def __init__(self, A, space, integrator, measure):
self.A = A
self.DL = tril(A).tocsr()
self.U = triu(A, k=1).tocsr()
self.DU = triu(A).tocsr()
self.L = tril(A, k=-1).tocsr()
linspace = LagrangeFiniteElementSpace(space.mesh, 1)
# construct amg solver for linear
A1 = stiff_matrix(linspace, integrator, measure)
isBdDof = linspace.boundary_dof()
bdIdx = np.zeros((A1.shape[0], ), np.int)
bdIdx[isBdDof] = 1
Tbd = spdiags(bdIdx, 0, A1.shape[0], A1.shape[0])
T = spdiags(1-bdIdx, 0, A1.shape[0], A1.shape[0])
A1 = T@A1@T + Tbd
self.ml = pyamg.ruge_stuben_solver(A1)
# Get interpolation matrix
NC = space.mesh.number_of_cells()
bc = space.dof.multiIndex/space.p
val = np.tile(bc, (NC, 1))
c2d0 = space.cell_to_dof()
c2d1 = linspace.cell_to_dof()
I = np.einsum('ij, k->ijk', c2d0, np.ones(3))
J = np.einsum('ik, j->ijk', c2d1, np.ones(len(bc)))
gdof = space.number_of_global_dofs()
lgdof = linspace.number_of_global_dofs()
self.PI = csr_matrix((val.flat, (I.flat, J.flat)), shape=(gdof, lgdof))
def split_matrix_by_relation(mat):
'''
split the sparse adjacency matrix by different relations
'''
length = mat.shape[0]
a, b, c, d = sp.triu(mat, 1), sp.triu(mat, 2), sp.tril(mat, -1), sp.tril(mat, -2)
rel_mats = [(a - b).tocoo(), (c - d).tocoo(), b, d]
return [rm.data for rm in rel_mats], \
[np.stack([rm.row, rm.col], axis=1) for rm in rel_mats], length
def main():
# create the matrix and the right hand sides
N = 1000
A = sp.coo_matrix(
hilbert(N) + np.identity(N)
) # a well-condition, symmetric, positive-definite matrix with off-diagonal entries
true_x1 = np.arange(N)
true_x2 = np.array(list(reversed(np.arange(N))))
b1 = A * true_x1
b2 = A * true_x2
# solve
solver = MumpsCentralizedAssembledLinearSolver()
x1, res = solver.solve(A, b1)
x2, res = solver.solve(A, b2)
assert np.allclose(x1, true_x1)
assert np.allclose(x2, true_x2)
# only perform factorization once
solver = MumpsCentralizedAssembledLinearSolver()
solver.do_symbolic_factorization(A)
solver.do_numeric_factorization(A)
x1, res = solver.do_back_solve(b1)
x2, res = solver.do_back_solve(b2)
assert np.allclose(x1, true_x1)
assert np.allclose(x2, true_x2)
# Tell Mumps the matrix is symmetric
# Note that the answer will be incorrect if both the lower
# and upper portions of the matrix are given.
solver = MumpsCentralizedAssembledLinearSolver(sym=2)
A_lower_triangular = sp.tril(A)
x1, res = solver.solve(A_lower_triangular, b1)
assert np.allclose(x1, true_x1)
# Tell Mumps the matrix is symmetric and positive-definite
solver = MumpsCentralizedAssembledLinearSolver(sym=1)
A_lower_triangular = sp.tril(A)
x1, res = solver.solve(A_lower_triangular, b1)
assert np.allclose(x1, true_x1)
# Set options
solver = MumpsCentralizedAssembledLinearSolver(
icntl_options={11: 2}) # compute error stats
solver.set_cntl(2,
1e-4) # set the stopping criteria for iterative refinement
solver.set_icntl(
10, 5
) # set the maximum number of iterations for iterative refinement to 5
x1, res = solver.solve(A, b1)
assert np.allclose(x1, true_x1)
# Get information after the solve
print('Number of iterations of iterative refinement performed: ',
solver.get_infog(15))
print('scaled residual: ', solver.get_rinfog(6))
def plotDispatchCurve(outData, eta, t, isoName):
chargeData = outData[0:t, ]
dischargeData = outData[t:2 * t, ]
A1 = hstack([eta * tril(np.ones([t, t])), -1 * tril(np.ones([t, t]))])
soc = A1 * outData
plt.plot(np.arange(0, t, 1), chargeData, 'r-', np.arange(0, t, 1),
dischargeData, 'b-', np.arange(0, t, 1), soc, 'g-')
plt.xlabel('Hour')
plt.legend(('Charging', 'Discharging', 'Stored Energy'))
plt.savefig(isoName + 'dispatchCurve.png')
plt.close()
def two_grid(A_, f_, al, be, N, max_iter, tol):
n = int((N - 1) / 2)
xc = np.linspace(0, 1, n + 2)[1:-1]
xf = np.linspace(0, 1, 2 * n + 1 + 2)[1:-1]
Ac = A_(n)
Af = A_(2 * n + 1)
def b_(f, n, al, be):
b = f
b[0] -= al * (n + 1)**2
b[-1] -= be * (n + 1)**2
return b
bf = b_(f_(xf, 0.5), n, 0, 0)
Icf = Icf_(n)
Ifc = Ifc_(n)
# Gauss-Seidel
Mc = sparse.tril(Ac, 0, format='csc')
Mf = sparse.tril(Af, 0, format='csc')
# jacobi
w = 3 / 2
Mc = w * sparse.diags(Ac.diagonal(), format='csc')
Mf = w * sparse.diags(Af.diagonal(), format='csc')
res_norm = []
uf = 0 * xf
rf = bf - Af.dot(uf)
for k in range(max_iter):
res_norm.append(np.linalg.norm(rf) / np.linalg.norm(bf))
if res_norm[-1] < tol:
return uf, k
# project to coarse grid
rc = Ifc.dot(rf)
# solve on coarse grid
zc = sparse.linalg.spsolve(Ac, rc)
# interpolate to fine grid and update fine solution
uf += Icf.dot(zc)
# smooth on fine grid
rf = bf - Af.dot(uf)
uf += sparse.linalg.spsolve_triangular(Mf, rf)
return uf, -1
def testProblemStiffness2(self):
width = 4
height = 4
[nodes, boundary_nodes, tris] = generateRectangularMesh((width, height), (0,0), (1,1))
p = Problem(nodes, boundary_nodes, tris)
K = p.getStiffnessMatrix(lambda x,y: x*y)
self.assertTrue((sparse.triu(K, 1).T.toarray() == sparse.tril(K,-1).toarray()).all())
width = 5
height = 5
[nodes, boundary_nodes, tris] = generateRectangularMesh((width, height), (0,0), (1,1))
p = Problem(nodes, boundary_nodes, tris)
K = p.getStiffnessMatrix(lambda x,y: x*y)
self.assertTrue((sparse.triu(K, 1).T.toarray() == sparse.tril(K,-1).toarray()).all())
def test_tril_behavior(self):
mat = get_base_matrix(use_tril=True)
mat2 = tril(mat)
self.assertTrue(np.all(mat.row == mat2.row))
self.assertTrue(np.all(mat.col == mat2.col))
self.assertTrue(np.allclose(mat.data, mat2.data))
mat = get_base_matrix_wrong_order(use_tril=True)
self.assertFalse(np.all(mat.row == mat2.row))
self.assertFalse(np.allclose(mat.data, mat2.data))
mat2 = tril(mat)
self.assertTrue(np.all(mat.row == mat2.row))
self.assertTrue(np.all(mat.col == mat2.col))
self.assertTrue(np.allclose(mat.data, mat2.data))
def __init__(self,H,g,A,b,l,u,x=None,lam=None,mu=None,pi=None):
"""
Quadratic program class.
