp.Variable(N)
yhat = cp.conv(h, y_mono)[:-3]
objective = cp.Minimize(cp.sum_squares(yhat - y))
constraints = [y_mono[0] >= 0, y_mono[:-1] <= y_mono[1:]]
prob = cp.Problem(objective, constraints)
_ = prob.solve()
error = np.sum((yhat.value - ytrue)**2)
return [
1 / (sigma * sqrt(2 * pi)) * exp(-float(x)**2 / (2 * sigma**2))
for x in r
]
np.random.seed(5)
random.seed(5)
DENSITY = 0.1
n = 1000
x = Variable(n)
# Create sparse signal.
signal = np.zeros(n)
for i in range(n):
if random.random() < DENSITY:
signal[i] = random.uniform(1, 100)
# Gaussian kernel.
m = 100
kernel = gauss(m)
# Noisy signal.
noisy_signal = conv(kernel, signal).value + np.random.normal(n + m - 1)
obj = norm(conv(kernel, x) - noisy_signal)
constraints = [x >= 0]
prob = Problem(Minimize(obj), constraints)
result = prob.solve(solver=SCS, verbose=True)
print(norm(signal - x.value, 1).value)
# Create sparse signal.
signal = np.zeros(n)
nnz = 0
for i in range(n):
if random.random() < DENSITY:
signal[i] = random.uniform(0, 100)
nnz += 1
# Gaussian kernel.
m = 1001
kernel = gauss(m, m/10)
# Noisy signal.
std = 1
noise = np.random.normal(scale=std, size=n+m-1)
noisy_signal = conv(kernel, signal) #+ noise
gamma = Parameter(nonneg=True)
fit = norm(conv(kernel, x) - noisy_signal, 2)
regularization = norm(x, 1)
constraints = [x >= 0]
gamma.value = 0.06
prob = Problem(Minimize(fit), constraints)
solver_options = {"NORMALIZE": True, "MAX_ITERS": 2500,
"EPS":1e-3}
result = prob.solve(solver=SCS,
verbose=True,
NORMALIZE=True,
MAX_ITERS=2500)
# Get problem matrix.
data, dims = prob.get_problem_data(solver=SCS)
# pl.figure(figsize=(6, 6))
# legend(loc='upper center', bbox_to_anchor=(0.5, 1.05), prop={'size': 18})
# print(dir(ct))
H = np.zeros((ct.N, ct.N))
for idx in np.arange(ct.h.size):
Htemp = np.diag(np.ones(ct.N) * ct.h[idx])
Htemp = np.roll(Htemp, idx, axis=0)
Htemp[:idx, :] = 0
H = H + Htemp
# print(H)
S = np.roll(np.eye(ct.N), 1, axis=0)
S[0, :] = 0
# print(S)
x = cv.Variable(ct.N)
# obj = cv.Minimize(cv.norm(H * x - ct.y))
obj = cv.Minimize(cv.norm(cv.conv(ct.h, x)[:-3] - ct.y))
# cst = [S * x <= x]
cst = [0 <= x[0], x[:-1] <= x[1:]]
prb = cv.Problem(obj, cst)
prb.solve()
# print(prb.status)
pl.figure(figsize=(6, 6))
pl.plot(x.value, label=r'$x_{ml}$')
pl.plot(ct.xtrue, label=r'$x_{true}$')
prb = cv.Problem(obj)
prb.solve()
# print(prb.status)
pl.plot(x.value, label=r'$x_{ml,free}$')
pl.legend(loc='upper left', bbox_to_anchor=(1.03, 1.03), prop={'size': 18})
