def _check_symmetric_graph_laplacian(mat, normed, copy=True):
if not hasattr(mat, 'shape'):
mat = eval(mat, dict(np=np, sparse=sparse))
if sparse.issparse(mat):
sp_mat = mat
mat = sp_mat.toarray()
else:
sp_mat = sparse.csr_matrix(mat)
mat_copy = np.copy(mat)
sp_mat_copy = sparse.csr_matrix(sp_mat, copy=True)
n_nodes = mat.shape[0]
explicit_laplacian = _explicit_laplacian(mat, normed=normed)
laplacian = csgraph.laplacian(mat, normed=normed, copy=copy)
sp_laplacian = csgraph.laplacian(sp_mat, normed=normed, copy=copy)
if copy:
assert_allclose(mat, mat_copy)
_assert_allclose_sparse(sp_mat, sp_mat_copy)
else:
if not (normed and check_int_type(mat)):
assert_allclose(laplacian, mat)
if sp_mat.format == 'coo':
_assert_allclose_sparse(sp_laplacian, sp_mat)
assert_allclose(laplacian, sp_laplacian.toarray())
for tested in [laplacian, sp_laplacian.toarray()]:
if not normed:
assert_allclose(tested.sum(axis=0), np.zeros(n_nodes))
assert_allclose(tested.T, tested)
assert_allclose(tested, explicit_laplacian)
def _check_laplacian(A, desired_L, desired_d, normed, use_out_degree):
for arr_type in np.array, sparse.csr_matrix, sparse.coo_matrix:
for t in int, float, complex:
adj = arr_type(A, dtype=t)
L = csgraph.laplacian(adj, normed=normed, return_diag=False, use_out_degree=use_out_degree)
_assert_allclose_sparse(L, desired_L, atol=1e-12)
L, d = csgraph.laplacian(adj, normed=normed, return_diag=True, use_out_degree=use_out_degree)
_assert_allclose_sparse(L, desired_L, atol=1e-12)
_assert_allclose_sparse(d, desired_d, atol=1e-12)
def compute_laplacian(self, laplacian_type='simple', square=False):
am = nx.adjacency_matrix(self.graph)
if square:
am = am.dot(am)
if laplacian_type == 'simple':
return laplacian(am).asfptype()
elif laplacian_type == 'normalized':
return laplacian(am, normed=True).asfptype()
raise Exception('Unknown Laplacian type: {}'.format(laplacian_type))
def ler(X, Y, n_components=2, affinity='nearest_neighbors',
n_neighbors=None, gamma=None, mu=1.0, y_gamma=None,
eigen_solver='auto', tol=1e-6, max_iter=100,
random_state=None):
if eigen_solver not in ('auto', 'arpack', 'dense'):
raise ValueError("unrecognized eigen_solver '%s'" % eigen_solver)
nbrs = NearestNeighbors(n_neighbors=n_neighbors + 1)
nbrs.fit(X)
X = nbrs._fit_X
Nx, d_in = X.shape
Ny = Y.shape[0]
if n_components > d_in:
raise ValueError("output dimension must be less than or equal "
"to input dimension")
if Nx != Ny:
raise ValueError("X and Y must have same number of points")
if affinity == 'nearest_neighbors':
if n_neighbors >= Nx:
raise ValueError("n_neighbors must be less than number of points")
if n_neighbors == None or n_neighbors <= 0:
raise ValueError("n_neighbors must be positive")
elif affinity == 'rbf':
if gamma != None and gamma <= 0:
raise ValueError("n_neighbors must be positive")
else:
raise ValueError("affinity must be 'nearest_neighbors' or 'rbf' must be positive")
if Y.ndim == 1:
Y = Y[:, None]
if y_gamma is None:
dists = pairwise_distances(Y)
y_gamma = 1.0 / median(dists)
if affinity == 'nearest_neighbors':
affinity = kneighbors_graph(X, n_neighbors, include_self=True)
else:
if gamma == None:
dists = pairwise_distances(X)
gamma = 1.0 / median(dists)
affinity = kneighbors_graph(X, n_neighbors, mode='distance', include_self=True)
affinity.data = exp(-gamma * affinity.data ** 2)
K = rbf_kernel(Y, gamma=y_gamma)
lap = laplacian(affinity, normed=True)
lapK = laplacian(K, normed=True)
embedding, _ = null_space(lap + mu * lapK, n_components,
k_skip=1, eigen_solver=eigen_solver,
tol=tol, max_iter=max_iter,
random_state=random_state)
return embedding
def _isoParition(img_graph, ground=0, algCode='full', verbose=False):
"""Returns the isoperimetric parition.