Parameters
----------
H : symmetric matrix
g : vector
A : matrix
l : vector
u : vector
x : vector
"""
OptProblem.__init__(self)
self.H = coo_matrix(H)
self.Hphi = tril(self.H) # lower triangular
self.g = g
self.A = coo_matrix(A)
self.b = b
self.u = u
self.l = l
self.x = x
self.lam = lam
self.mu = mu
self.pi = pi
def texture_boundary(mesh, atex, val):
"""
compute indexes that are the boundary of a region defined by value
in a texture
:param mesh:
:param atex:
:param val:
:return:
"""
# see mesh.facets_boundary()
tex_val_indices = np.where(atex == val)[0]
if not tex_val_indices.size:
print('no value ' + str(val) + ' in the input texture!!')
return list()
else:
bound_verts = texture_boundary_vertices(atex, val,
mesh.vertex_neighbors)
# select the edges that are on the boundary in the polygons
adja = edges_to_adjacency_matrix(mesh)
adja_tri = sparse.triu(adja) + sparse.tril(adja).transpose()
r = sparse.extract.find(adja_tri)
inr0 = []
inr1 = []
for v in bound_verts:
inr0.extend(np.where(r[0] == v)[0])
inr1.extend(np.where(r[1] == v)[0])
r[2][inr0] = r[2][inr0] + 1
r[2][inr1] = r[2][inr1] + 1
li = r[0][np.where(r[2] == 4)]
lj = r[1][np.where(r[2] == 4)]
return edges_to_boundary(li, lj)
def ssor(A, b, x0=None, w=1., maxiter=200, tol=1E-6):
'''For symmetric matrices combine forward and backward SOR.'''
assert is_symmetric(A, tol=1E-6)
L, D, U = tril(A, k=-1), diags(A.diagonal(), 0), triu(A, k=1)
# Forward
MF = L + D/w
NF = (1/w - 1)*D - U
# Backward
MB = U + D/w
NB = (1/w - 1)*D - L
# Start from 0 initial guess
if x0 is None: x0 = np.zeros(A.shape[1])
r = b - A.dot(x0)
residuals = [np.linalg.norm(r)]
count = 0
while residuals[-1] > tol and count < maxiter:
# Update
x0 = spsolve(MF, NF.dot(x0) + b)
x0 = spsolve(MB, NB.dot(x0) + b)
# Error
r = b - A.dot(x0)
residuals.append(np.linalg.norm(r))
# Count
count += 1
converged = residuals[-1] < tol
n_iters = len(residuals) - 1
data = {'status': converged, 'iter count': n_iters, 'residuals': residuals}
return x0, data
def sparse_power_iteration(P, x, tol=10e-16, maxiter=200):
"""Preconditioned power iteration for a sparse stochastic matrix
Parameters
---------------
P : array, shape (n, n), sparse
transition matrix of a Markov Chain
x : array, shape (n, )
On entry, the initial guess. On exit, the final solution.
"""
t = 0
eps = tol + 1
n = P.shape[0]
# ILU factorization
LU = ilu0_factor(P)
L = sparse.tril(LU)
U = sparse.triu(LU)
# New matrix Q
Q = P.copy()
Q.setdiag(1 - Q.diagonal())
Q *= -1
Q = Q.T
info = -1
t = -1
for t in range(maxiter):
## dot() is matrix multiplication
dx = spla.spsolve(U, spla.spsolve(L, Q.matvec(x)))
x -= dx
relres = tvnorm(dx)
if relres < tol:
info = 0
break
t += 1
return (info, t, relres)
def make_graph(scores, threshold=True, k_closest=False, k=3):
print('==> Getting edges')
no_pts = len(scores)
if k_closest:
k_ = k
else:
k_ = 1 # ensure there is at least one edge per node in the threshold graph
edges = sparse.lil_matrix((no_pts, no_pts), dtype=np.uint8)
scores.fill_diagonal_(0) # get rid of self connection scores
for patient in range(no_pts):
k_highest = torch.sort(scores[patient].flatten()).indices[-k_:]
for i in k_highest:
edges[patient, i] = 1
del scores
edges = edges + edges.transpose() # make it symmetric again
# do upper triangle again and then save
edges = sparse.tril(edges, k=-1)
if threshold:
scores_lower = torch.tril(scores, diagonal=-1)
del scores
desired_no_edges = k * no_pts
threshold_value = torch.sort(scores_lower.flatten()).values[-desired_no_edges]
#for batch in batch(no_pts, n=10):
for batch in torch.split(scores_lower, 100, dim=0):
batch[batch < threshold_value] = 0
edges = edges + sparse.lil_matrix(scores_lower)
del scores_lower
v, u, _ = sparse.find(edges)
return u, v, k
def get_edge_list(self):
r"""Return an edge list, an alternative representation of the graph.
The weighted adjacency matrix is the canonical form used in this
package to represent a graph as it is the easiest to work with when
considering spectral methods.
Returns
-------
v_in : vector of int
v_out : vector of int
weights : vector of float
Examples
--------
>>> G = graphs.Logo()
>>> v_in, v_out, weights = G.get_edge_list()
>>> v_in.shape, v_out.shape, weights.shape
((3131,), (3131,), (3131,))
"""
if self.is_directed():
raise NotImplementedError('Directed graphs not supported yet.')
else:
v_in, v_out = sparse.tril(self.W).nonzero()
weights = self.W[v_in, v_out]
weights = np.asarray(weights).squeeze()
# TODO G.ind_edges = sub2ind(size(G.W), G.v_in, G.v_out)
assert self.Ne == v_in.size == v_out.size == weights.size
return v_in, v_out, weights
def eval_h(x, lagrange, obj_factor, flag, user_data=None):
"""Calculates the Hessian matrix (optional).
If omitted, set nnzh to 0 and Ipopt will use approximated Hessian
which will make the convergence slower.
"""
Hs = user_data['Hs']
if flag:
return (Hs.row, Hs.col)
else:
neqnln = user_data['neqnln']
niqnln = user_data['niqnln']
om = user_data['om']
Ybus = user_data['Ybus']
Yf = user_data['Yf']
Yt = user_data['Yt']
ppopt = user_data['ppopt']
il = user_data['il']
lam = {}
lam['eqnonlin'] = lagrange[:neqnln]
lam['ineqnonlin'] = lagrange[arange(niqnln) + neqnln]
H = opf_hessfcn(x, lam, om, Ybus, Yf, Yt, ppopt, il, obj_factor)
Hl = tril(H, format='csc')
## FIXME: Extend PyIPOPT to handle changes in sparsity structure
nnzh = Hs.nnz
Hd = zeros(nnzh)
for i in range(nnzh):
Hd[i] = Hl[Hs.row[i], Hs.col[i]]
return Hd
def _neighbour_list(graph, separator, secondary, attributes, **kwargs):
'''
Generate a string containing the neighbour list of the graph as well as a
dict containing the notifiers as key and the associated values.
@todo: speed this up!
