def test_beta():
for matrix in [real_matrix(), complex_matrix()]:
for beta in [0, 1, 3.4]:
matrix_plus_beta = matrix + beta * sparse.eye(*matrix.shape)
for mode in ["auto", "supernodal", "simplicial"]:
L = cholesky(matrix, beta=beta).L()
assert factor_of(cholesky(matrix, beta=beta), matrix_plus_beta)
def __init__(self, verts, tris, m=1.0):
self._verts = verts
self._tris = tris
# precompute some stuff needed later on
e01 = verts[tris[:, 1]] - verts[tris[:, 0]]
e12 = verts[tris[:, 2]] - verts[tris[:, 1]]
e20 = verts[tris[:, 0]] - verts[tris[:, 2]]
self._triangle_area = .5 * veclen(np.cross(e01, e12))
unit_normal = normalized(np.cross(normalized(e01), normalized(e12)))
self._un = unit_normal
self._unit_normal_cross_e01 = np.cross(unit_normal, -e01)
self._unit_normal_cross_e12 = np.cross(unit_normal, -e12)
self._unit_normal_cross_e20 = np.cross(unit_normal, -e20)
# parameters for heat method
h = np.mean(map(veclen, [e01, e12, e20]))
t = m * h**2
# pre-factorize poisson systems
Lc, vertex_area = compute_mesh_laplacian(verts,
tris,
area_type='lumped_mass')
A = sparse.spdiags(vertex_area, 0, len(verts), len(verts))
#self._factored_AtLc = splu((A - t * Lc).tocsc()).solve
self._factored_AtLc = cholesky((A - t * Lc).tocsc(), mode='simplicial')
#self._factored_L = splu(Lc.tocsc()).solve
self._factored_L = cholesky(Lc.tocsc(), mode='simplicial')
def test_beta():
for matrix in [real_matrix(), complex_matrix()]:
for beta in [0, 1, 3.4]:
matrix_plus_beta = matrix + beta * sparse.eye(*matrix.shape)
for mode in ["auto", "supernodal", "simplicial"]:
L = cholesky(matrix, beta=beta).L()
assert factor_of(cholesky(matrix, beta=beta),
matrix_plus_beta)
def sparse_build_np_models(kernel, trans_samples, ter_samples, ter_rew_samples, lamb):
Xa, Ra, Xpa = zip(*trans_samples)
Xa_term, Ra_term = zip(*ter_samples)
Xa = [ np.vstack((xa, xa_term)) if xa_term.size > 0 else xa for xa, xa_term in izip(Xa, Xa_term) ]
Ra = [ np.hstack((ra, ra_term)) if ra_term.size > 0 else ra for ra, ra_term in izip(Ra, Ra_term) ]
k = len(trans_samples)
# build the K_a,b matrices
Kab = dict()
KabT = dict()
for a,b in product(xrange(k), xrange(k)):
if Xa_term[b].size > 0:
Kab[(a,b)] = np.hstack((kernel(Xa[a], Xpa[b]),
np.zeros((Xa[a].shape[0], Xa_term[b].shape[0]))))
else:
Kab[(a,b)] = kernel(Xa[a], Xpa[b])
Kab[(a,b)] = csr_matrix(Kab[(a,b)] * (np.abs(Kab[(a,b)]) > 1e-3))
KabT[(a,b)] = Kab[(a,b)].T.tocsr()
# build the K_a, D_a matrices
Ka = [kernel(Xa[i], Xa[i]) for i in xrange(k)]
Dainv = [csc_matrix(Ka[i] * (np.abs(Ka[i]) > 1e-3)) + lamb*scipy.sparse.eye(*Ka[i].shape) for i in xrange(k)]
# print np.linalg.matrix_rank(Dainv[2].toarray()), Dainv[2].shape, np.linalg.cond(Dainv[2].toarray())
# print np.linalg.eig(Dainv[2].toarray())[0]
# print [np.linalg.eig(Dainv[i].toarray())[0].min() for i in xrange(3)]
# plt.spy(Dainv[2].toarray())
# plt.show()
# print Dainv[2].shape
index = (squareform(pdist(Xa[2])) == 0.0).nonzero()
# print (squareform(pdist(Xa[2])) == 0.0).nonzero()
# print Xa[2][index[0][:5],:]
# print Xa[2][index[1][:5],:]
#
# splu(Dainv[0])
# cholesky(Dainv[0])
# splu(Dainv[1])
# cholesky(Dainv[1])
# splu(Dainv[2])
cholesky(Dainv[2])
Da= [cholesky(Dainv[i]) for i in xrange(k)]
# Da = [splu(Dainv[i]) for i in xrange(k)]
# build K_ter matrix
Kterma = [ np.hstack((kernel(ter_rew_samples[0], Xpa[i]),
np.zeros((ter_rew_samples[0].shape[0], Xa_term[i].shape[0])))) if Xa_term[i].size > 0
else kernel(ter_rew_samples[0], Xpa[i]) for i in xrange(k)]
K_ter = kernel(ter_rew_samples[0], ter_rew_samples[0])
D_ter = cholesky(csc_matrix(K_ter*(np.abs(K_ter) > 1e-3)) + lamb*scipy.sparse.eye(*K_ter.shape))
# D_ter = splu(csc_matrix(K_ter*(np.abs(K_ter) > 1e-3)) + lamb*scipy.sparse.eye(*K_ter.shape))
R_ter = ter_rew_samples[1]
return kernel, Kab, KabT, Da, Dainv, Ra, Kterma, D_ter, R_ter, Xa
def getModel(img,
var,
omat,
cmat,
rmat,
reg,
niter=10,
fixreg=False,
isin=None,
removeedges=True,
regmin=1e-5,
regacc=0.001):
from scikits.sparse.cholmod import cholesky
from scikits.sparse.cholmod import CholmodError
import numpy
if ((removeedges) & (isin != None)):
omat = omat.tocsr()[:, isin]
rmat = rmat[isin][:, isin]
rhs = omat.T * (img / var)
B = omat.T * cmat * omat
res = 0.
if fixreg == False:
regs = [reg]
else:
regs = [fixreg]
lhs = B + regs[-1] * rmat
F = cholesky(lhs)
fit = F(rhs)
if fixreg != False:
finalreg = fixreg
else:
finalreg = solveSPregularization(B,
rmat,
reg,
rhs,
niter=niter,
regmin=regmin,
regacc=regacc)
lhs = B + finalreg * rmat
F = cholesky(lhs)
fit = F(rhs)
model = omat * fit
return model, fit, finalreg, B
def get_potentials(conductivity_laplacian, io_index_pair):
"""
Recovers voltages based on the conductivity Laplacian and the IO array
:param conductivity_laplacian:
:param io_index_pair:
:return: array of potential in each node
"""
# TODO: technically, Cholesky is not the best solver. Change is needed, but in approximation
# it should be good enough
if switch_to_splu:
io_array = build_sink_source_current_array(
io_index_pair, conductivity_laplacian.shape, splu=True)
local_conductivity_laplacian = trim_matrix(conductivity_laplacian,
io_index_pair[1])
log.info('starting splu computation')
solver = splu(csc_matrix(local_conductivity_laplacian))
log.info('splu computation done')
return solver.solve(io_array.toarray())
else:
io_array = build_sink_source_current_array(
io_index_pair, conductivity_laplacian.shape)
solver = cholesky(csc_matrix(conductivity_laplacian), fudge)
return solver(io_array)
def test_spinv():
X = sparse.rand(500,500,density=0.03)
K = (X.T * X).tocsc() + sparse.identity(500)
for mode in ("simplicial", "supernodal"):
L = cholesky(K, mode=mode)
# Full inverse
inv_K = L.inv().todense()
for form in ("lower", "upper", "full"):
# Sparse inverse
spinv_K = L.spinv(form=form)
# 1) Check that the non-zero elements of spinv_K are
# correct
spinv_K = spinv_K.tocoo()
i = spinv_K.row
j = spinv_K.col
x = spinv_K.data
assert(np.allclose(x, inv_K[i,j]))
# 2) Check that the elements that are non-zero in K
# are correct in spinv_K
K = K.tocoo()
i = K.row
j = K.col
spinv_K = spinv_K.todense()
if form == "lower":
Z = np.tril(inv_K)
elif form == "upper":
Z = np.triu(inv_K)
else:
Z = inv_K
assert(np.allclose(spinv_K[i,j], Z[i,j]))
def getModel(img,var,lmat,pmat,cmat,rmat,reg,niter=10,onlyRes=False):
from scikits.sparse.cholmod import cholesky
import numpy
omat = pmat*lmat
rhs = omat.T*(img/var)
B = omat.T*cmat*omat
res = 0.