"""
n_comp, dummy = connected_components(img_graph)
d = img_graph.sum(axis=1)
ground = np.argmax(d)
# Get the laplacian on which to calculate the solution based on algCode
if algCode == 'full':
img_laplacian = csr_matrix(laplacian(img_graph))
elif algCode == 'umst':
img_graph_umst = get_umst(img_graph)
img_laplacian = csr_matrix(laplacian(img_graph_umst))
elif algCode == 'mst' or 'mstGrady':
img_graph_mst = get_mst(img_graph)
img_laplacian = csr_matrix(laplacian(img_graph_mst))
else:
raise Exception("algCode should be one of {'full', 'umst', 'mst'. 'mstGrady'}")
# get the seeded laplacian
ind = np.arange(img_graph.shape[0], dtype = np.int32)
ind = np.hstack([ind[:ground], ind[(ground+1):]])
# Remove the row and column indicated by ground
img_laplacian_seeded = (img_laplacian[ind]).transpose()[ind]
# Solve the isoperimetric equation
d = np.ones(img_laplacian_seeded.shape[0], dtype=np.float64)
if algCode == 'mstGrady':
x0 = solve(img_laplacian,ground)
x0 = x0[ind]
else:
x0 = spsolve(img_laplacian_seeded, d)
minVal = np.min(x0)
if minVal < 0:
x0[x0<0] = np.max(x0) + 1
if verbose:
print("Error is {:4f}".format(norm(img_laplacian_seeded.dot(x0) - d)/norm(d)))
x0 = x0 - np.min(x0) + 1e-6
x0 = x0/np.max(x0) # Normalize to get values between [0,1]
# Get the total answer
ans = np.zeros(img_graph.shape[0], dtype=np.float64)
ans[ind]= x0
# Compute the threshold
img_laplacian = csr_matrix(laplacian(img_graph))
part1, part2, val = _compute_isoperimetric_partition(ans, img_laplacian)
return part1, part2, val, ans
def spectral_distance(G1, G2):
adj1 = nx.to_numpy_array(G1)
adj2 = nx.to_numpy_array(G2)
L1 = laplacian(adj1, normed=False)
L2 = laplacian(adj2, normed=False)
v1 = np.sort(eigvals(L1))
v2 = np.sort(eigvals(L2))
u1 = np.sort(eigvals(adj1))
u2 = np.sort(eigvals(adj2))
return np.abs(np.sqrt(np.sum(np.square(u1-u2)+np.square(v1-v2))))
def get_laplacian(A, type="unnormalized"):
"""Returns a sparse Laplacian from a given adjecancy matrix. It can either be of type unnormalized,
normalized or random-walk"""
if type == "unnormalized":
return laplacian(A, normed=False)
elif type == "normalized":
return laplacian(A, normed=True)
elif type == "randomwalk":
D_inv = spdiags(1.0 / A.sum(axis=1).flatten(), [0], A.shape[0], A.shape[0], format='csr')
return eye(A.shape[0], A.shape[0]) - np.dot(D_inv, A)
def _check_laplacian(A, desired_L, desired_d, normed, use_out_degree):
for arr_type in np.array, sparse.csr_matrix, sparse.coo_matrix:
for t in int, float, complex:
adj = arr_type(A, dtype=t)
L = csgraph.laplacian(adj, normed=normed, return_diag=False,
use_out_degree=use_out_degree)
_assert_allclose_sparse(L, desired_L, atol=1e-12)
L, d = csgraph.laplacian(adj, normed=normed, return_diag=True,
use_out_degree=use_out_degree)
_assert_allclose_sparse(L, desired_L, atol=1e-12)
_assert_allclose_sparse(d, desired_d, atol=1e-12)
def von_Neumann_distance(G1, G2):
A1 = __get_adjacency_sparse(G1)
A2 = __get_adjacency_sparse(G2)
L1 = csgraph.laplacian(A1, normed=False)
L2 = csgraph.laplacian(A2, normed=False)
L = L1 / L1.diagonal().sum() + L2 / L2.diagonal().sum()
S1 = __von_Neumann_entropy(L1)
S2 = __von_Neumann_entropy(L2)
S12 = __von_Neumann_entropy(L)
return np.sqrt(S12 - (S1 + S2) / 2)
def spectral_clustering(adj, sim, alpha, ncluster=10, v0=None):
"""
individually fair spectral clustering
:param adj: adjacency matrix
:param sim: similarity matrix
:param alpha: regularization parameter
:param ncluster: number of clusters
:param v0: starting vector for eigen-decomposition
:return: soft cluster membership matrix of fair spectral clustering
"""
lap = laplacian(adj) + alpha * laplacian(sim)
lap *= -1
_, u = eigsh(lap, which='LM', k=ncluster, sigma=1.0, v0=v0)
return u
def _spectral_embedding(self, affinity_matrix):
""" Computes spectral embedding.