'''
lst_neighbours = None
if graph.is_directed():
lst_neighbours = list(graph.adjacency_matrix(mformat="lil").rows)
else:
import scipy.sparse as ssp
lst_neighbours = list(
ssp.tril(graph.adjacency_matrix(), format="lil").rows)
for v1 in range(graph.node_nb()):
for i, v2 in enumerate(lst_neighbours[v1]):
str_edge = str(v2)
eattr = graph.get_edge_attributes((v1, v2))
for attr in attributes:
str_edge += "{}{}".format(secondary, eattr[attr])
lst_neighbours[v1][i] = str_edge
lst_neighbours[v1] = "{}{}{}".format(
v1, separator, separator.join(lst_neighbours[v1]))
str_neighbours = "\n".join(lst_neighbours)
return str_neighbours
def Jacobi(A, B, x, e):
'''Jacobi è¿ä»£æ–¹æ³•'''
T0 = time.time()
D = sparse.diags(A.diagonal())
L = -sparse.tril(A, -1)
U = -sparse.triu(A, 1)
print('Solving B_J')
B_J = sparse.linalg.inv(D) * (L + U)
print('Solved B_J')
x_next = x.copy()
times = 0
err_record = np.array([])
while True:
x_next = B_J * x + sparse.linalg.inv(D) * B
err = max(abs(x_next - x))
err_record = np.append(err_record, err)
times += 1
print('err = ', err)
x = x_next.copy()
if err < e:
break
elif times > 1000:
print('Times out of range')
break
T1 = time.time()
print('Iteration Times = ', times)
print('err = ', err)
print('CPU time = ', T1 - T0)
#print(A * x)
#print('X = ', x)
return x, err_record
def SOR(A, b, w, tol, max_iter=20, opt=False):
n = np.shape(A)[1]
x = sparse.csc_matrix((n, 1))
D = sparse.diags(A.diagonal(), format='csc')
L = sparse.tril(A, -1, format='csc')
M = D / w + L
if opt:
G = sparse.eye(n) - sparse.diags(1 / A.diagonal(), format='csc').dot(A)
p = np.linalg.norm(sparse.linalg.eigs(G, 1, which='LM')[0])
w = 2 / (1 + np.sqrt(1 - p**2))
print(w)
residual_norm = np.zeros(max_iter)
iter_tol = -1
for iter in range(max_iter):
residual = b - A.dot(x)
x += sparse.linalg.spsolve_triangular(M, residual)
residual_norm[iter] = np.linalg.norm(residual) / np.linalg.norm(b)
if residual_norm[iter] < tol:
iter_tol = iter
break
return (x, residual_norm, iter_tol)
def sor(A, b, x0=None, w=1., maxiter=200, tol=1E-6, direction='forward'):
'''
SOR iteration has M = L + D/w, N = (1/w-1)*D - U for forward
and M = U + D/w, N = (1/w-1)*D - L for bacward.
'''
L, D, U = tril(A, k=-1), diags(A.diagonal(), 0), triu(A, k=1)
if direction == 'forward':
M = L + D/w
N = (1/w - 1)*D - U
else:
M = U + D/w
N = (1/w - 1)*D - L
# Start from 0 initial guess
if x0 is None: x0 = np.zeros(A.shape[1])
r = b - A.dot(x0)
residuals = [np.linalg.norm(r)]
count = 0
while residuals[-1] > tol and count < maxiter:
# Update
x0 = spsolve(M, N.dot(x0) + b)
# Error
r = b - A.dot(x0)
residuals.append(np.linalg.norm(r))
# Count
count += 1
converged = residuals[-1] < tol
n_iters = len(residuals) - 1
data = {'status': converged, 'iter count': n_iters, 'residuals': residuals}
return x0, data
def deg_vec(W, deg_type='out', sparse_flag=True):
r"""
Create the degree vector
Parameters
----------
W : array
Adjacency matrix
deg_type : string
Degree type to use in case the graph is directed.
sparse_flag : bool
Use sparse matrices (True) or not (False).
"""
if is_directed(W):
if deg_type == 'in':
d = sparse.tril(W, k=0, format='csr').sum(
1) if sparse_flag else np.sum(np.tril(W), axis=1)
elif deg_type == 'out':
d = sparse.triu(W, k=0, format='csr').sum(
1) if sparse_flag else np.sum(np.triu(W), axis=1)
elif deg_type == 'average':
d = 0.5 * (np.sum(W, axis=0) + np.sum(W, axis=1))
else:
d = np.sum(W, axis=1)
return np.ravel(d)
def do_symbolic_factorization(self, matrix, raise_on_error=True):
if not isspmatrix_coo(matrix):
matrix = matrix.tocoo()
matrix = tril(matrix)
nrows, ncols = matrix.shape
self._dim = nrows
try:
self._mumps.do_symbolic_factorization(matrix)
self._prev_allocation = max(self.get_infog(16), self.get_icntl(23))
# INFOG(16) is the Mumps estimate for memory usage; ICNTL(23)
# is the override used in increase_memory_allocation. Both are
# already rounded to MB, so neither should every be negative.
except RuntimeError as err:
if raise_on_error:
raise err
stat = self.get_infog(1)
res = LinearSolverResults()
if stat == 0:
res.status = LinearSolverStatus.successful
elif stat in {-6, -10}:
res.status = LinearSolverStatus.singular
elif stat < 0:
res.status = LinearSolverStatus.error
else:
res.status = LinearSolverStatus.warning
return res
def do_numeric_factorization(self,
matrix,
raise_on_error=True,
timer=None):
if not isspmatrix_coo(matrix):
matrix = matrix.tocoo()
matrix = tril(matrix)
if (not np.array_equal(matrix.row, self._row)) or (not np.array_equal(
matrix.col, self._col)):
self.do_symbolic_factorization(matrix=matrix,
raise_on_error=raise_on_error,
timer=timer)
try:
self._mumps.do_numeric_factorization(matrix)
except RuntimeError as err:
if raise_on_error:
raise err
stat = self.get_infog(1)
res = LinearSolverResults()
if stat == 0:
res.status = LinearSolverStatus.successful
elif stat in {-6, -10}:
res.status = LinearSolverStatus.singular
elif stat in {-8, -9}:
res.status = LinearSolverStatus.not_enough_memory
elif stat < 0:
res.status = LinearSolverStatus.error
else:
res.status = LinearSolverStatus.warning
return res
def do_symbolic_factorization(self, matrix, raise_on_error=True):
if not isspmatrix_coo(matrix):
matrix = matrix.tocoo()
matrix = tril(matrix)
nrows, ncols = matrix.shape
self._dim = nrows
try:
self._mumps.do_symbolic_factorization(matrix)
self._prev_allocation = self.get_infog(16)
except RuntimeError as err:
if raise_on_error:
raise err
stat = self.get_infog(1)
res = LinearSolverResults()
if stat == 0:
res.status = LinearSolverStatus.successful
elif stat in {-6, -10}:
res.status = LinearSolverStatus.singular
elif stat < 0:
res.status = LinearSolverStatus.error
else:
res.status = LinearSolverStatus.warning
return res
def eval_h(x, lagrange, obj_factor, flag, user_data=None):
"""Calculates the Hessian matrix (optional).
If omitted, set nnzh to 0 and Ipopt will use approximated Hessian
which will make the convergence slower.
"""
Hs = user_data['Hs']
if flag:
return (Hs.row, Hs.col)
else:
neqnln = user_data['neqnln']
niqnln = user_data['niqnln']
om = user_data['om']
Ybus = user_data['Ybus']
Yf = user_data['Yf']
Yt = user_data['Yt']
ppopt = user_data['ppopt']
il = user_data['il']
lam = {}
lam['eqnonlin'] = lagrange[:neqnln]
lam['ineqnonlin'] = lagrange[arange(niqnln) + neqnln]
H = opf_hessfcn(x, lam, om, Ybus, Yf, Yt, ppopt, il, obj_factor)
Hl = tril(H, format='csc')
## FIXME: Extend PyIPOPT to handle changes in sparsity structure
nnzh = Hs.nnz
Hd = zeros(nnzh)
for i in range(nnzh):
Hd[i] = Hl[Hs.row[i], Hs.col[i]]
return Hd
def do_numeric_factorization(self, matrix, raise_on_error=True):
if not isspmatrix_coo(matrix):
matrix = matrix.tocoo()
matrix = tril(matrix)
nrows, ncols = matrix.shape
if nrows != ncols:
raise ValueError('Matrix must be square')
if nrows != self._dim:
raise ValueError(
'Matrix dimensions do not match the dimensions of '
'the matrix used for symbolic factorization')
stat = self._ma27.do_numeric_factorization(irn=matrix.row,
icn=matrix.col,
dim=self._dim,
entries=matrix.data)
res = LinearSolverResults()
if stat == 0:
res.status = LinearSolverStatus.successful
else:
if raise_on_error:
raise RuntimeError(
'Numeric factorization was not successful; return code: ' +
str(stat))
if stat in {-3, -4}:
res.status = LinearSolverStatus.not_enough_memory
elif stat in {-5, 3}:
res.status = LinearSolverStatus.singular
else:
res.status = LinearSolverStatus.error
self._num_status = res.status
return res
def diag_trim(mat, n):
"""
Trim an upper triangle sparse matrix so that only the first n diagonals
are kept.