regs = [reg]
lhs = B+regs[-1]*rmat
F = cholesky(lhs)
fit = F(rhs)
for i in range(niter):
res = fit.dot(rmat*fit)
delta = reg*1e3
lhs2 = B+(reg+delta)*rmat
T = (2./delta)*(numpy.log(F.cholesky(lhs2).L().diagonal()).sum()-numpy.log(F.L().diagonal()).sum())
reg = (omat.shape[0]-T*reg)/res
if abs(reg-regs[-1])/reg<0.005:
break
regs.append(reg)
lhs = B+regs[-1]*rmat
F = F.cholesky(lhs)
fit = F(rhs)
print reg,regs
res = -0.5*res*regs[-1] + -0.5*((omat*fit-img)**2/var).sum()
if onlyRes:
return res,reg
model = (omat*fit)
return res,reg,fit,model
def test_complex():
c = complex_matrix()
fc = cholesky(c)
r = real_matrix()
fr = cholesky(r)
assert factor_of(fc, c)
assert np.allclose(fc(np.arange(4)),
(c.todense().I * np.arange(4)[:, np.newaxis]).ravel())
assert np.allclose(fc(np.arange(4) * 1j),
(c.todense().I * (np.arange(4) * 1j)[:, np.newaxis]).ravel())
assert np.allclose(fr(np.arange(4)),
(r.todense().I * np.arange(4)[:, np.newaxis]).ravel())
# If we did a real factorization, we can't do solves on complex arrays:
assert_raises(CholmodError, fr, np.arange(4) * 1j)
def test_spinv():
X = sparse.rand(500, 500, density=0.03)
K = (X.T * X).tocsc() + sparse.identity(500)
for mode in ("simplicial", "supernodal"):
L = cholesky(K, mode=mode)
# Full inverse
inv_K = L.inv().todense()
for form in ("lower", "upper", "full"):
# Sparse inverse
spinv_K = L.spinv(form=form)
# 1) Check that the non-zero elements of spinv_K are
# correct
spinv_K = spinv_K.tocoo()
i = spinv_K.row
j = spinv_K.col
x = spinv_K.data
assert (np.allclose(x, inv_K[i, j]))
# 2) Check that the elements that are non-zero in K
# are correct in spinv_K
K = K.tocoo()
i = K.row
j = K.col
spinv_K = spinv_K.todense()
if form == "lower":
Z = np.tril(inv_K)
elif form == "upper":
Z = np.triu(inv_K)
else:
Z = inv_K
assert (np.allclose(spinv_K[i, j], Z[i, j]))
def sample_mult_normal_given_precision(mu, precision, num_of_samples): from scikits.sparse import cholmod factor = cholmod.cholesky(precision) D = factor.D(); D = numpy.reshape(D, (len(D),1)) samples = factor.solve_Lt(numpy.random.normal(size=(len(mu),num_of_samples))/numpy.sqrt(D)) samples = numpy.repeat(mu, num_of_samples, axis=1)+ factor.apply_Pt(samples) return samples
def manifold_harmonics(verts,
tris,
K,
scaled=True,
return_D=False,
return_eigenvalues=False):
Q, vertex_area = compute_mesh_laplacian(verts,
tris,
'cotangent',
return_vertex_area=True,
area_type='lumped_mass')
if scaled:
D = sparse.spdiags(vertex_area, 0, len(verts), len(verts))
else:
D = sparse.spdiags(np.ones_like(vertex_area), 0, len(verts),
len(verts))
try:
lambda_dense, Phi_dense = eigsh(-Q, M=D, k=K, sigma=0)
except RuntimeError, e:
if e.message == 'Factor is exactly singular':
logging.warn(
"factor is singular, trying some regularization and cholmod")
chol_solve = cholesky(-Q + sparse.eye(Q.shape[0]) * 1.e-9,
mode='simplicial')
OPinv = sparse.linalg.LinearOperator(Q.shape, matvec=chol_solve)
lambda_dense, Phi_dense = eigsh(-Q, M=D, k=K, sigma=0, OPinv=OPinv)
else:
raise e
def test_complex():
c = complex_matrix()
fc = cholesky(c)
r = real_matrix()
fr = cholesky(r)
assert factor_of(fc, c)
assert np.allclose(fc(np.arange(4)),
(c.todense().I * np.arange(4)[:, np.newaxis]).ravel())
assert np.allclose(fc(np.arange(4) * 1j),
(c.todense().I *
(np.arange(4) * 1j)[:, np.newaxis]).ravel())
assert np.allclose(fr(np.arange(4)),
(r.todense().I * np.arange(4)[:, np.newaxis]).ravel())
# If we did a real factorization, we can't do solves on complex arrays:
assert_raises(CholmodError, fr, np.arange(4) * 1j)
def IRLS(Y, W, nore, reData, estCov, curr_bbeta):
N = W.shape[0]
nofe = W.shape[1] - nore
err = 1
logistic_input = W*curr_bbeta;
curr_pred = np.asarray(np.divide(np.exp(logistic_input),(1+np.exp(logistic_input))))
inv_curr_covar = invrecovar(reData, estCov); # Construct the covariance matrix b given the estimate of covariance matrix of random effects
hessian_from_logprior = block_diag((inv_curr_covar,np.zeros((nofe,nofe))))
start_time = time.time()
curr_b = curr_bbeta[np.arange(nore)]
while err>0.00001:
old_b = curr_b;
old_bbeta = curr_bbeta;
curr_gradient = -((W.T)*(Y-curr_pred) -sp.vstack(((inv_curr_covar*curr_b), np.zeros((nofe,1)))));
hessain_from_loglik = (W.T)*(sp.csr_matrix(((curr_pred*(1-curr_pred)).ravel(), (np.arange(N), np.arange(N)))))*W;
curr_hessian = hessain_from_loglik+hessian_from_logprior;
factor = cholesky(curr_hessian);
delta_bbeta = factor(curr_gradient);
delta_bbeta = np.reshape(delta_bbeta, (len(delta_bbeta),1));
curr_bbeta = curr_bbeta - delta_bbeta;
logistic_input = W*curr_bbeta;
curr_pred = np.asarray(np.divide(np.exp(logistic_input),(1+np.exp(logistic_input))));
curr_b = curr_bbeta[0:nore];
curr_beta = curr_bbeta[nore:];
err = np.sqrt(sum(np.square(curr_bbeta - old_bbeta))/len(old_bbeta));
#print repr(err)
pred_error = np.sqrt(sum(np.square(curr_pred - Y))/len(curr_pred)); # current prediction mean square error
print 'Current training RMSE = '+repr(pred_error);
hessian_from_loglik = (W.T)*(sp.csr_matrix(((curr_pred*(1-curr_pred)).ravel(), (np.arange(N), np.arange(N)))))*W;
curr_hessian = hessian_from_logprior+hessian_from_loglik;
curr_b = sp.csr_matrix(curr_b);
log_c1 = sp.csr_matrix((W*curr_bbeta).T)*Y - sum(np.log(1+np.exp(W*curr_bbeta))) - 0.5*curr_b.T*(inv_curr_covar)*(curr_b); # refer to sampling paper
# log_lik_part = - 0.5*inv_curr_covar.shape[1]*np.log(2*np.pi);
# for i in range(len(reData)):
# p = len(np.unique(reData[i][1]));
# (sign, logdet) = np.linalg.slogdet(estCov[i]);
# log_lik_part = log_lik_part - p*logdet;
# log_lik = log_c1 + log_lik_part;
factor1 = cholesky(curr_hessian);
factor2 = cholesky(inv_curr_covar);
log_lik = -factor1.logdet()+factor2.logdet();
return ([curr_bbeta, curr_pred, curr_hessian, log_c1, log_lik])
def logdet_cholmod(R,U,rows,cols, psd_tolerance=1e-6, factor=None):
from scikits.sparse.cholmod import cholesky
if not is_psd_cholmod(R, U, rows, cols, psd_tolerance, factor):
return -np.Inf
X = build_sparse(R, U, rows, cols)
# print factor
filled_factor = cholesky(X) if factor is None else factor.cholesky(X)
D = filled_factor.D()
return np.sum(np.log(D))
def cholsolve(A, b, fac=None):
if fac is None:
fac = cholmod.cholesky(A)
else:
# store a representation of the factorised matrix and update it in
# place
fac.cholesky_inplace(A)
x = fac.solve_A(b)
return x, fac
def block_cov_matrix(W, s):
"""
extract block inverse from W.