First calculates normalized laplacian
Then does the eigenvalue decomposition
"""
numComponents, labels = connected_components(affinity_matrix)
if numComponents > 1:
# for each component figure out embedding, return the complete embedding
embedding = []
connected_component = np.zeros(affinity_matrix.shape)
for i in xrange(numComponents):
for j in affinity_matrix.shape[0]:
if labels[j] == i:
connected_component[:, i]
embedding.append(self._spectral_embedding(connected_component))
return embedding
self.n_components += 1
L, diag_vector = laplacian(affinity_matrix, normed=True, return_diag=True)
D = np.diag(diag_vector)
# eigvals, eigvects = eigsh(-L, k=self.n_components, sigma=1.0, which='LM')
eigvals, eigvects = eigh(L)
embedding = eigvects.T[: self.n_components] * diag_vector
return embedding[1 : self.n_components].T
def compute_tda_for_graphs(graph_folder, filtrations):
diag_repo = graph_folder + "diagrams/"
if os.path.exists(diag_repo) and os.path.isdir(diag_repo):
shutil.rmtree(diag_repo)
[os.makedirs(diag_repo + dtype) for dtype in [""] + graph_dtypes]
pad_size = 1
for graph_name in os.listdir(graph_folder + "mat/"):
A = np.array(loadmat(graph_folder + "mat/" + graph_name)["A"],
dtype=np.float32)
pad_size = np.max((A.shape[0], pad_size))
print("Pad size for eigenvalues in this dataset is: %i" % pad_size)
for graph_name in os.listdir(graph_folder + "mat/"):
A = np.array(loadmat(graph_folder + "mat/" + graph_name)["A"],
dtype=np.float32)
name = graph_name.split("_")
gid = int(name[name.index("gid") + 1]) - 1
egvals, egvectors = eigh(csgraph.laplacian(A, normed=True))
for filtration in filtrations:
time = float(filtration.split("-")[0])
filtration_val = np.square(egvectors).dot(
np.diag(np.exp(-time * egvals))).sum(axis=1)
dgmOrd0, dgmExt0, dgmRel1, dgmExt1 = apply_graph_extended_persistence(
A, filtration_val, get_base_simplex(A))
[
np.savetxt(diag_repo + "%s/graph_%06i_filt_%s.csv" %
(dtype, gid, filtration),
diag,
delimiter=',') for diag, dtype in
zip([dgmOrd0, dgmExt0, dgmRel1, dgmExt1], graph_dtypes)
]
return
def main(args: [str]) -> None:
'''
Generate eigenvalue distribution chart from graph <input-file>.