Parameters
----------
mat : scipy.sparse.csr_matrix or numpy.array
The sparse matrix to be trimmed
n : int
The number of diagonals from the center to keep (0-based).
Returns
-------
scipy.sparse.dia_matrix or numpy.array:
The diagonally trimmed upper triangle matrix with only the first
n diagonal.
"""
if not sp.issparse(mat):
trimmed = mat.copy()
n_diags = trimmed.shape[0]
for diag in range(n, n_diags):
set_mat_diag(trimmed, diag, 0)
return trimmed
if mat.format != "csr":
raise ValueError("input type must be scipy.sparse.csr_matrix")
# Trim diagonals by removing all elements further than n in the
# upper triangle
trimmed = sp.tril(mat, n, format="csr")
trimmed = sp.triu(trimmed, format="csr")
return trimmed
def prune_adj(oriadj, non_zero_idx: int, percent: int):
original_prune_num = int(
((non_zero_idx - oriadj.size()[0]) / 2) * (percent / 100))
adj = SparseTensor.from_torch_sparse_coo_tensor(oriadj).to_scipy()
# find the lower half of the matrix
low_adj = tril(adj, -1)
non_zero_low_adj = low_adj.data[low_adj.data != 0]
low_pcen = np.percentile(abs(non_zero_low_adj), percent)
under_threshold = abs(low_adj.data) < low_pcen
before = len(non_zero_low_adj)
low_adj.data[under_threshold] = 0
non_zero_low_adj = low_adj.data[low_adj.data != 0]
after = len(non_zero_low_adj)
rest_pruned = original_prune_num - (before - after)
if rest_pruned > 0:
mask_low_adj = (low_adj.data != 0)
low_adj.data[low_adj.data == 0] = 2000000
flat_indices = np.argpartition(low_adj.data,
rest_pruned - 1)[:rest_pruned]
low_adj.data = np.multiply(low_adj.data, mask_low_adj)
low_adj.data[flat_indices] = 0
low_adj.eliminate_zeros()
new_adj = low_adj + low_adj.transpose()
new_adj = new_adj + sparse.eye(new_adj.shape[0])
return SparseTensor.from_scipy(new_adj).to_torch_sparse_coo_tensor().to(
device)
def do_symbolic_factorization(self, matrix, raise_on_error=True):
self._num_status = None
if not isspmatrix_coo(matrix):
matrix = matrix.tocoo()
matrix = tril(matrix)
nrows, ncols = matrix.shape
if nrows != ncols:
raise ValueError('Matrix must be square')
self._dim = nrows
stat = self._ma27.do_symbolic_factorization(dim=self._dim,
irn=matrix.row,
icn=matrix.col)
res = LinearSolverResults()
if stat == 0:
res.status = LinearSolverStatus.successful
else:
if raise_on_error:
raise RuntimeError(
'Symbolic factorization was not successful; return code: '
+ str(stat))
if stat in {-3, -4}:
res.status = LinearSolverStatus.not_enough_memory
elif stat in {-5, 3}:
res.status = LinearSolverStatus.singular
else:
res.status = LinearSolverStatus.error
return res
def setUp(self):
n = 50
nrhs = 20
self.A = sp.rand(n, n, 0.4) + sp.identity(n)
self.sol = np.ones((n, nrhs))
self.rhsU = sp.triu(self.A) * self.sol
self.rhsL = sp.tril(self.A) * self.sol
def test_directLower_python(self):
from pymatsolver import _ForwardSolver
ALinv = _ForwardSolver(sp.tril(self.A))
X = ALinv * self.rhsL
x = ALinv * self.rhsL[:,0]
self.assertLess(np.linalg.norm(self.sol-X,np.inf), TOL)
self.assertLess(np.linalg.norm(self.sol[:,0]-x,np.inf), TOL)
def cooler2csr(cooleruri):
'''
loads a cooler into a csr matrix
taken from HiCMatrix cool.py see also
https://github.com/deeptools/HiCMatrix/blob/master/hicmatrix/lib/cool.py
:param cooleruri: uri to a given cooler
:return: data in cooler as scipy.sparse.csr_matrix
'''
cooler_file = cooler.Cooler(cooleruri)
matrixDataFrame = cooler_file.matrix(balance=False,
sparse=True,
as_pixels=True)
used_dtype = np.int32
if np.iinfo(np.int32).max < cooler_file.info['nbins']:
used_dtype = np.int64
count_dtype = matrixDataFrame[0]['count'].dtype
data = np.empty(cooler_file.info['nnz'], dtype=count_dtype)
instances = np.empty(cooler_file.info['nnz'], dtype=used_dtype)
features = np.empty(cooler_file.info['nnz'], dtype=used_dtype)
i = 0
size = cooler_file.info['nbins'] // 32
if size == 0:
size = 1
start_pos = 0
while i < cooler_file.info['nbins']:
matrixDataFrameChunk = matrixDataFrame[i:i + size]
_data = matrixDataFrameChunk['count'].values.astype(count_dtype)
_instances = matrixDataFrameChunk['bin1_id'].values.astype(used_dtype)
_features = matrixDataFrameChunk['bin2_id'].values.astype(used_dtype)
data[start_pos:start_pos + len(_data)] = _data
instances[start_pos:start_pos + len(_instances)] = _instances
features[start_pos:start_pos + len(_features)] = _features
start_pos += len(_features)
i += size
del _data
del _instances
del _features
matrix = csr_matrix((data, (instances, features)),
shape=(np.int(cooler_file.info['nbins']),
np.int(cooler_file.info['nbins'])),
dtype=count_dtype)
del data
del instances
del features
gc.collect()
# filling lower triangle in case only upper triangle was saved
if tril(matrix, k=-1).sum() == 0:
# this case means that the lower triangle of the
# symmetric matrix (below the main diagonal)
# is zero. In this case, replace the lower
# triangle using the upper triangle
matrix = matrix + triu(matrix, 1).T
return matrix
def adjacency_split_naive(A, p_val, neg_mul=1, max_num_val=None):
edges = np.column_stack(sp.tril(A).nonzero())
num_edges = edges.shape[0]
num_val_edges = int(num_edges * p_val)
if max_num_val is not None:
num_val_edges = min(num_val_edges, max_num_val)
shuffled = np.random.permutation(num_edges)
which_val = shuffled[:num_val_edges]
which_train = shuffled[num_val_edges:]
train_ones = edges[which_train]
val_ones = edges[which_val]
A_train = sp.coo_matrix((np.ones_like(train_ones.T[0]), (train_ones.T[0], train_ones.T[1])),
shape=A.shape).tocsr()
A_train = A_train.maximum(A_train.T)
num_nodes = A.shape[0]
num_val_nonedges = neg_mul * num_val_edges
candidate_zeros = np.random.choice(np.arange(num_nodes, dtype=np.int32),
size=(2 * num_val_nonedges, 2), replace=True)
cne1, cne2 = candidate_zeros[:, 0], candidate_zeros[:, 1]
to_keep = (1 - A[cne1, cne2]).astype(np.bool).A1
val_zeros = candidate_zeros[to_keep][:num_val_nonedges]
if to_keep.sum() < num_val_nonedges:
raise ValueError("Couldn't produce enough non-edges")
return A_train, val_ones, val_zeros