the indices of the block elements are specified in the array s
"""
A, B, C, D = extract_blocks(W, s)
f = cholesky(D)
w = A - B * f(C)
return np.linalg.inv(w.todense())
def is_psd_cholmod(R,U,rows,cols,tolerance=1e-6,factor=None):
X = build_sparse(R, U, rows, cols)
from scikits.sparse.cholmod import cholesky
try:
full_factor = cholesky(X) if factor is None else factor.cholesky(X)
except:
return False
D = full_factor.D()
return np.all(D>tolerance)
def adjust_pose_graph(x_hat, constraints):
num_of_nodes = len(x_hat)
jacobian = np.empty((3, 6), dtype=float)
F = 0
threshold = 0.1
while not (F < threshold):
hessian = np.empty((num_of_nodes * 3, num_of_nodes * 3), dtype=float)
info_vector = np.empty((num_of_nodes * 3), dtype=float)
F = 0
for c in constraints:
i = c[0]._prev_node._id
j = c[0]._post_node._id
node_i = x_hat[i]
node_j = x_hat[j]
# Following section is programmed with motion model
calc_jacobian(c, jacobian)
A_ij = jacobian[0:3, 0:3]
B_ij = jacobian[0:3, 3:6]
Sigma_ij = c[1]
# update H
hessian[i * 3:(i + 1) * 3,
i * 3:(i + 1) * 3] += np.dot(np.dot(A_ij.T, Sigma_ij),
A_ij)
hessian[i * 3:(i + 1) * 3,
j * 3:(j + 1) * 3] += np.dot(np.dot(A_ij.T, Sigma_ij),
B_ij)
hessian[j * 3:(j + 1) * 3,
i * 3:(i + 1) * 3] += np.dot(np.dot(B_ij.T, Sigma_ij),
A_ij)
hessian[j * 3:(j + 1) * 3,
j * 3:(j + 1) * 3] += np.dot(np.dot(B_ij.T, Sigma_ij),
B_ij)
# update b
info_vector[i * 3:(i + 1) * 3] += np.dot(np.dot(A_ij.T, Sigma_ij),
A_ij)
info_vector[j * 3:(j + 1) * 3] += np.dot(np.dot(A_ij.T, Sigma_ij),
A_ij)
hessian[0][0] += 10000 # to keep initial coordinates
factor = cholesky(hessian)
delta_x = factor(info_vector)
x_hat += delta_x
# calculate F(x) = e.T * Omega * e
error = (node_i.pose - node_j.pose) - c[0].edge.pose
F += np.dot(np.dot(error.T, c[1]), error)
def test_deprecation():
f = cholesky(sparse.eye(5, 5))
b = np.ones(5)
for dep_method in "solve_P", "solve_Pt":
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
getattr(f, dep_method)(b)
assert len(w) == 1
assert issubclass(w[-1].category, DeprecationWarning)
assert "deprecated" in str(w[-1].message)
def test_deprecation():
f = cholesky(sparse.eye(5, 5))
b = np.ones(5)
for dep_method in "solve_P", "solve_Pt":
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
getattr(f, dep_method)(b)
assert len(w) == 1
assert issubclass(w[-1].category, DeprecationWarning)
assert "deprecated" in str(w[-1].message)
def _random_correlated_image(mean, sigma, image_shape, alpha=0.3, rng=None):
"""
Creates a random image with correlated neighbors.
pixel covariance is sigma^2, direct neighors pixel covariance is alpha * sigma^2.
Parameters
----------
mean : the mean value of the image pixel values.
sigma : the std dev of image pixel values.
image_shape : tuple, shape = (3, )
alpha : the neighbors correlation factor.
rng : random number generator (a numpy.random.RandomState instance).
"""
dim_x, dim_y, dim_z = image_shape
dim_image = dim_x * dim_y * dim_z
correlated_image = 0
for neighbor in [(1, 0, 0), (0, 1, 0), (0, 0, 1)]:
corr_data = []
corr_i = []
corr_j = []
for i, j, k in [(0, 0, 0), neighbor]:
d2 = 1.0 * (i*i + j*j + k*k)
ind = np.asarray(np.mgrid[0:dim_x-i, 0:dim_y-j, 0:dim_z-k], dtype=np.int)
ind = ind.reshape((3, (dim_x - i) * (dim_y - j) * (dim_z - k)))
corr_i.extend(np.ravel_multi_index(ind, (dim_x, dim_y, dim_z)).tolist())
corr_j.extend(np.ravel_multi_index(ind + np.asarray([i, j, k])[:, None],
(dim_x, dim_y, dim_z)).tolist())
if i>0 or j>0 or k>0:
corr_i.extend(np.ravel_multi_index(ind + np.asarray([i, j, k])[:, None],
(dim_x, dim_y, dim_z)).tolist())
corr_j.extend(np.ravel_multi_index(ind, (dim_x, dim_y, dim_z)).tolist())
if i==0 and j==0 and k==0:
corr_data.extend([3.0] * ind.shape[1])
else:
corr_data.extend([alpha * 3.0] * 2 * ind.shape[1])
correlation = scisp.csc_matrix((corr_data, (corr_i, corr_j)), shape=(dim_image, dim_image))
factor = cholesky(correlation)
L = factor.L()
P = factor.P()[None, :]
P = scisp.csc_matrix((np.ones(dim_image),
np.vstack((P, np.asarray(range(dim_image))[None, :]))),
shape=(dim_image, dim_image))
sq_correlation = P.dot(L)
X = rng.normal(0, 1, dim_image)
Y = sq_correlation.dot(X)
Y = Y.reshape((dim_x, dim_y, dim_z))
X = X.reshape((dim_x, dim_y, dim_z))
correlated_image += Y
correlated_image /= 3
return correlated_image * sigma + mean
def test_spinv_complex():
# Complex matrices do not work yet, but this test checks that some
# reasonable error is produced
C = complex_matrix()
L = cholesky(C)
try:
# This should result CholmodError because spinv is not
# implemented for complex matrices - yet
invC = L.spinv()
assert (False)
except CholmodError:
assert (True)
def get_posterior_params(W, V, M, K_inv): """ INPUT: W, sparse matrix format, same dimension as K_inv V, a numpy vector of size K_inv.shape[0] M: the mean vector in numpy format K_inv: precision matrix in sparse form """ from scikits.sparse import cholmod posterior_precision = K_inv + W posterior_precision_factor = cholmod.cholesky(posterior_precision) posterior_mean = posterior_precision_factor(V + K_inv*M) return (posterior_mean, posterior_precision)
def test_spinv_complex():
# Complex matrices do not work yet, but this test checks that some
# reasonable error is produced
C = complex_matrix()
L = cholesky(C)
try:
# This should result CholmodError because spinv is not
# implemented for complex matrices - yet
invC = L.spinv()
