def _compute(self): start = datetime.datetime.now() C = self.C gamma = self.gamma Kcount = len( gamma ) (N,d) = self.data.shape X = self.data # CMF of observations X Xcmf = ( (X.reshape(N,1,d) > transpose(X.reshape(N,1,d),[1,0,2])).prod(2).sum(1,dtype=float) / N ).reshape([N,1]) # epsilon of observations X e = sqrt( (1./N) * ( Xcmf ) * (1.-Xcmf) ).reshape([N,1]) K = self._K( Xcmf.reshape(N,1,d), transpose(Xcmf.reshape(N,1,d), [1,0,2]), gamma ) xipos = cvxmod.optvar( 'xi+', N,1) xipos.pos = True xineg = cvxmod.optvar( 'xi-', N,1) xineg.pos = True alphas = list() expr = ( C*cvxmod.sum(xipos) ) + ( C*cvxmod.sum(xineg) ) ineq = 0 eq = 0 for i in range( Kcount ): alpha = cvxmod.optvar( 'alpha(%s)' % i, N,1) alpha.pos = True alphas.append( alpha ) expr += ( float(1./gamma[i]) * cvxmod.sum( alpha ) ) ineq += ( cvxopt.matrix( K[i], (N,N) ) * alpha ) eq += cvxmod.sum( alpha ) objective = cvxmod.minimize( expr ) ineq1 = ineq <= cvxopt.matrix( Xcmf + e ) + xineg ineq2 = ineq >= cvxopt.matrix( Xcmf - e ) - xipos eq1 = eq == cvxopt.matrix( 1.0 ) # Solve! p = cvxmod.problem( objective = objective, constr = [ineq1,ineq2,eq1] ) start = datetime.datetime.now() p.solve() duration = datetime.datetime.now() - start print "optimized in %ss" % (float(duration.microseconds)/1000000) self.Fl = Xcmf self.betas = [ ma.masked_less( alpha.value, 1e-4) for alpha in alphas ] print "SV's found: %s" % [ len( beta.compressed()) for beta in self.betas ]
def _compute(self):
C = self.C
gamma = self.gamma
(N, d) = self.data.shape
X = self.data
Xcmf = (
(X.reshape(N, 1, d) > transpose(X.reshape(N, 1, d), [1, 0, 2])).prod(2).sum(1, dtype=float) / N
).reshape([N, 1])
sigma = 0.75 / sqrt(N)
K = self._K(X.reshape(N, 1, d), transpose(X.reshape(N, 1, d), [1, 0, 2]), gamma).reshape([N, N])
# NOTE: this integral depends on K being the gaussian kernel
Kint = (1.0 / gamma) * scipy.special.ndtr((X - X.T) / gamma)
alpha = cvxmod.optvar("alpha", N, 1)
alpha.pos = True
xi = cvxmod.optvar("xi", N, 1)
xi.pos = True
pXcmf = cvxmod.param("Xcmf", N, 1)
pXcmf.pos = True
pXcmf.value = cvxopt.matrix(Xcmf, (N, 1))
pKint = cvxmod.param("Kint", N, N)
pKint.value = cvxopt.matrix(Kint, (N, N))
objective = cvxmod.minimize(cvxmod.sum(cvxmod.atoms.power(alpha, 2)) + (C * cvxmod.sum(xi)))