def project_dot_onemode(superclass, biadjmatrix, onemode):
""" Get the weigthed adjacency matrix of the onemode graph by matrix multiplication """
t_start = time.time()
if onemode == "t":
wmatrix = np.dot(biadjmatrix, biadjmatrix.T)
elif onemode == "b":
wmatrix = np.dot(biadjmatrix.T, biadjmatrix)
print(
f"[Time] onemode-dotproduct {onemode} {time.time() - t_start:.3f} sec")
print(f"[Info] wmatrix {onemode} type {type(wmatrix)}")
print(f"[Info] wmatrix {onemode} dtype {wmatrix.dtype}")
print(
f"[Info] wmatrix {onemode} nbytes in GB {(wmatrix.data.nbytes + wmatrix.indptr.nbytes + wmatrix.indices.nbytes) / (1000 ** 3):.6f}"
)
print(f"[Info] wmatrix {onemode} shape {wmatrix.shape}")
print(f"[Info] wmatrix {onemode} maxelement {wmatrix.max()}")
count_nonzeroes = wmatrix.nnz
max_nonzeroes = wmatrix.shape[0] * (wmatrix.shape[0] - 1)
matrix_density = count_nonzeroes / max_nonzeroes
print(f"[Info] wmatrix {onemode} density {matrix_density:.4f}")
t_start = time.time()
wmatrix = sparse.tril(wmatrix, k=-1)
print(f"[Time] wmatrix {onemode} tril {time.time() - t_start:.3f} sec")
wmatrix = wmatrix.tocsr()
print(f"[Info] wmatrix {onemode} tril type {type(wmatrix)}")
print(
f"[Info] wmatrix {onemode} tril nbytes data in GB {(wmatrix.data.nbytes) / (1000 ** 3):.6f}"
)
t_start = time.time()
sparse.save_npz(f"out/{superclass}.{onemode}.npz", wmatrix)
print(f"[Time] save-npz {onemode} {time.time() - t_start:.3f} sec")
def test_topological_nodes(n=100):
g = dgl.DGLGraph()
a = sp.random(n, n, 3 / n, data_rvs=lambda n: np.ones(n))
b = sp.tril(a, -1).tocoo()
g.from_scipy_sparse_matrix(b)
layers_dgl = dgl.topological_nodes_generator(g)
adjmat = g.adjacency_matrix()
def tensor_topo_traverse():
n = g.number_of_nodes()
mask = F.copy_to(F.ones((n, 1)), F.cpu())
degree = F.spmm(adjmat, mask)
while F.reduce_sum(mask) != 0.:
v = F.astype((degree == 0.), F.float32)
v = v * mask
mask = mask - v
frontier = F.copy_to(F.nonzero_1d(F.squeeze(v, 1)), F.cpu())
yield frontier
degree -= F.spmm(adjmat, v)
layers_spmv = list(tensor_topo_traverse())
assert len(layers_dgl) == len(layers_spmv)
assert all(toset(x) == toset(y) for x, y in zip(layers_dgl, layers_spmv))
def support_dropout(adj, p, drop_edge=False):
assert 0.0 < p < 1.0
lower_adj = sp.tril(adj)
n_nodes = lower_adj.shape[0]
# find nodes to isolate
isolate = np.random.choice(range(n_nodes),
size=int(n_nodes * p),
replace=False)
s_idx, e_idx = lower_adj.nonzero()
# mask the nodes that have been selected
# here mask contains all the nodes that have been selected in isolated
# regardless whether it is source node or end node of an edge
mask = np.in1d(s_idx, isolate)
mask += np.in1d(e_idx, isolate)
# csr_matrix.data is the array storing the non-zero elements, it is usually much
# fewer than csr_matrix.shape[0] * csr_matrix.shape[1]
assert mask.shape[0] == lower_adj.data.shape[0]
lower_adj.data[mask] = 0
lower_adj.eliminate_zeros()
if drop_edge:
prob = np.random.uniform(0, 1, size=lower_adj.data.shape)
remove = prob < p
lower_adj.data[remove] = 0
lower_adj.eliminate_zeros()
lower_adj = lower_adj + lower_adj.transpose()
return lower_adj
def fast_solve(self):
self.precondieitoner()
tgdof = self.tensorspace.number_of_global_dofs()
vgdof = self.vectorspace.number_of_global_dofs()
gdof = tgdof + vgdof
start = timer()
print("Construting linear system ......!")
self.M, self.B = self.get_left_matrix()
S = self.B@spdiags(1/self.D, 0, tgdof, tgdof)@self.B.transpose()
self.SL = tril(S).tocsc()
self.SU = triu(S, k=1).tocsr()
self.SUT = self.SL.transpose().tocsr()
self.SLT = self.SU.transpose().tocsr()
b = self.get_right_vector()
AA = bmat([[self.M, self.B.transpose()], [self.B, None]]).tocsr()
bb = np.r_[np.zeros(tgdof), b]
end = timer()
print("Construct linear system time:", end - start)
start = timer()
P = LinearOperator((gdof, gdof), matvec=self.linear_operator)
x, exitCode = gmres(AA, bb, M=P, tol=1e-8)
print(exitCode)
end = timer()
print("Solve time:", end-start)
self.sh[:] = x[0:tgdof]
self.uh[:] = x[tgdof:]
def _lower_neighbors(dist_mat, max_scale):
"""
Converts a distance matrix to neighbor information.
Takes a square, possibly lower triangular, and returns a list
of lists of neighbor indices, for neighbors up to the specified scale.
Parameters
----------
dist_mat: 2D array
the distance matrix, which may be lower triangular
max_scale: float
the highest scale (distance) to consider
Returns
-------
neighbors: list of lists of int
"""
d = sp.lil_matrix(dist_mat)
d[d == 0] = sys.float_info.epsilon
d[np.diag_indices(d.shape[0])] = 0
d[d > max_scale] = 0
d = sp.tril(d)
result = [[] for i in range(d.shape[0])]
for k, v in np.transpose(d.nonzero()):
result[k].append(v)
return result
def nbr_idx_pairs(self):
"""
Returns a 2d array with 2 columns. Each row contains the node idxs of
a pair of neighbors in the graph. Each pair is only returned once, so
for example only one of (0,3) and (3,0) could appear as rows.
"""
return np.argwhere(sparse.tril(self.is_nbr))
def setUp(self):
n = 50
nrhs = 20
self.A = sp.rand(n, n, 0.4) + sp.identity(n)
self.sol = np.ones((n, nrhs))
self.rhsU = sp.triu(self.A) * self.sol
self.rhsL = sp.tril(self.A) * self.sol
def get_Qc(self,x0):
if hasattr(self.F, 'shape'):
FT = self.F.T
else:
FT = self.F
Q = sparse.tril(FT * self.P * self.F)
c = FT*(self.q + self.P*x0)
return Q,c
def is_tri(X):
diag = X.diagonal().sum()
if sparse.issparse(X):
if not (sparse.tril(X).sum() - diag) or \
not (sparse.triu(X).sum() - diag):
return True
elif not np.triu(X, 1).sum() or not np.tril(X, -1).sum():
return True
else:
return False
def _h(self, x, lagrange, obj_factor, flag, user_data=None):
if flag:
# return (self._Hrow, self._Hcol)
return (self._Hcol, self._Hrow)
else:
neqnln = user_data["neqnln"]
niqnln = user_data["niqnln"]
lmbda = {"eqnonlin": lagrange[:neqnln],
"ineqnonlin": lagrange[neqnln:neqnln + niqnln]}
H = tril(self._hessfcn(x, lmbda), format="coo")
return H.data
def triangular_lower(self, k=0):
"""
Returns the lower triangular portion of this matrix.