assert(False)
except CholmodError:
assert(True)
def test_update_downdate():
m = real_matrix()
f = cholesky(m)
L = f.L()[f.P(), :]
assert factor_of(f, m)
f.update_inplace(L)
assert factor_of(f, 2 * m)
f.update_inplace(L)
assert factor_of(f, 3 * m)
f.update_inplace(L, subtract=True)
assert factor_of(f, 2 * m)
f.update_inplace(L, subtract=True)
assert factor_of(f, m)
def test_update_downdate():
m = real_matrix()
f = cholesky(m)
L = f.L()[f.P(), :]
assert factor_of(f, m)
f.update_inplace(L)
assert factor_of(f, 2 * m)
f.update_inplace(L)
assert factor_of(f, 3 * m)
f.update_inplace(L, subtract=True)
assert factor_of(f, 2 * m)
f.update_inplace(L, subtract=True)
assert factor_of(f, m)
def test_convenience():
A_dense_seed = np.array([[10, 0, 3, 0], [0, 5, 0, -2], [3, 0, 5, 0],
[0, -2, 0, 2]])
for mode in ("simplicial", "supernodal"):
for dtype in (float, complex):
A_dense = np.array(A_dense_seed, dtype=dtype)
A_sp = sparse.csc_matrix(A_dense)
f = cholesky(A_sp, mode=mode)
assert np.allclose(f.det(), np.linalg.det(A_dense))
assert np.allclose(f.logdet(), np.log(np.linalg.det(A_dense)))
assert np.allclose(f.slogdet(),
[1, np.log(np.linalg.det(A_dense))])
assert np.allclose((f.inv() * A_sp).todense(), np.eye(4))
def get_potentials(conductivity_laplacian, io_index_pair):
"""
Recovers voltages based on the conductivity Laplacian and the IO array
:param conductivity_laplacian:
:param io_index_pair:
:return: array of potential in each node
"""
# TODO: technically, Cholesky is not the best solver => change it if needed
solver = cholesky(csc_matrix(conductivity_laplacian), fudge)
io_array = build_sink_source_current_array(io_index_pair,
conductivity_laplacian.shape)
return solver(io_array)
def test_convenience():
A_dense_seed = np.array([[10, 0, 3, 0],
[0, 5, 0, -2],
[3, 0, 5, 0],
[0, -2, 0, 2]])
for mode in ("simplicial", "supernodal"):
for dtype in (float, complex):
A_dense = np.array(A_dense_seed, dtype=dtype)
A_sp = sparse.csc_matrix(A_dense)
f = cholesky(A_sp, mode=mode)
assert np.allclose(f.det(), np.linalg.det(A_dense))
assert np.allclose(f.logdet(), np.log(np.linalg.det(A_dense)))
assert np.allclose(f.slogdet(), [1, np.log(np.linalg.det(A_dense))])
assert np.allclose((f.inv() * A_sp).todense(), np.eye(4))
def __init__(self, verts, tris, m=1.0):
self._verts = verts
self._tris = tris
# precompute some stuff needed later on
e01 = verts[tris[:,1]] - verts[tris[:,0]]
e12 = verts[tris[:,2]] - verts[tris[:,1]]
e20 = verts[tris[:,0]] - verts[tris[:,2]]
self._triangle_area = .5 * veclen(np.cross(e01, e12))
unit_normal = normalized(np.cross(normalized(e01), normalized(e12)))
self._un = unit_normal
self._unit_normal_cross_e01 = np.cross(unit_normal, -e01)
self._unit_normal_cross_e12 = np.cross(unit_normal, -e12)
self._unit_normal_cross_e20 = np.cross(unit_normal, -e20)
# parameters for heat method
h = np.mean(map(veclen, [e01, e12, e20]))
t = m * h ** 2
# pre-factorize poisson systems
Lc, vertex_area = compute_mesh_laplacian(verts, tris, area_type='lumped_mass')
A = sparse.spdiags(vertex_area, 0, len(verts), len(verts))
#self._factored_AtLc = splu((A - t * Lc).tocsc()).solve
self._factored_AtLc = cholesky((A - t * Lc).tocsc(), mode='simplicial')
#self._factored_L = splu(Lc.tocsc()).solve
self._factored_L = cholesky(Lc.tocsc(), mode='simplicial')
def random_projection_cholmod_csc(X,k,factor=None): from scikits.sparse.cholmod import cholesky # TODO: add factor update full_factor = cholesky(X) if factor is None else factor.cholesky(X) (n,_) = X.shape D = full_factor.D() D12 = np.sqrt(D) Q = np.random.rand(k,n) Q[Q<0.5] = -1.0/np.sqrt(k) Q[Q>=0.5] = 1.0/np.sqrt(k) zA = (Q / D12).T zB = full_factor.solve_Lt(zA) A = full_factor.apply_Pt(zB).T return A
def test_cholesky_smoke_test():
f = cholesky(sparse.eye(10, 10) * 1.)
d = np.arange(20).reshape(10, 2)
print "dense"
assert np.allclose(f(d), d)
print "sparse"
s_csc = sparse.csc_matrix(np.eye(10)[:, :2] * 1.)
assert sparse.issparse(f(s_csc))
assert np.allclose(f(s_csc).todense(), s_csc.todense())
print "csr"
s_csr = s_csc.tocsr()
assert sparse.issparse(f(s_csr))
assert np.allclose(f(s_csr).todense(), s_csr.todense())
print "extract"
assert np.all(f.P() == np.arange(10))
def test_cholesky_smoke_test():
f = cholesky(sparse.eye(10, 10) * 1.)
d = np.arange(20).reshape(10, 2)
print "dense"
assert np.allclose(f(d), d)
print "sparse"
s_csc = sparse.csc_matrix(np.eye(10)[:, :2] * 1.)
assert sparse.issparse(f(s_csc))
assert np.allclose(f(s_csc).todense(), s_csc.todense())
print "csr"
s_csr = s_csc.tocsr()
assert sparse.issparse(f(s_csr))
assert np.allclose(f(s_csr).todense(), s_csr.todense())
print "extract"
assert np.all(f.P() == np.arange(10))
def solve (self, recover_cov=False):
n = len (self.nodes)
L = sparse.coo_matrix ((self._data, (self._rows,self._cols)),
shape=(self._Ns*n,self._Ns*n))
e = np.asarray (self._vector)
factor = cholmod.cholesky (L.tocsc ())
x = factor (e)
self.x = x.reshape (n,self._Ns)
if recover_cov:
S = factor (np.eye (self._Ns*n))
Smarginal = [S[ii:ii+2,ii:ii+2].ravel ().tolist () for ii in range (0,self._Ns*n,self._Ns)]
self.cov = np.asarray (Smarginal)
else: self.cov = np.zeros ((n,4))
def __init__(self, A, **kwargs):
"""CholeskyOperator corresponding to the sparse matrix `A`.
CholeskyOperator(A)
The sparse matrix `A` must be in one of the SciPy sparse
matrix formats.