eq1 = cvxmod.abs((pKint * alpha) - pXcmf) <= sigma + xi
eq2 = cvxmod.sum(alpha) == 1.0
# Solve!
p = cvxmod.problem(objective=objective, constr=[eq1, eq2])
p.solve()
beta = ma.masked_less(alpha.value, 1e-7)
mask = ma.getmask(beta)
data = ma.array(X, mask=mask)
self.beta = beta.compressed().reshape([1, len(beta.compressed())])
self.SV = data.compressed().reshape([len(beta.compressed()), 1])
print "%s SV's found" % len(self.SV)
def interior_point(X, y, lam):
"""
solve lasso using an interior point method
requires cvxmod (Jacob Mattingley and Stephen Boyd)
http://cvxmod.net/
"""
import cvxmod as cvx
n, m = X.shape
X_cvx = cvx.matrix(np.array(X))
y_cvx = cvx.matrix(np.array(y))
theta = cvx.optvar('theta', m)
p = cvx.problem(cvx.minimize(cvx.sum(cvx.atoms.power(X_cvx*theta - y_cvx, 2)) +
(2*lam)*cvx.norm1(theta)))
p.solve()
return np.array(cvx.value(theta))
def _compute(self): start = datetime.datetime.now() gamma = self.gamma (N,d) = self.data.shape X = self.data Xcmf = ( (X.reshape(N,1,d) > transpose(X.reshape(N,1,d),[1,0,2])).prod(2).sum(1,dtype=float) / N ).reshape([N,1]) sigma = .75 / sqrt(N) K = self._K( X.reshape(N,1,d), transpose(X.reshape(N,1,d), [1,0,2]), gamma ).reshape([N,N]) #NOTE: this integral depends on K being the gaussian kernel Kint = ( (1.0/gamma)*scipy.special.ndtr( (X-X.T)/gamma ) ) alpha = cvxmod.optvar( 'alpha',N,1) alpha.pos = True pK = cvxmod.param( 'K',N,N ) pK.psd = True pK.value = cvxopt.matrix(K,(N,N) ) pKint = cvxmod.param( 'Kint',N,N ) pKint.value = cvxopt.matrix(Kint,(N,N)) #pKint.pos = True pXcmf = cvxmod.param( 'Xcmf',N,1) pXcmf.value = cvxopt.matrix(Xcmf, (N,1)) #pXcmf.pos = True objective = cvxmod.minimize( cvxmod.atoms.quadform(alpha, pK) ) eq1 = cvxmod.abs( pXcmf - ( pKint * alpha ) ) <= sigma eq2 = cvxmod.sum( alpha ) == 1.0 # Solve! p = cvxmod.problem( objective = objective, constr = [eq1, eq2] ) start = datetime.datetime.now() p.solve() duration = datetime.datetime.now() - start print "optimized in %ss" % (float(duration.microseconds)/1000000) beta = ma.masked_less( alpha.value, 1e-7 ) mask = ma.getmask( beta ) data = ma.array(X,mask=mask) self.Fl = Xcmf self.beta = beta.compressed().reshape([ 1, len(beta.compressed()) ]) self.SV = data.compressed().reshape([len(beta.compressed()),1]) print "%s SV's found" % len(self.SV)
def run_opt(feature_lists, reference_indices, alpha):
"""
run_opt( feature_lists ) -> weights
feature_lists is a list of I image_feature_sets
image_feature_sets are a list of P stacked_features
stacked_features are a list of N different feature types
reference_indices is a set of indices which will be held out as "reference"
performs the opt:
min sum_{i_r in ref_idx} sum_{i < I not in ref_idx} sum_{p < P}
w' * ||f_{i_r,p,n} - f_{i,p,n}||^2 - alpha/(P-1) * sum_{p'<p} || f_{i_r,p,n} - f_{i,p',n} ||^2
"""
I = len(feature_lists)
P = len(feature_lists[0])
N = len(feature_lists[0][0])
non_reference_indices = [i for i in range(I) if not i in reference_indices]
closeness_reward = np.zeros(N)
uniqueness_penalty = np.zeros(N)
f = feature_lists
for i_r in reference_indices:
for i in non_reference_indices:
for p in range(P):
for n in range(N):
closeness_reward[n] += feature_distance(
f[i_r][p][n], f[i][p][n])
for p_false in range(p):
uniqueness_penalty[n] += feature_distance(
f[i_r][p][n], f[i][p_false][n])
c = cvx.param('c',
value=cvx.matrix(closeness_reward -
alpha / float(P - 1) * uniqueness_penalty))
print c.value
w = cvx.optvar('w', N)
w.pos = True
w | cvx.In | cvx.norm1ball(N)
p = cvx.problem()
p.objective = cvx.minimize(cvx.tp(c) * w)
p.constr = [cvx.sum(w) == 1]
print "Running solver"
p.solve()
print "Ran!"