:param k:
- k = 0 corresponds to the main diagonal
- k > 0 is above the main diagonal
- k < 0 is below the main diagonal
TODO: Add unit tests
"""
return self._new_instance(sp.tril(self.matrix, k=k))
def gseidel(A, b, x, maxit=1000, tol=10e-13, relax=1., normalizer=None):
""" Gauss-Jacobi iterative linear solver for sparse matrices
Parameters
------------
A : sparse matrix, shape (n, n)
Left hand side of linear system.
b : array, array (n, )
Right hand side of linear system.
x : array, array (n, )
On entry, `x` holds the initial guess. On exit `x` holds the final solution.
tol : float, optional
Requested error tolerance for convergence.
maxit : int, optional
Maximum number of iterations.
relax : float, optional
Relaxation parameter. Default is 1 in Gauss-Seidel. Set
to values of less than or greater to 1 for under or over relaxation.
Returns
-----------
info : int
Exit status. 0 if converged. -1 if it did not.
iter : int
Number of iterations
relres : float
total variance norm of the final solution.
See Also
---------
gjacobi, sparse_power_iteration
Notes
--------
Code based on gseidel in the compecon Matlab toolbox.
"""
Q = sparse.tril(A).tocsr()
info = -1
for iter in range(maxit):
print iter
dx = spla.spsolve(Q, b - A.dot(x))
x += dx * relax
if normalizer:
normalizer(x)
relres = tvnorm(dx)
print relres, tol
if relres < tol:
info = 0
break
iter += 1
return (info, iter, relres)
def test_sparse_cholesky(A):
# Perform decompositions
np_chlsky = np.linalg.cholesky(A)
L = tril(csr_matrix(np_chlsky), format='csr')
# Check L is actually sparse!
assert len(L.data) < np.product(A.shape)
assert issparse(L)
# Test sparse implementation
py_chlsky = sparse_ldl.cholesky(A, L.indptr, L.indices)
# Check implementation of Cholesky above is the same as numpy
np.testing.assert_allclose(L.data, py_chlsky)
def __init__(self, A, *args, **kwargs):
"""Initialization routine for the smoother
Args:
A (scipy.sparse.csc_matrix): sparse matrix A of the system to solve
*args: Variable length argument list
**kwargs: Arbitrary keyword arguments
"""
super(GaussSeidel, self).__init__(A, *args, **kwargs)
self.P = sp.tril(self.A, format='csc')
# precompute inverse of the preconditioner for later usage
self.Pinv = sp.linalg.inv(self.P)
def test_sparse_modified_ldl(m, n):
A = sparse_rand(m, n, density=0.025).toarray()
A = np.dot(A, A.T)
# Perform decompositions
np_chlsky = cholesky(A)
D, L = modified_ldl(A)
py_chlsky = L*np.sqrt(D)
L = tril(csr_matrix(L), format='csr')
# Check L is actually sparse!
assert len(L.data) < np.product(A.shape)
assert issparse(L)
spD, spL = sparse_ldl.modified_ldl(A, L.indptr, L.indices)
# Check implementation of Cholesky above is the same as numpy
np.testing.assert_allclose(spD, D)
np.testing.assert_allclose(spL, L.data, atol=1e-7)
def __init__(self, relaxer_factory, fine_level, aggregate_index):
# Galerkin coarsening
self.P = util.caliber_one_interpolation(aggregate_index)
self.R = np.transpose(self.P)
nc = self.R.shape[0]
inv_agg_size = 1./np.sum(self.R, axis=1).flatten()
self.T = spdiags(inv_agg_size, 0, nc, nc) * self.R
self.A = self.R * fine_level.A * self.P
super(_CoarseLevel, self).__init__(relaxer_factory)
self.fine_level = fine_level
self.aggregate_index = aggregate_index
# Recover the graph and adjacency matrix from A.
# TODO: replace building W by the proper sub-edgelist of A (more efficient?)
self.W = -tril(self.A, k=-1) - triu(self.A, k=1)
self.g = nx.from_scipy_sparse_matrix(self.W)
def test_1d_fused_lasso(n=100):
l1 = 1.
sparsity1 = l1norm(n, l=l1)
D = (np.identity(n) - np.diag(np.ones(n-1),-1))[1:]
extra = np.zeros(n)
extra[0] = 1.
D = np.vstack([D,extra])
D = sparse.csr_matrix(D)
fused = seminorm(l1norm(D, l=l1))
X = np.random.standard_normal((2*n,n))
Y = np.random.standard_normal((2*n,))
regloss = squaredloss(X,Y)
p=regloss.add_seminorm(fused)
solver=FISTA(p)
solver.debug = True
vals1 = solver.fit(max_its=25000, tol=1e-12)
soln1 = solver.problem.coefs
B = np.array(sparse.tril(np.ones((n,n))).todense())
X2 = np.dot(X,B)
time.sleep(3)
D2 = np.diag(np.ones(n))
p2 = lasso.gengrad((X2, Y))
p2.assign_penalty(l1=l1)
opt = FISTA(p2)
opt.debug = True
opt.fit(tol=1e-12,max_its=25000)
beta = opt.problem.coefs
soln2 = np.dot(B,beta)
print soln1[range(10)]
print soln2[range(10)]
print p.obj(soln1), p.obj(soln2)
#np.testing.assert_almost_equal(soln1,soln2)
return vals1
def _solve(self, x0, A, l, u, xmin, xmax):
""" Solves using the Interior Point OPTimizer.
"""
# Indexes of constrained lines.
il = [i for i,ln in enumerate(self._ln) if 0.0 < ln.rate_a < 1e10]
nl2 = len(il)
neqnln = 2 * self._nb # no. of non-linear equality constraints
niqnln = 2 * len(il) # no. of lines with constraints
user_data = {"A": A, "neqnln": neqnln, "niqnln": niqnln}
self._f(x0)
Jdata = self._dg(x0, False, user_data)
# Hdata = self._h(x0, ones(neqnln + niqnln), None, False, user_data)
lmbda = {"eqnonlin": ones(neqnln),
"ineqnonlin": ones(niqnln)}
H = tril(self._hessfcn(x0, lmbda), format="coo")
self._Hrow, self._Hcol = H.row, H.col
n = len(x0) # the number of variables
xl = xmin
xu = xmax
gl = r_[zeros(2 * self._nb), -Inf * ones(2 * nl2), l]
gu = r_[zeros(2 * self._nb), zeros(2 * nl2), u]
m = len(gl) # the number of constraints
nnzj = len(Jdata) # the number of nonzeros in Jacobian matrix
nnzh = 0#len(H.data) # the number of non-zeros in Hessian matrix
f_fcn, df_fcn, g_fcn, dg_fcn, h_fcn = \
self._f, self._df, self._g, self._dg, self._h
nlp = pyipopt.create(n, xl, xu, m, gl, gu, nnzj, nnzh,
f_fcn, df_fcn, g_fcn, dg_fcn)#, h_fcn)
# print dir(nlp)
# nlp.str_option("print_options_documentation", "yes")
# nlp.int_option("max_iter", 10)
# x, zl, zu, obj = nlp.solve(x0)
success = nlp.solve(x0, user_data)
nlp.close()
def GaussSeidel(A, b, MAXITER, TOLL):
n = len(b)
xk = np.ones(shape = n,dtype = float)
D = sparse.diags(A.diagonal(), 0, format = 'csc',)
L = sparse.tril(A, format = 'csc')
U = sparse.triu(A, format = 'csc')
T = -(linalg.inv(D+L))* U
c = (linalg.inv(D+L))* b
i = 0
err = TOLL + 1
while i < MAXITER and err > TOLL:
x = T*xk + c
err = np.linalg.norm(x-xk, 1)/np.linalg.norm(x,1)
xk = x
i += 1
return xk, i
def __init__(self, block_problem, set_tril=[0], *args, **kwargs):
"""Initialization routine for the smoother
Args:
block_problem (scipy.sparse.csc_matrix): A BlockProblem object
*args: Variable length argument list
**kwargs: Arbitrary keyword arguments
"""
super(BlockGaussSeidel, self).__init__(block_problem, *args, **kwargs)
assert len(set_tril) <= block_problem.n_layer, "not enough layers, choose another dimensions to set diagonal"
assert min(set_tril) >= 0 and max(set_tril) < block_problem.n_layer, "set_tril is wrong"
As = block_problem.As
for a_row in As:
for i in set_tril:
a_row[i] = sp.tril(a_row[i], format='csc')
self.P = BlockProblemBase.generate_A(As)
self.Pinv = sp.linalg.inv(self.P)
def test_create_hint_mask_matrix(tensor, symmetric=False, keepConnections=False):
# use only atom to atom indices for hint connection prediction
atom_indices = np.zeros(tensor.getMatrixShape()[0])
atom_indices[tensor.getAtomIndices()] = 1
atom_vector = atom_indices[np.newaxis]
atom_vector = lil_matrix(atom_vector)
mask_matrix = atom_vector.multiply(atom_vector.T).tolil()
mask_matrix.setdiag(0)
# optionally exclude already existing connections from prediction
if not keepConnections:
connection_array = np.asarray(tensor.getSliceMatrix(SparseTensor.CONNECTION_SLICE).toarray())