"""
m, n = A.shape
if m != n:
raise ValueError("Input matrix must be square")
self.__factor = cholesky(A)
super(CholeskyOperator, self).__init__(n, n, matvec=self.cholesky_matvec, symmetric=True, **kwargs)
def test_solve_edge_cases():
m = real_matrix()
f = cholesky(m)
# sparse matrices give a sparse back:
assert sparse.issparse(f(sparse.eye(*m.shape).tocsc()))
# dense matrices give a dense back:
assert not sparse.issparse(f(np.eye(*m.shape)))
# 1d dense matrices are accepted and a 1d vector is returned (this matches
# the behavior of np.dot):
assert f(np.arange(m.shape[0])).shape == (m.shape[0],)
# 2d dense matrices are also accepted:
assert f(np.arange(m.shape[0])[:, np.newaxis]).shape == (m.shape[0], 1)
# But not if the dimensions are wrong...:
assert_raises(CholmodError, f, np.arange(m.shape[0] + 1)[:, np.newaxis])
assert_raises(CholmodError, f, np.arange(m.shape[0])[np.newaxis, :])
assert_raises(CholmodError, f, np.arange(m.shape[0])[:, np.newaxis, np.newaxis])
# And ditto for the sparse version:
assert_raises(CholmodError, f, sparse.eye(m.shape[0] + 1, m.shape[1]).tocsc())
def getModelG(img,var,mat,cmat,rmat=None,reg=None,niter=10):
from scikits.sparse.cholmod import cholesky
from scipy.sparse import hstack
import numpy
rhs = mat.T*(img/var)
B = fastMult(mat.T,cmat*mat)
if rmat is not None:
lhs = B+rmat*reg
regs = [reg]
else:
niter = 0
lhs = B
reg = 0
i = 0
import time
t = time.time()
F = cholesky(lhs)
print time.time()-t
t = time.time()
fit = F(rhs)
print time.time()-t
res = 0.
for i in range(niter):
res = fit.dot(rmat*fit)
delta = reg*1e-3
lhs2 = B+(reg+delta)*rmat
T = (2./delta)*(numpy.log(F.cholesky(lhs2).L().diagonal()).sum()-numpy.log(F.L().diagonal()).sum())
reg = abs(mat.shape[1]-T*reg)/res
if abs(reg-regs[-1])/reg<0.005:
break
regs.append(reg)
lhs = B+regs[-1]*rmat
F = F.cholesky(lhs)
fit = F(rhs)
res = -0.5*res*regs[-1]
res += -0.5*((mat*fit-img)**2/var).sum()
return res,fit,mat*fit,rhs,(reg,i)
def scipy_solver(A, M, is_hermitian):
'''Smallest eigenvalues'''
A, M = map(mat_to_csr, (A, M))
chol = cholesky(M)
Minv = LinearOperator(matvec=lambda x, mat=chol: mat.solve_A(x),
shape=M.shape)
if is_hermitian:
try:
eigw = eigsh(A, k=5, M=M, sigma=None, which='SM', v0=None, ncv=None,
maxiter=None, tol=1E-10, return_eigenvectors=False)
except ArpackError as e:
print '\tDivergence'
eigw = e.eigenvalues
else:
raise NotImplementedError
return eigw
def __init__(self, A, **kwargs):
"""CholeskyOperator corresponding to the sparse matrix `A`.
CholeskyOperator(A)
The sparse matrix `A` must be in one of the SciPy sparse
matrix formats.
"""
m, n = A.shape
if m != n:
raise ValueError("Input matrix must be square")
self.__factor = cholesky(A)
super(CholeskyOperator, self).__init__(n,
n,
matvec=self.cholesky_matvec,
symmetric=True,
**kwargs)
def manifold_harmonics(verts, tris, K, scaled=True, return_D=False, return_eigenvalues=False):
Q, vertex_area = compute_mesh_laplacian(
verts, tris, 'cotangent',
return_vertex_area=True, area_type='lumped_mass'
)
if scaled:
D = sparse.spdiags(vertex_area, 0, len(verts), len(verts))
else:
D = sparse.spdiags(np.ones_like(vertex_area), 0, len(verts), len(verts))
try:
lambda_dense, Phi_dense = eigsh(-Q, M=D, k=K, sigma=0)
except RuntimeError, e:
if e.message == 'Factor is exactly singular':
logging.warn("factor is singular, trying some regularization and cholmod")
chol_solve = cholesky(-Q + sparse.eye(Q.shape[0]) * 1.e-9, mode='simplicial')
OPinv = sparse.linalg.LinearOperator(Q.shape, matvec=chol_solve)
lambda_dense, Phi_dense = eigsh(-Q, M=D, k=K, sigma=0, OPinv=OPinv)
else:
raise e
def test_solve_edge_cases():
m = real_matrix()
f = cholesky(m)
# sparse matrices give a sparse back:
assert sparse.issparse(f(sparse.eye(*m.shape).tocsc()))
# dense matrices give a dense back:
assert not sparse.issparse(f(np.eye(*m.shape)))
# 1d dense matrices are accepted and a 1d vector is returned (this matches
# the behavior of np.dot):
assert f(np.arange(m.shape[0])).shape == (m.shape[0], )
# 2d dense matrices are also accepted:
assert f(np.arange(m.shape[0])[:, np.newaxis]).shape == (m.shape[0], 1)
# But not if the dimensions are wrong...:
assert_raises(CholmodError, f, np.arange(m.shape[0] + 1)[:, np.newaxis])
assert_raises(CholmodError, f, np.arange(m.shape[0])[np.newaxis, :])
assert_raises(CholmodError, f,
np.arange(m.shape[0])[:, np.newaxis, np.newaxis])
# And ditto for the sparse version:
assert_raises(CholmodError, f,
sparse.eye(m.shape[0] + 1, m.shape[1]).tocsc())
def compute_mfpt_symmetric(self, Peq):
"""make the matrix symmetric by multiplying both sides of the equation by Peq
sum_v Peq_u k_uv = -Peq_u
"""
intermediates = self.nodes - self.B
node_list = list(intermediates)
n = len(node_list)
matrix = scipy.sparse.dok_matrix((n,n))
node2i = dict([(node,i) for i, node in enumerate(node_list)])
right_side = -np.array([Peq[u] for u in node_list])
for iu, u in enumerate(node_list):
matrix[iu,iu] = -self.sum_out_rates[u] * Peq[u]
for uv, rate in self.rates.iteritems():
u, v = uv
if u in intermediates and v in intermediates:
ui = node2i[u]
vi = node2i[v]
assert ui != vi
matrix[ui,vi] = rate * Peq[u]
node_list = node_list
node2i = node2i
matrix = matrix.tocsc()
t0 = time.clock()
from scikits.sparse.cholmod import cholesky
factor = cholesky(matrix)
times = factor(right_side)
# times = scikits.sparse.spsolve(matrix, right_side,
# use_umfpack=True)
print "time solving symmetric linalg", time.clock() - t0
self.time_solve += time.clock() - t0
self.mfpt_dict = dict(((node, time) for node, time in izip(node_list, times)))
if np.any(times < 0):
raise LinalgError("error the mean first passage times are not all greater than zero")
return self.mfpt_dict
def sample_v(hessian, m_factor, nore, bbeta, Y, W, inv_curr_covar, log_c1, num_v_samples, num_parallel_loop, parallel_loop_flag):
##### b_opt is same as \theta*
# First we sample b's using cholesky decomposition of hessian
#### start of line 11
np.random.seed()
start_time = time.time()
warnings.filterwarnings("ignore")
factor = cholesky(hessian)
D = factor.D(); D = np.reshape(D, (len(D),1));
bbeta_propP = np.sqrt(m_factor)*factor.solve_Lt(np.random.normal(size=(W.shape[1],num_v_samples))/np.sqrt(D))
bbeta_prop = np.repeat(bbeta.toarray(), num_v_samples, axis=1)+ factor.apply_Pt(bbeta_propP)
bbeta_prop = sp.csr_matrix(bbeta_prop, dtype=np.float64)
##### end of line 11
##### Calculate v's in parallel given the \theta's (b_prop)
###### start of line 12
if parallel_loop_flag:
range_list = []
for i in range(num_parallel_loop):
range_list.append(np.arange(int(i*num_v_samples/num_parallel_loop), int((i+1)*num_v_samples/num_parallel_loop)))
bbeta_prop_list = []
for i in range(num_parallel_loop):
bbeta_prop_list.append(bbeta_prop[:,range_list[i]])
result = (Parallel(n_jobs=num_parallel_loop)(delayed(calc_vm)(W, Y, bbeta_prop_list[i], bbeta.toarray(), inv_curr_covar, hessian,log_c1, m_factor, nore) for i in range(num_parallel_loop)))
v_m = ();
for temp in result:
v_m = v_m+(temp,)
v_m = np.vstack(v_m)
v_m = v_m.ravel()
v_m = np.squeeze(np.asarray(v_m))
else:
result = calc_vm(W, Y, bbeta_prop, bbeta.toarray(), inv_curr_covar, hessian,log_c1, m_factor, nore)
v_m = np.asarray(result);
v_m = np.reshape(v_m, (len(v_m),))
############ End of line 12
return (v_m, bbeta_prop)
def get_potentials(conductivity_laplacian, io_index_pair):
"""
Recovers voltages based on the conductivity Laplacian and the IO array
:param conductivity_laplacian:
:param io_index_pair:
:return: array of potential in each node
"""
# TODO: technically, Cholesky is not the best solver. Change is needed, but in approximation
# it should be good enough
if switch_to_splu:
io_array = build_sink_source_current_array(io_index_pair, conductivity_laplacian.shape, splu=True)
local_conductivity_laplacian = trim_matrix(conductivity_laplacian, io_index_pair[1])
log.info('starting splu computation')
solver = splu(csc_matrix(local_conductivity_laplacian))
log.info('splu computation done')
return solver.solve(io_array.toarray())
else:
io_array = build_sink_source_current_array(io_index_pair, conductivity_laplacian.shape)
solver = cholesky(csc_matrix(conductivity_laplacian), line_loss)
return solver(io_array)
def getModel(img, var, lmat, pmat, cmat, rmat, reg, niter=10, onlyRes=False):
from scikits.sparse.cholmod import cholesky
import numpy
omat = pmat * lmat
rhs = omat.T * (img / var)
B = omat.T * cmat * omat
res = 0.