return np.array(w.value)
def run_opt( feature_lists, reference_indices, alpha ):
"""
run_opt( feature_lists ) -> weights
feature_lists is a list of I image_feature_sets
image_feature_sets are a list of P stacked_features
stacked_features are a list of N different feature types
reference_indices is a set of indices which will be held out as "reference"
performs the opt:
min sum_{i_r in ref_idx} sum_{i < I not in ref_idx} sum_{p < P}
w' * ||f_{i_r,p,n} - f_{i,p,n}||^2 - alpha/(P-1) * sum_{p'<p} || f_{i_r,p,n} - f_{i,p',n} ||^2
"""
I = len( feature_lists )
P = len( feature_lists[0] )
N = len( feature_lists[0][0] )
non_reference_indices = [i for i in range(I) if not i in reference_indices]
closeness_reward = np.zeros( N )
uniqueness_penalty = np.zeros( N )
f = feature_lists
for i_r in reference_indices:
for i in non_reference_indices:
for p in range(P):
for n in range(N):
closeness_reward[ n ] += feature_distance( f[i_r][p][n], f[i][p][n] )
for p_false in range(p):
uniqueness_penalty[ n ] += feature_distance( f[i_r][p][n], f[i][p_false][n] )
c = cvx.param('c', value = cvx.matrix( closeness_reward - alpha / float(P-1) * uniqueness_penalty ) )
print c.value
w = cvx.optvar('w', N )
w.pos = True
w | cvx.In | cvx.norm1ball(N)
p = cvx.problem()
p.objective = cvx.minimize( cvx.tp(c) * w )
p.constr = [ cvx.sum(w) == 1]
print "Running solver"
p.solve()
print "Ran!"
return np.array( w.value )
def fit_ellipse_stack_squared(dx, dy, dz, di):
"""
fit ellipoid using squared loss
idea to learn all stacks together including smoothness
"""
# sanity check
assert len(dx) == len(dy)
assert len(dx) == len(dz)
assert len(dx) == len(di)
# unique zs
dat = defaultdict(list)
# resort data
for idx in range(len(dx)):
dat[dz[idx]].append( [dx[idx], dy[idx], di[idx]] )
# init ret
ellipse_stack = []
for idx in range(max(dz)):
ellipse_stack.append(Ellipse(0, 0, idx, 1, 1, 0))
total_N = len(dx)
M = len(dat.keys())
D = 5
X_matrix = []
thetas = []
for z in dat.keys():
x = numpy.array(dat[z])[:,0]
y = numpy.array(dat[z])[:,1]
# intensities
i = numpy.array(dat[z])[:,2]
# log intensities
i = numpy.log(i)
# create matrix
ity = numpy.diag(i)
# dimensionality
N = len(x)
d = numpy.zeros((N, D))
d[:,0] = x*x
d[:,1] = y*y
#d[:,2] = x*y
d[:,2] = x
d[:,3] = y
d[:,4] = numpy.ones(N)
#d[:,0] = x*x
#d[:,1] = y*y
#d[:,2] = x*y
#d[:,3] = x
#d[:,4] = y
#d[:,5] = numpy.ones(N)
# consider intensities
old_shape = d.shape
d = numpy.dot(ity, d)
assert d.shape == old_shape
print d.shape
d = cvxmod.matrix(d)
#### parameters
# da
X = cvxmod.param("X" + str(z), N, D)
X.value = d
X_matrix.append(X)
#### varibales
# parameter vector
theta = cvxmod.optvar("theta" + str(z), D)
thetas.append(theta)
# contruct objective
objective = 0
for (i,X) in enumerate(X_matrix):
#TODO try abs loss here!