connection_indices = connection_array > 0.0
mask_matrix[connection_indices] = 0
# symmetric mask needed?
if not symmetric:
mask_matrix = tril(mask_matrix)
return mask_matrix
def is_symmetric_sparse(mat):
from pyomo.contrib.pynumero.sparse.block_matrix import BlockMatrix
# Note: this check is expensive
flag = False
if isinstance(mat, np.ndarray):
flag = is_symmetric_dense(mat)
elif isscalarlike(mat):
flag = True
elif isspmatrix(mat) or isinstance(mat, BlockMatrix):
if mat.shape[0] != mat.shape[1]:
flag = False
else:
if isinstance(mat, BlockMatrix):
mat = mat.tocoo()
# get upper and lower triangular
l = tril(mat)
u = triu(mat)
diff = l - u.transpose()
z = np.zeros(diff.nnz)
flag = np.allclose(diff.data, z, atol=1e-6)
else:
raise RuntimeError("Format not recognized {}".format(type(mat)))
return flag
def test_1d_fused_lasso():
"""
Check the 1d fused lasso solution using an equivalent lasso formulation
"""
n = 100
l1 = 1.
D = (np.identity(n) - np.diag(np.ones(n-1),-1))[1:]
extra = np.zeros(n)
extra[0] = 1.
D = np.vstack([D,extra])
D = sparse.csr_matrix(D)
fused = R.l1norm.linear(D, lagrange=l1)
X = np.random.standard_normal((2*n,n))
Y = np.random.standard_normal((2*n,))
loss = R.quadratic.affine(X, -Y, coef=0.5)
fused_lasso = R.container(loss, fused)
solver=R.FISTA(fused_lasso)
vals1 = solver.fit(max_its=25000, tol=1e-10)
soln1 = solver.composite.coefs
B = np.array(sparse.tril(np.ones((n,n))).todense())
X2 = np.dot(X,B)
loss = R.quadratic.affine(X2, -Y, coef=0.5)
sparsity = R.l1norm(n, lagrange=l1)
lasso = R.container(loss, sparsity)
solver = R.FISTA(lasso)
solver.fit(tol=1e-10)
soln2 = np.dot(B, solver.composite.coefs)
npt.assert_array_almost_equal(soln1, soln2, 3)
def test_1d_fused_lasso(n=100):
l1 = 1.
sparsity1 = R.l1norm(n, lagrange=l1)
D = (np.identity(n) - np.diag(np.ones(n-1),-1))[1:]
extra = np.zeros(n)
extra[0] = 1.
D = np.vstack([D,extra])
D = sparse.csr_matrix(D)
fused = R.l1norm.linear(D, lagrange=l1)
X = np.random.standard_normal((2*n,n))
Y = np.random.standard_normal((2*n,))
loss = R.quadratic.affine(X, -Y, coef=0.5)
fused_lasso = R.container(loss, fused)
solver=R.FISTA(fused_lasso)
solver.debug = True
vals1 = solver.fit(max_its=25000, tol=1e-12)
soln1 = solver.composite.coefs
B = np.array(sparse.tril(np.ones((n,n))).todense())
X2 = np.dot(X,B)
def adj2vec(G):
r"""
Prepare the graph for the gradient computation.
Parameters
----------
G : Graph structure
"""
if not hasattr(G, 'directed'):
G.is_directed()
if G.directed:
raise NotImplementedError("Not implemented yet.")
else:
v_i, v_j = (sparse.tril(G.W)).nonzero()
weights = G.W[v_i, v_j]
# TODO G.ind_edges = sub2ind(size(G.W), G.v_in, G.v_out)
G.v_in = v_i
G.v_out = v_j
G.weights = weights
G.Ne = np.shape(v_i)[0]
def basic(A,pname=None,omega=1.0):
""" Jacobi, Gauss-Seidel, SOR and symmetric version
J, GS, SGS, SOR, SSOR
"""
n = A.shape[0]
if pname is None:
I = eye(n,n).tocsr()
L = I
U = I
elif pname=='J':
# weigthed Jacobi
I = eye(n,n).tocsr()
D = spdiags(A.diagonal(),0,n,n).tocsr()
#
L = omega * D
U = I
elif pname=='GS':
# Gauss Seidel
I = eye(n,n).tocsr()
D = spdiags(A.diagonal(),0,n,n).tocsr()
E = tril(A,-1).tocsr()
#
L = D + E
U = I
elif pname=='SGS':
# Symmetric Gauss-Seidel
D = spdiags(A.diagonal(),0,n,n).tocsr()
Dinv = spdiags(1/A.diagonal(),0,n,n).tocsr()
E = tril(A,-1).tocsr()
F = triu(A,1).tocsr()
#
L = (D+E)*Dinv
U = D+F
elif pname=='SOR':
# Successive Overrelaxation
I = eye(n,n).tocsr()
D = spdiags(A.diagonal(),0,n,n).tocsr()
E = tril(A,-1).tocsr()
#
L = D + omega * E
U = I
elif pname=='SSOR':
# Symmetric Successive Overrelaxation
D = spdiags(A.diagonal(),0,n,n).tocsr()
Dinv = spdiags(1/A.diagonal(),0,n,n).tocsr()
E = tril(A,-1).tocsr()
F = triu(A,1).tocsr()
#
L = (D+omega*E)*Dinv
U = D+omega*F
else:
print ">>>>>>>>>>>>> Problem with the preconditioner name..."
I = eye(n,n).tocsr()
L = I
U = I
L.sort_indices()
U.sort_indices()
return preconditioner_matvec(L,U)
def graph_sparsify(M, epsilon, maxiter=10):
r"""
Sparsify a graph using Spielman-Srivastava algorithm.