regs = [reg]
lhs = B + regs[-1] * rmat
F = cholesky(lhs)
fit = F(rhs)
for i in range(niter):
res = fit.dot(rmat * fit)
delta = reg * 1e3
lhs2 = B + (reg + delta) * rmat
T = (2. / delta) * (numpy.log(F.cholesky(lhs2).L().diagonal()).sum() -
numpy.log(F.L().diagonal()).sum())
reg = (omat.shape[0] - T * reg) / res
if abs(reg - regs[-1]) / reg < 0.005:
break
regs.append(reg)
lhs = B + regs[-1] * rmat
F = F.cholesky(lhs)
fit = F(rhs)
print reg, regs
res = -0.5 * res * regs[-1] + -0.5 * ((omat * fit - img)**2 / var).sum()
if onlyRes:
return res, reg
model = (omat * fit)
return res, reg, fit, model
#means = {'1':1,'3':3,'2':2}
#precisions={('1','2'):12,('2','3'):23,('1','1'):11,('3','3'):33,('2','2'):22}
means = {'1':1,'3':3,'2':2}
n = len(means)
precisions={('1','2'):1,('2','3'):1,('1','1'):2,('3','3'):2,('2','2'):2}
gmrf = GMRF(means, precisions)
sig_complete = inv(gmrf.precision.todense())
def fexact(idxs):
return np.sum(sig_complete[np.ix_(idxs, idxs)])
factor = cholesky(gmrf.precision)
D = factor.D()
D12 = np.sqrt(D)
#X = gmrf.precision.todense()
#P=np.eye(n)
#P=P[factor.P()].T
#L=factor.L().todense()
#D=np.diag(factor.D())
#X2 = P.dot(L).dot(L.T).dot(P.T)
#I = P.dot(inv(L.T)).dot(inv(D)).dot(inv(L)).dot(P.T)
def fexact2(idxs):
z = np.zeros(n)
z[idxs] = 1.0
z_perm = factor.solve_P(z)
C_inv = sp.bsr_matrix((Cp_inv_arr, indices, indptr), blocksize=(point_param, point_param)) indptr = np.arange(num_cameras+1) indices = np.arange(num_cameras) B = sp.bsr_matrix((Bp_arr, indices, indptr), blocksize=(cam_param, cam_param)) indices = E_dict['point_id'] u, indptr = np.unique(E_dict['cam_id'], return_index=True) indptr = np.hstack([indptr, len(indices)]) E = sp.bsr_matrix((E_arr, indices, indptr), blocksize=(cam_param, point_param)) EC_inv = E*C_inv S = B + EC_inv*E.T k = v - EC_inv*w factor = cholesky(S) y = factor(k)[:,0] z = C_inv * (w - E.T*y) camera_new = np.empty_like(cameras) points_new = np.empty_like(points) camera_new.flat = cameras.flat - y points_new.flat = points.flat - z new_error_sum = 0 for observation_id in range(num_observations): o = observations[observation_id] cam_id = o['cam_id'] point_id = o['point_id'] cam = camera_new[cam_id]
for number in range(1, 21):
print 'Domain {}:'.format(number)
B = get_matrix(
'./results/matrices_Z/domains_64_0_12.25_mass_{}.txt'.format(number))
A = get_matrix(
'./results/matrices_Z/domains_64_0_12.25_stiff_{}.txt'.format(number))
# To get eigenvalue about 12.25 we need to:
# * 2883 : integer entries in A, B
# / 128^2 : mesh of size 93 x 128 instead of 93/128 x 1
# We scale again to get integer entries after subtracting 12.25 + 3/256,
positive = 128 * 128 * 256 * A - (49 * 64 + 3) * 2883 * B
# 128^2 * A - (12.25 + 3/256) * 2883 * B
# everything * 256 to get integers
# cholesky factorization
factor = cholesky(positive)
diag = factor.D()
L = factor.L_D()[0]
P = factor.P()
print >>text, 'Number of entries: ', len(positive.data)
# the following should be positive
print >>text, "Smallest diagonal element in LDL': ", diag.min()
print >>text, 'Ratio largest/smallest : ', diag.max() / diag.min()
print >>text, 'Number of entries in L : ', len(L.data)
print >>text, 'Permutation : ', P
# save permuted positive
posP = positive[:, P].tocsr()[P].tocsc()
f = open('results/CHOLMOD_permuted/permuted{}.txt'.format(number), 'w')
M = posP.tocoo()
print >>f, zip(M.row, M.col, M.data)
def als_baseline_scikits_sparse(signal_input, smoothness_param=1e3, asym_param=1e-4, print_iteration=False):
"""
als_baseline_scikits_sparse(signal_input [,smoothness_param , asym_param]
Compute the baseline_current of signal_input using an asymmetric least squares
algorithm designed by P. H. C. Eilers. This implementation uses
CHOLMOD through the scikits.sparse toolkit wrapper.
Parameters
----------
signal_input : ndarray (1D)
smoothness_param : float, optional (default, 1e3)
Smoothness parameter
asym_param : float, optional (default, 1e-4)
Assymetry parameter
Returns
-------
out : ndarray
Baseline vector
Notes
-----
This is the first attempt at converting MATLAB (Mathworks, Inc)
scripts into Python code; thus, there will be bugs, the efficiency
will be low(-ish), and I appreciate any useful suggestions or
bug-finds.
References
----------
[1] P. H. C. Eilers, "A perfect smoother," Anal. Chem. 75,
3631-3636 (2003).
[2] P. H. C. Eilers and H. F. M. Boelens, "Baseline correction with
asymmetric least squares smoothing," Report. October 21, 2005.