objective += cvxmod.sum(cvxmod.atoms.square(X*thetas[i]))
#objective += cvxmod.sum(cvxmod.atoms.abs(X*thetas[i]))
# add smoothness regularization
reg_const = float(total_N) / float(M-1)
for i in xrange(M-1):
objective += reg_const * cvxmod.sum(cvxmod.atoms.square(thetas[i] - thetas[i+1]))
print objective
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# add constraints
for i in xrange(M):
p.constr.append(thetas[i][0] + thetas[i][1] == 1)
###### set values
p.solve()
# wrap up result
ellipse_stack = {}
active_layers = dat.keys()
assert len(active_layers) == M
for i in xrange(M):
theta_ = numpy.array(cvxmod.value(thetas[i]))
z_layer = active_layers[i]
ellipse_stack[z_layer] = conic_to_ellipse(theta_)
ellipse_stack[z_layer].cz = z_layer
return ellipse_stack
def fit_ellipse_stack_abs(dx, dy, dz, di):
"""
fit ellipoid using squared loss
idea to learn all stacks together including smoothness
"""
# sanity check
assert len(dx) == len(dy)
assert len(dx) == len(dz)
assert len(dx) == len(di)
# unique zs
dat = defaultdict(list)
# resort data
for idx in range(len(dx)):
dat[dz[idx]].append( [dx[idx], dy[idx], di[idx]] )
# init ret
ellipse_stack = []
for idx in range(max(dz)):
ellipse_stack.append(Ellipse(0, 0, idx, 1, 1, 0))
total_N = len(dx)
M = len(dat.keys())
D = 5
X_matrix = []
thetas = []
slacks = []
eps_slacks = []
mean_di = float(numpy.mean(di))
for z in dat.keys():
x = numpy.array(dat[z])[:,0]
y = numpy.array(dat[z])[:,1]
# intensities
i = numpy.array(dat[z])[:,2]
# log intensities
i = numpy.log(i)
# create matrix
ity = numpy.diag(i)# / mean_di
# dimensionality
N = len(x)
d = numpy.zeros((N, D))
d[:,0] = x*x
d[:,1] = y*y
#d[:,2] = x*y
d[:,2] = x
d[:,3] = y
d[:,4] = numpy.ones(N)
#d[:,0] = x*x
#d[:,1] = y*y
#d[:,2] = x*y
#d[:,3] = x
#d[:,4] = y
#d[:,5] = numpy.ones(N)
print "old", d
# consider intensities
old_shape = d.shape
d = numpy.dot(ity, d)
print "new", d
assert d.shape == old_shape
print d.shape
d = cvxmod.matrix(d)
#### parameters
# da
X = cvxmod.param("X" + str(z), N, D)
X.value = d
X_matrix.append(X)
#### varibales
# parameter vector
theta = cvxmod.optvar("theta" + str(z), D)
thetas.append(theta)
# construct obj
objective = 0
# loss term
for i in xrange(M):
objective += cvxmod.atoms.norm1(X_matrix[i] * thetas[i])
# add smoothness regularization
reg_const = 5 * float(total_N) / float(M-1)
for i in xrange(M-1):
objective += reg_const * cvxmod.norm1(thetas[i] - thetas[i+1])
# create problem
prob = cvxmod.problem(cvxmod.minimize(objective))
# add constraints
"""
for (i,X) in enumerate(X_matrix):
p.constr.append(X*thetas[i] <= slacks[i])
p.constr.append(-X*thetas[i] <= slacks[i])
#eps = 0.5
#p.constr.append(slacks[i] - eps <= eps_slacks[i])
#p.constr.append(0 <= eps_slacks[i])
"""
# add non-degeneracy constraints
for i in xrange(1, M-1):
prob.constr.append(thetas[i][0] + thetas[i][1] == 1.0) # A + C = 1
# pinch ends
prob.constr.append(cvxmod.sum(thetas[0]) >= -0.01)
prob.constr.append(cvxmod.sum(thetas[-1]) >= -0.01)
print prob
###### set values
from cvxopt import solvers
solvers.options['reltol'] = 1e-1
solvers.options['abstol'] = 1e-1
print solvers.options
prob.solve()
# wrap up result
ellipse_stack = {}
active_layers = dat.keys()
assert len(active_layers) == M
# reconstruct original parameterization
for i in xrange(M):
theta_ = numpy.array(cvxmod.value(thetas[i]))
z_layer = active_layers[i]
ellipse_stack[z_layer] = conic_to_ellipse(theta_)
ellipse_stack[z_layer].cz = z_layer
return ellipse_stack
def solve_boosting(out, labels, nu, solver):
'''
solve boosting formulation used by gelher and novozin