Parameters
----------
M : Graph or sparse matrix
Graph structure or a Laplacian matrix
epsilon : int
Sparsification parameter
Returns
-------
Mnew : Graph or sparse matrix
New graph structure or sparse matrix
Note
----
Epsilon should be between 1/sqrt(N) and 1
Examples
--------
>>> from pygsp import graphs, operators
>>> G = graphs.Sensor(256, Nc=20, distribute=True)
>>> epsilon = 0.4
>>> G2 = operators.graph_sparsify(G, epsilon)
Reference
---------
See :cite: `spielman2011graph` `rudelson1999random` `rudelson2007sampling`
for more informations
"""
# Test the input parameters
if isinstance(M, Graph):
if not M.lap_type == 'combinatorial':
raise NotImplementedError
L = M.L
else:
L = M
N = np.shape(L)[0]
if not 1./np.sqrt(N) <= epsilon < 1:
raise ValueError('GRAPH_SPARSIFY: Epsilon out of required range')
# pas sparse
resistance_distances = resistance_distance(L).toarray()
# Get the Weight matrix
if isinstance(M, Graph):
W = M.W
else:
W = np.diag(L.diagonal()) - L.toarray()
W[W < 1e-10] = 0
W = sparse.csc_matrix(W)
start_nodes, end_nodes, weights = sparse.find(sparse.tril(W))
# Calculate the new weights.
weights = np.maximum(0, weights)
Re = np.maximum(0, resistance_distances[start_nodes, end_nodes])
Pe = weights * Re
Pe = Pe / np.sum(Pe)
for i in range(maxiter):
# Rudelson, 1996 Random Vectors in the Isotropic Position
# (too hard to figure out actual C0)
C0 = 1 / 30.
# Rudelson and Vershynin, 2007, Thm. 3.1
C = 4 * C0
q = round(N * np.log(N) * 9 * C**2 / (epsilon**2))
results = stats.rv_discrete(values=(np.arange(np.shape(Pe)[0]), Pe)).rvs(size=q)
spin_counts = stats.itemfreq(results).astype(int)
per_spin_weights = weights / (q * Pe)
counts = np.zeros(np.shape(weights)[0])
counts[spin_counts[:, 0]] = spin_counts[:, 1]
new_weights = counts * per_spin_weights
sparserW = sparse.csc_matrix((new_weights, (start_nodes, end_nodes)),
shape=(N, N))
sparserW = sparserW + sparserW.T
sparserL = sparse.diags(sparserW.diagonal(), 0) - sparserW
if Graph(W=sparserW).is_connected():
break
elif i == maxiter - 1:
logger.warning('Despite attempts to reduce epsilon, sparsified graph is disconnected')
else:
epsilon -= (epsilon - 1/np.sqrt(N)) / 2.
if isinstance(M, Graph):
sparserW = sparse.diags(sparserL.diagonal(), 0) - sparserL
if not M.directed:
sparserW = (sparserW + sparserW.T) / 2.
Mnew = Graph(W=sparserW)
M.copy_graph_attributes(Mnew)
else:
Mnew = sparse.lil_matrix(sparserL)
return Mnew
def cluster(data, k=30, directed=False, prune=False, min_cluster_size=10, jaccard=True,
primary_metric='euclidean', n_jobs=-1, q_tol=1e-3, louvain_time_limit=2000,
nn_method='kdtree'):
"""
PhenoGraph clustering
:param data: Numpy ndarray of data to cluster, or sparse matrix of k-nearest neighbor graph
If ndarray, n-by-d array of n cells in d dimensions
If sparse matrix, n-by-n adjacency matrix
:param k: Number of nearest neighbors to use in first step of graph construction
:param directed: Whether to use a symmetric (default) or asymmetric ("directed") graph
The graph construction process produces a directed graph, which is symmetrized by one of two methods (see below)
:param prune: Whether to symmetrize by taking the average (prune=False) or product (prune=True) between the graph
and its transpose
:param min_cluster_size: Cells that end up in a cluster smaller than min_cluster_size are considered outliers
and are assigned to -1 in the cluster labels
:param jaccard: If True, use Jaccard metric between k-neighborhoods to build graph.
If False, use a Gaussian kernel.
:param primary_metric: Distance metric to define nearest neighbors.
Options include: {'euclidean', 'manhattan', 'correlation', 'cosine'}
Note that performance will be slower for correlation and cosine.
:param n_jobs: Nearest Neighbors and Jaccard coefficients will be computed in parallel using n_jobs. If n_jobs=-1,
the number of jobs is determined automatically
:param q_tol: Tolerance (i.e., precision) for monitoring modularity optimization
:param louvain_time_limit: Maximum number of seconds to run modularity optimization. If exceeded
the best result so far is returned
:param nn_method: Whether to use brute force or kdtree for nearest neighbor search. For very large high-dimensional
data sets, brute force (with parallel computation) performs faster than kdtree.
:return communities: numpy integer array of community assignments for each row in data
:return graph: numpy sparse array of the graph that was used for clustering
:return Q: the modularity score for communities on graph
"""
# NB if prune=True, graph must be undirected, and the prune setting takes precedence
if prune:
print("Setting directed=False because prune=True")
directed = False
if n_jobs == 1:
kernel = jaccard_kernel
else:
kernel = parallel_jaccard_kernel
kernelargs = {}
# Start timer
tic = time.time()
# Go!
if isinstance(data, sp.spmatrix) and data.shape[0] == data.shape[1]:
print("Using neighbor information from provided graph, rather than computing neighbors directly", flush=True)
lilmatrix = data.tolil()
d = np.vstack(lilmatrix.data).astype('float32') # distances
idx = np.vstack(lilmatrix.rows).astype('int32') # neighbor indices by row
del lilmatrix
assert idx.shape[0] == data.shape[0]
k = idx.shape[1]
else:
d, idx = find_neighbors(data, k=k, metric=primary_metric, method=nn_method, n_jobs=n_jobs)
print("Neighbors computed in {} seconds".format(time.time() - tic), flush=True)
subtic = time.time()
kernelargs['idx'] = idx
# if not using jaccard kernel, use gaussian
if not jaccard:
kernelargs['d'] = d
kernelargs['sigma'] = 1.
kernel = gaussian_kernel
graph = neighbor_graph(kernel, kernelargs)
print("Gaussian kernel graph constructed in {} seconds".format(time.time() - subtic), flush=True)
else:
del d
graph = neighbor_graph(kernel, kernelargs)
print("Jaccard graph constructed in {} seconds".format(time.time() - subtic), flush=True)
if not directed:
if not prune:
# symmetrize graph by averaging with transpose
sg = (graph + graph.transpose()).multiply(.5)
else:
# symmetrize graph by multiplying with transpose
sg = graph.multiply(graph.transpose())
# retain lower triangle (for efficiency)
graph = sp.tril(sg, -1)
# write to file with unique id
uid = uuid.uuid1().hex
graph2binary(uid, graph)
communities, Q = runlouvain(uid, tol=q_tol, time_limit=louvain_time_limit)
print("PhenoGraph complete in {} seconds".format(time.time() - tic), flush=True)
communities = sort_by_size(communities, min_cluster_size)
# clean up
for f in os.listdir():
if re.search(uid, f):
os.remove(f)
return communities, graph, Q
rowptr = self.A.indptr
colind = self.A.indices
x = np.empty_like(rhs)
for i in reversed(xrange(self.A.shape[0])):
ith_row = vals[rowptr[i] : rowptr[i+1]]
cols = colind[rowptr[i] : rowptr[i+1]]
x_vals = x[cols]
x[i] = (rhs[i] - np.dot(ith_row[1:], x_vals[1:])) / ith_row[0]
return x
_solve1 = _solveM
if __name__ == '__main__':
TOL = 1e-12
n = 30
A = sp.rand(n, n, 0.4) + sp.identity(n)
AL = sp.tril(A)
ALinv = ForwardSolver(AL)
e = np.ones((n,5))
rhs = AL * e
x = ALinv * rhs
print np.linalg.norm(e-x,np.inf), TOL
AU = sp.triu(A)
AUinv = BackwardSolver(AU)
e = np.ones((n,5))
rhs = AU * e
x = AUinv * rhs
print np.linalg.norm(e-x,np.inf), TOL