[3] C. H. Camp Jr, Y. J. Lee, and M. T. Cicerone, "Quantitative,
Comparable Coherent Anti-Stokes Raman Scattering (CARS)
Spectroscopy: Correcting Errors in Phase Retrieval"
"""
signal_shape_orig = signal_input.shape
signal_length = signal_shape_orig[-1]
dim = _np.ndim(signal_input)
assert dim <= 3, "The input signal_input needs to be 1D, 2D, or 3D"
if dim == 1:
num_to_detrend = 1
elif dim == 2:
num_to_detrend = signal_input.shape[0]
else:
space_shp = list(signal_shape_orig)[0:-1]
num_to_detrend = signal_input.shape[0]*signal_input.shape[1]
baseline_output = _np.zeros(_np.shape(signal_input))
difference_matrix = _sparse.coo_matrix(_np.diff(_np.eye(signal_length),\
n=ORDER, axis=0))
# Iterate over spatial dimension (always 2D)
for count_spectra in range(num_to_detrend):
if dim == 1:
signal_current = signal_input
elif dim == 2:
signal_current = signal_input[count_spectra,:]
else:
# Calculate equivalent row- and column-count
rc, cc = _row_col_from_lin(count_spectra, space_shp)
signal_current = signal_input[rc, cc, :]
if count_spectra == 0:
penalty_vector = _np.ones(signal_length)
baseline_current = _np.zeros([signal_length])
baseline_last = _np.zeros([signal_length])
else: # Start with the previous spectral baseline to seed
penalty_vector = _np.squeeze(asym_param*(signal_current >=\
baseline_current)+(1-asym_param)*\
(signal_current < baseline_current))
# Iterative asymmetric least squares smoothing
for count_iterate in range(MAX_ITER):
penalty_matrix = _sparse.diags(penalty_vector, 0, shape=(signal_length,\
signal_length), format='csc')
minimazation_matrix = penalty_matrix + (smoothness_param*difference_matrix.T)*difference_matrix
# Cholesky factorization A = LL'
factor = _scikits_cholmod.cholesky(minimazation_matrix, beta=0, mode="auto")
if (count_iterate > 0 or count_spectra > 0):
baseline_last = baseline_current
# Solve A * baseline_current = penalty vector * Signal
baseline_current = factor.solve_A(penalty_vector[:]*signal_current)
# Difference check b/w iterations
if count_iterate > 0 or count_spectra > 0:
differ = _np.abs(_np.sum(baseline_current -\
baseline_last, axis=0))
if differ < MIN_DIFF:
break
# Apply asymmetric penalization
penalty_vector = _np.squeeze(asym_param*(signal_current >=\
baseline_current)+(1-asym_param)*\
(signal_current < baseline_current))
if dim == 1:
baseline_output = baseline_current
elif dim == 2:
baseline_output[count_spectra,:] = baseline_current
else:
baseline_output[rc, cc, :] = baseline_current
if print_iteration:
if dim < 3:
print('Finished detrending spectra {}/{}'.format(
count_spectra + 1, num_to_detrend))
elif rc + 1 == space_shp[0]:
print('Finished detrending spectra {}/{}'.format(
count_spectra + 1, num_to_detrend))
return baseline_output
def create_biharmonic_solver(self, boundary_verts, clip_D=0.1):
r"""Set up biharmonic equation with Dirichlet boundary conditions on the cortical
mesh and precompute Cholesky factorization for solving it. The vertices listed in
`boundary_verts` are considered part of the boundary, and will not be included in
the factorization.
To facilitate Cholesky decomposition (which requires a symmetric matrix), the
squared Laplace-Beltrami operator is separated into left-hand-side (L2) and
right-hand-side (Dinv) parts. If we write the L-B operator as the product of
the stiffness matrix (V-W) and the inverse mass matrix (Dinv), the biharmonic
problem is as follows (with `u` denoting non-boundary vertices)
.. math::
:nowrap:
\begin{eqnarray}
L^2_{u} \phi &=& -\rho_{u} \\
\left[ D^{-1} (V-W) D^{-1} (V-W) \right]_{u} \phi &=& -\rho_{u} \\
\left[ (V-W) D^{-1} (V-W) \right]_{u} \phi &=& -\left[D \rho\right]_{u}
\end{eqnarray}
Parameters
----------
boundary_verts : list or ndarray of length V
Indices of vertices that will be part of the Dirichlet boundary.
Returns
-------
lhs : sparse matrix
Left side of biharmonic problem, (V-W) D^{-1} (V-W)
rhs : sparse matrix, dia
Right side of biharmonic problem, D
Dinv : sparse matrix, dia
Inverse mass matrix, D^{-1}
lhsfac : cholesky Factor object
Factorized left side, solves biharmonic problem
notboundary : ndarray, int
Indices of non-boundary vertices
"""
try:
from scikits.sparse.cholmod import cholesky
factorize = lambda x: cholesky(x).solve_A
except ImportError:
factorize = sparse.linalg.dsolve.factorized
B, D, W, V = self.laplace_operator
npt = len(D)
g = np.nonzero(D > 0)[0] # Find vertices with non-zero mass
#g = np.nonzero((L.sum(0) != 0).A.ravel())[0] # Find vertices with non-zero mass
notboundary = np.setdiff1d(np.arange(npt)[g], boundary_verts) # find non-boundary verts
D = np.clip(D, clip_D, D.max())
Dinv = sparse.dia_matrix((D**-1,[0]), (npt,npt)).tocsr() # construct Dinv
L = Dinv.dot((V-W)) # construct Laplace-Beltrami operator
lhs = (V-W).dot(L) # construct left side, almost squared L-B operator
#lhsfac = cholesky(lhs[notboundary][:,notboundary]) # factorize
lhsfac = factorize(lhs[notboundary][:,notboundary]) # factorize
return lhs, D, Dinv, lhsfac, notboundary
def __init__(self, A):
self.n = A.shape[0]
self.factor = factor = cholmod.cholesky(A)
self.L = factor.L()
self.p = np.argsort(factor.P())
def master_edge_current(conductivity_laplacian,
index_list,
cancellation=True,
potential_dominated=True,
sampling=False,
sampling_depth=10,
memory_source=None,
memoization=None):
"""
master method for all the required edge current calculations
:param conductivity_laplacian:
:param index_list:
:param cancellation:
:param potential_dominated:
:param sampling:
:param sampling_depth:
:param memory_source:
:param memoization:
:return:
"""
# TODO: remove memoization in order to reduce overhead nad mongo database load
# generate index list in agreement with the sampling strategy
if sampling:
list_of_pairs = []
for _ in repeat(None, sampling_depth):
idx_list_c = copy(index_list)
random.shuffle(idx_list_c)
list_of_pairs += zip(idx_list_c[:len(idx_list_c) / 2],
idx_list_c[len(idx_list_c) / 2:])
else:
list_of_pairs = [(i, j) for i, j in combinations(set(index_list), 2)]
total_pairs = len(list_of_pairs)
up_pair_2_voltage_current = {}
current_accumulator = lil_matrix(conductivity_laplacian.shape)
# dump(csc_matrix(conductivity_laplacian), open('debug_for_cholesky.dmp', 'w'))
# log.exception('debug dump occured here!')
# raw_input('press enter to continue!')
solver = cholesky(csc_matrix(conductivity_laplacian), fudge)
# run the main loop on the list of indexes in agreement with the memoization strategy:
for counter, (i, j) in enumerate(list_of_pairs):
if counter % total_pairs / 100 == 0:
log.debug('getting pairwise flow %s out of %s', counter + 1,
total_pairs)
if memory_source and tuple(sorted((i, j))) in memory_source.keys():
potential_diff, current_upper = memory_source[tuple(sorted(
(i, j)))]
else:
potential_diff, current_upper = \
edge_current_iteration(conductivity_laplacian, (i, j),
solver=solver)
if memoization:
up_pair_2_voltage_current[tuple(sorted((i, j)))] = \
(potential_diff, current_upper)
# normalize to potential, if needed
if potential_dominated:
if potential_diff != 0:
current_upper = current_upper / potential_diff
# warn if potential difference is null or close to it
else:
log.warning(
'pairwise flow. On indexes %s %s potential difference is null. %s',
i, j, 'Tension-normalization was aborted')
current_accumulator += sparse_abs(current_upper)
if cancellation:
current_accumulator /= total_pairs
return current_accumulator, up_pair_2_voltage_current
def inv_mat(M):
fac = cholesky(M)
uni = sparse.coo_matrix(
((np.arange(M.shape[0]), np.arange(M.shape[0])), np.ones(M.shape[0])))
return fac(uni)
def _random_correlated_image(mean, sigma, image_shape, alpha=0.3, rng=None):
"""
Creates a random image with correlated neighbors.
pixel covariance is sigma^2, direct neighors pixel covariance is alpha * sigma^2.