@param out: matrix (N,F) of predictions (for each f_i) for all examples
@param y: vector (N,1) label for each example
@param p: regularization constant
'''
N = out.size[0]
F = out.size[1]
assert(N==len(labels))
norm_fact = 1.0 / (nu * float(N))
print norm_fact
label_matrix = cvxmod.zeros((N,N))
# avoid point-wise product
for i in xrange(N):
label_matrix[i,i] = labels[i]
#### parameters
f = cvxmod.param("f", N, F)
y = cvxmod.param("y", N, N, symm=True)
norm = cvxmod.param("norm", 1)
#### varibales
# rho
rho = cvxmod.optvar("rho", 1)
# dim = (N x 1)
chi = cvxmod.optvar("chi", N)
# dim = (F x 1)
beta = cvxmod.optvar("beta", F)
#objective = -rho + cvxmod.sum(chi) * norm_fact + square(norm2(beta))
objective = -rho + cvxmod.sum(chi) * norm_fact
print objective
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# create contraint for probability simplex
#p.constr.append(beta |cvxmod.In| probsimp(F))
p.constr.append(cvxmod.sum(beta)==1.0)
#p.constr.append(square(norm2(beta)) <= 1.0)
p.constr.append(beta >= 0.0)
# y f beta y f*beta y*f*beta
# (N x N) (N x F) (F x 1) --> (N x N) (N x 1) --> (N x 1)
p.constr.append(y * (f * beta) + chi >= rho)
###### set values
f.value = out
y.value = label_matrix
norm.value = norm_fact
p.solve(lpsolver=solver)
weights = numpy.array(cvxmod.value(beta))
#print weights
cvxmod.printval(chi)
cvxmod.printval(beta)
cvxmod.printval(rho)
return p
def solve_svm(out, labels, nu, solver):
'''
solve boosting formulation used by gelher and nowozin
@param out: matrix (N,F) of predictions (for each f_i) for all examples
@param labels: vector (N,1) label for each example
@param nu: regularization constant
@param solver: which solver to use. options: 'mosek', 'glpk'
'''
# get dimension
N = out.size[0]
F = out.size[1]
assert N == len(labels), str(N) + " " + str(len(labels))
norm_fact = 1.0 / (nu * float(N))
print "normalization factor %f" % (norm_fact)
# avoid point-wise product
label_matrix = cvxmod.zeros((N, N))
for i in xrange(N):
label_matrix[i, i] = labels[i]
#### parameters
f = cvxmod.param("f", N, F)
y = cvxmod.param("y", N, N, symm=True)
norm = cvxmod.param("norm", 1)
#### varibales
# rho
rho = cvxmod.optvar("rho", 1)
# dim = (N x 1)
chi = cvxmod.optvar("chi", N)
# dim = (F x 1)
beta = cvxmod.optvar("beta", F)
#objective = -rho + cvxmod.sum(chi) * norm_fact + square(norm2(beta))
objective = -rho + cvxmod.sum(chi) * norm_fact
print objective
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# create contraints for probability simplex
#p.constr.append(beta |cvxmod.In| probsimp(F))
p.constr.append(cvxmod.sum(beta) == 1.0)
p.constr.append(beta >= 0.0)
p.constr.append(chi >= 0.0)
# attempt to perform non-sparse boosting
#p.constr.append(square(norm2(beta)) <= 1.0)
# y f beta y f*beta y*f*beta
# (N x N) (N x F) (F x 1) --> (N x N) (N x 1) --> (N x 1)
p.constr.append(y * (f * beta) + chi >= rho)
# set values for parameters
f.value = out
y.value = label_matrix
norm.value = norm_fact
print "solving problem"
print "============================================="
print p
print "============================================="
# start solver
p.solve(lpsolver=solver)
# print variables
cvxmod.printval(chi)
cvxmod.printval(beta)
cvxmod.printval(rho)
return numpy.array(cvxmod.value(beta))
def fit_ellipse_linear(x, y):
"""
fit ellipse stack using absolute loss
"""
x = numpy.array(x)
y = numpy.array(y)
print "shapes", x.shape, y.shape
assert len(x) == len(y)
N = len(x)
D = 6
dat = numpy.zeros((N, D))
dat[:,0] = x*x
dat[:,1] = y*y
dat[:,2] = y*x
dat[:,3] = x
dat[:,4] = y
dat[:,5] = numpy.ones(N)
print dat.shape
dat = cvxmod.matrix(dat)