Parameters
----------
mean : the mean value of the image pixel values.
sigma : the std dev of image pixel values.
image_shape : tuple, shape = (3, )
alpha : the neighbors correlation factor.
rng : random number generator (a numpy.random.RandomState instance).
"""
dim_x, dim_y, dim_z = image_shape
dim_image = dim_x * dim_y * dim_z
correlated_image = 0
for neighbor in [(1, 0, 0), (0, 1, 0), (0, 0, 1)]:
corr_data = []
corr_i = []
corr_j = []
for i, j, k in [(0, 0, 0), neighbor]:
d2 = 1.0 * (i * i + j * j + k * k)
ind = np.asarray(np.mgrid[0:dim_x - i, 0:dim_y - j, 0:dim_z - k],
dtype=np.int)
ind = ind.reshape((3, (dim_x - i) * (dim_y - j) * (dim_z - k)))
corr_i.extend(
np.ravel_multi_index(ind, (dim_x, dim_y, dim_z)).tolist())
corr_j.extend(
np.ravel_multi_index(ind + np.asarray([i, j, k])[:, None],
(dim_x, dim_y, dim_z)).tolist())
if i > 0 or j > 0 or k > 0:
corr_i.extend(
np.ravel_multi_index(ind + np.asarray([i, j, k])[:, None],
(dim_x, dim_y, dim_z)).tolist())
corr_j.extend(
np.ravel_multi_index(ind, (dim_x, dim_y, dim_z)).tolist())
if i == 0 and j == 0 and k == 0:
corr_data.extend([3.0] * ind.shape[1])
else:
corr_data.extend([alpha * 3.0] * 2 * ind.shape[1])
correlation = scisp.csc_matrix((corr_data, (corr_i, corr_j)),
shape=(dim_image, dim_image))
factor = cholesky(correlation)
L = factor.L()
P = factor.P()[None, :]
P = scisp.csc_matrix((np.ones(dim_image),
np.vstack(
(P, np.asarray(range(dim_image))[None, :]))),
shape=(dim_image, dim_image))
sq_correlation = P.dot(L)
X = rng.normal(0, 1, dim_image)
Y = sq_correlation.dot(X)
Y = Y.reshape((dim_x, dim_y, dim_z))
X = X.reshape((dim_x, dim_y, dim_z))
correlated_image += Y
correlated_image /= 3
return correlated_image * sigma + mean
#system.params.structural_quench_params.debug = True
#system.params.structural_quench_params.iprint = 100
db = system.create_database()
bh = system.get_basinhopping(db)
bh.run(1)
m = db.minima()[0]
coords = m.coords
potential = system.get_potential()
energy, gradient, hessian = potential.getEnergyGradientHessian(coords)
dummy_vec = ts.gramm_schmidt(ts.zeroEV_cluster(coords))
shifted_hess = hessian.copy()
for i in range(6):
shifted_hess += np.outer(dummy_vec[i], dummy_vec[i])
shifted_eval, shifted_evec = get_sorted_eig(shifted_hess)
print "First log sum: ", np.sum(np.log(shifted_eval[6:]))
sparse_hess = scipy.sparse.csc_matrix(shifted_hess)
factor = cholmod.cholesky(sparse_hess)
diagonal = np.diagonal(factor.L().todense())
logar = 2 * np.log(diagonal)
log_sum = np.sum(logar)
print "Second log sum: ", log_sum
A = np.eye(nmatrix)*10
A[3,1] = 4
A[1,3] = 4
#A[5,4] = 2
#A[4,5] = 2
print A
E, VV = np.linalg.eig(A)
print E
print VV
AA = csc_matrix(A)
Ls = []
factor = cholesky(AA)
L = factor.L()
for i in range(50):
#print i
factor = cholesky(AA)
L = factor.L()
AA = L.conjugate().transpose().dot(L)
Ls.append(L.conjugate().transpose())
print np.diag(AA.todense())
LL = np.zeros_like(Ls[0].todense())
LL = Ls[0].todense()
for L in Ls:
LL = np.dot(L.todense(),LL)
def test_cholesky_matrix_market():
for problem in ("well1033", "illc1033", "well1850", "illc1850"):
X = mm_matrix(problem)
y = mm_matrix(problem + "_rhs1")
answer = np.linalg.lstsq(X.todense(), y)[0]
XtX = (X.T * X).tocsc()
Xty = X.T * y
for mode in ("auto", "simplicial", "supernodal"):
assert np.allclose(cholesky(XtX, mode=mode)(Xty), answer)
assert np.allclose(cholesky_AAt(X.T, mode=mode)(Xty), answer)
assert np.allclose(cholesky(XtX, mode=mode).solve_A(Xty), answer)
assert np.allclose(
cholesky_AAt(X.T, mode=mode).solve_A(Xty), answer)
f1 = analyze(XtX, mode=mode)
f2 = f1.cholesky(XtX)
assert np.allclose(f2(Xty), answer)
assert_raises(CholmodError, f1, Xty)
assert_raises(CholmodError, f1.solve_A, Xty)
assert_raises(CholmodError, f1.solve_LDLt, Xty)
assert_raises(CholmodError, f1.solve_LD, Xty)
assert_raises(CholmodError, f1.solve_DLt, Xty)
assert_raises(CholmodError, f1.solve_L, Xty)
assert_raises(CholmodError, f1.solve_D, Xty)
assert_raises(CholmodError, f1.apply_P, Xty)
assert_raises(CholmodError, f1.apply_Pt, Xty)
f1.P()
assert_raises(CholmodError, f1.L)
assert_raises(CholmodError, f1.LD)
assert_raises(CholmodError, f1.L_D)
assert_raises(CholmodError, f1.L_D)
f1.cholesky_inplace(XtX)
assert np.allclose(f1(Xty), answer)
f3 = analyze_AAt(X.T, mode=mode)
f4 = f3.cholesky(XtX)
assert np.allclose(f4(Xty), answer)
assert_raises(CholmodError, f3, Xty)
f3.cholesky_AAt_inplace(X.T)
assert np.allclose(f3(Xty), answer)
print problem, mode
for f in (f1, f2, f3, f4):
pXtX = XtX.todense()[f.P()[:, np.newaxis],
f.P()[np.newaxis, :]]
assert np.allclose(np.prod(f.D()),
np.linalg.det(XtX.todense()))
assert np.allclose((f.L() * f.L().T).todense(), pXtX)
L, D = f.L_D()
assert np.allclose((L * D * L.T).todense(), pXtX)
b = np.arange(XtX.shape[0])[:, np.newaxis]
assert np.allclose(f.solve_A(b), np.dot(XtX.todense().I, b))
assert np.allclose(f(b), np.dot(XtX.todense().I, b))
assert np.allclose(f.solve_LDLt(b),
np.dot((L * D * L.T).todense().I, b))
assert np.allclose(f.solve_LD(b), np.dot((L * D).todense().I,
b))
assert np.allclose(f.solve_DLt(b),
np.dot((D * L.T).todense().I, b))
assert np.allclose(f.solve_L(b), np.dot(L.todense().I, b))
assert np.allclose(f.solve_Lt(b), np.dot(L.T.todense().I, b))
assert np.allclose(f.solve_D(b), np.dot(D.todense().I, b))
assert np.allclose(f.apply_P(b), b[f.P(), :])
assert np.allclose(f.solve_P(b), b[f.P(), :])
# Pt is the inverse of P, and argsort inverts permutation
# vectors:
assert np.allclose(f.apply_Pt(b), b[np.argsort(f.P()), :])
assert np.allclose(f.solve_Pt(b), b[np.argsort(f.P()), :])