# norm
norm = numpy.zeros((N,N))
for i in range(N):
norm[i,i] = numpy.sqrt(numpy.dot(dat[i], numpy.transpose(dat[i])))
norm = cvxmod.matrix(norm)
#### parameters
# data
X = cvxmod.param("X", N, D)
Q_grad = cvxmod.param("Q_grad", N, N)
#### varibales
# parameter vector
theta = cvxmod.optvar("theta", D)
# dim = (N x 1)
s = cvxmod.optvar("s", N)
# simple objective
objective = cvxmod.sum(s)
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# add constraints
# (N x D) * (D X 1) = (N x N) * (N X 1)
p.constr.append(X*theta <= Q_grad*s)
p.constr.append(-X*theta <= Q_grad*s)
#p.constr.append(theta[4] == 1)
# trace constraint
p.constr.append(theta[0] + theta[1] == 1)
###### set values
X.value = dat
Q_grad.value = norm
#solver = "mosek"
#p.solve(lpsolver=solver)
p.solve()
cvxmod.printval(theta)
theta_ = numpy.array(cvxmod.value(theta))
ellipse = conic_to_ellipse(theta_)
return ellipse
def fit_ellipse_eps_insensitive(x, y):
"""
fit ellipse using epsilon-insensitive loss
"""
x = numpy.array(x)
y = numpy.array(y)
print "shapes", x.shape, y.shape
assert len(x) == len(y)
N = len(x)
D = 5
dat = numpy.zeros((N, D))
dat[:,0] = x*x
dat[:,1] = y*y
#dat[:,2] = y*x
dat[:,2] = x
dat[:,3] = y
dat[:,4] = numpy.ones(N)
print dat.shape
dat = cvxmod.matrix(dat)
#### parameters
# data
X = cvxmod.param("X", N, D)
# parameter for eps-insensitive loss
eps = cvxmod.param("eps", 1)
#### varibales
# parameter vector
theta = cvxmod.optvar("theta", D)
# dim = (N x 1)
s = cvxmod.optvar("s", N)
t = cvxmod.optvar("t", N)
# simple objective
objective = cvxmod.sum(t)
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# add constraints
# (N x D) * (D X 1) = (N X 1)
p.constr.append(X*theta <= s)
p.constr.append(-X*theta <= s)
p.constr.append(s - eps <= t)
p.constr.append(0 <= t)
#p.constr.append(theta[4] == 1)
# trace constraint
p.constr.append(theta[0] + theta[1] == 1)
###### set values
X.value = dat
eps.value = 0.0
#solver = "mosek"
#p.solve(lpsolver=solver)
p.solve()
cvxmod.printval(theta)
theta_ = numpy.array(cvxmod.value(theta))
ellipse = conic_to_ellipse(theta_)
return ellipse
def solve_boosting(out, labels, nu, solver):
'''
solve boosting formulation used by gelher and novozin
@param out: matrix (N,F) of predictions (for each f_i) for all examples
@param y: vector (N,1) label for each example
@param p: regularization constant
'''
N = out.size[0]
F = out.size[1]
assert(N==len(labels))
norm_fact = 1.0 / (nu * float(N))
print norm_fact
label_matrix = cvxmod.zeros((N,N))
# avoid point-wise product
for i in xrange(N):
label_matrix[i,i] = labels[i]
#### parameters
f = cvxmod.param("f", N, F)
y = cvxmod.param("y", N, N, symm=True)
norm = cvxmod.param("norm", 1)
#### varibales
# rho
rho = cvxmod.optvar("rho", 1)
# dim = (N x 1)
chi = cvxmod.optvar("chi", N)
# dim = (F x 1)
beta = cvxmod.optvar("beta", F)
#objective = -rho + cvxmod.sum(chi) * norm_fact + square(norm2(beta))
objective = -rho + cvxmod.sum(chi) * norm_fact
print objective
# create problem
p = cvxmod.problem(cvxmod.minimize(objective))
# create contraint for probability simplex
#p.constr.append(beta |cvxmod.In| probsimp(F))
p.constr.append(cvxmod.sum(beta)==1.0)
#p.constr.append(square(norm2(beta)) <= 1.0)
p.constr.append(beta >= 0.0)
# y f beta y f*beta y*f*beta
# (N x N) (N x F) (F x 1) --> (N x N) (N x 1) --> (N x 1)
p.constr.append(y * (f * beta) + chi >= rho)
###### set values
f.value = out
y.value = label_matrix
norm.value = norm_fact
p.solve(lpsolver=solver)
weights = numpy.array(cvxmod.value(beta))
#print weights
cvxmod.printval(chi)
cvxmod.printval(beta)
cvxmod.printval(rho)
return p