The input format of the graph, which consists of directed edges
[(a_1, b_1), ... (a_n, b_n)], where a_i, b_i are 0-indexed integers:
a_1 b_1
a_2 b_2
...
a_n b_n
The output will be generated in the <input-file>-eigval.png
'''
input_filename = args[0]
edges = []
max_node = 0
with open(input_filename, 'r') as inp:
for line in inp.readlines():
f, t = map(int, line.split())
edges.append((f, t))
max_node = max(max_node, f, t)
nodes = max_node + 1
mtx = np.zeros(shape=(nodes, nodes))
for f, t in edges:
mtx[f, t] += 1.0
mtx[t, f] += 1.0
laplacian_mtx = laplacian(mtx)
print(f'Laplacian: \n{laplacian_mtx}')
print(f'Nodes in graph: {nodes}')
vals = eigvals(laplacian_mtx)
print(f'Eigenvalues: {vals}')
plt.plot(sorted(vals))
# plt.show()
# plt.savefig()
plt.savefig(input_filename + '-eigval.svg')
def calc(self):
x, y, val = [], [], []
for item in self._wlist.get_all():
item.onlyFloatOn()
p = item.get_P()
if p > 800 and p < 1300:
x.append(item.get_lon())
y.append(item.get_lat())
val.append(item.get_P())
val = np.array(val)
lon = np.array(x)
lat = np.array(y)
self._xi_l = np.linspace(-179, 179, self._numcols)
self._yi_l = np.linspace(-88, 88, self._numrows)
xi, yi = np.meshgrid(self._xi_l, self._yi_l)
self._lon_lat = np.c_[lon.ravel(), lat.ravel()]
zi = griddata(self._lon_lat, val.ravel(), (xi, yi), method='linear')
self._lap = csgraph.laplacian(zi, normed=False)
# print(zi[56.3,45.4])
self._zi = zi
self._xi = xi
self._yi = yi
self._val = val
self._lon = lon
self._lat = lat
# print(self._lap)
return self
def RGG(node_num, dimension=2): RS=np.random.RandomState(seed=100) features=RS.uniform(low=0, high=1, size=(node_num, dimension)) adj_matrix=rbf_kernel(features, gamma=(1)/(2*(0.5)**2)) np.fill_diagonal(adj_matrix,0) laplacian=csgraph.laplacian(adj_matrix, normed=False) return adj_matrix, laplacian, features
def initializeGW(self,G, Gepsilon): n = len(self.users) L = csgraph.laplacian(G, normed = False) I = np.identity(n = G.shape[0]) GW = I + Gepsilon*L # W is a double stochastic matrix print 'GW', GW return GW.T
def initializeGW(self, G, Gepsilon):
n = len(self.users)
L = csgraph.laplacian(G, normed=False)
I = np.identity(n=G.shape[0])
GW = I + Gepsilon * L # W is a double stochastic matrix
print 'GW', GW
return GW.T
def _build_A(self, force=False):
s_dis = self.settings['s_scheme']
network = self.project.network
phase = self.project.phases()[self.settings['phase']]
conns = network['throat.conns']
P = phase[self.settings['pressure']]
gh = phase[self.settings['hydraulic_conductance']]
gd = phase[self.settings['diffusive_conductance']]
gd = np.tile(gd, 2)
Qij = -gh*np.diff(P[conns], axis=1).squeeze()
Qij = np.append(Qij, -Qij)
if force:
self._pure_A = None
if self._pure_A is None:
if (s_dis == 'upwind'):
w = gd + np.maximum(0, -Qij)
A = network.create_adjacency_matrix(weights=w)
elif (s_dis == 'hybrid'):
w = np.maximum(0, np.maximum(-Qij, gd-Qij/2))
A = network.create_adjacency_matrix(weights=w)
elif (s_dis == 'powerlaw'):
Peij = np.absolute(Qij/gd)
w = gd*np.maximum(0, (1-0.1*Peij)**5) + np.maximum(0, -Qij)
A = network.create_adjacency_matrix(weights=w)
A = laplacian(A)
self._pure_A = A
self.A = self._pure_A.copy()
def initializeGW0(self, Gepsilon):
G0 = self.W0
L = csgraph.laplacian(G0, normed=False)
I = np.identity(n=G0.shape[0])
GW0 = I + Gepsilon * L # W is a double stochastic matrix
print 'GW0', GW0
return GW0
def manifold_regularize(feature,predict,size):
alpha = 0.99
sigma = 0.2
dm = cdist(feature, feature, 'euclidean')
matrix = laplacian(dm, normed=False)
u_matrix = np.diag(matrix)
s_1 = np.dot(np.dot(np.transpose(predict),matrix),predict)
s_1_value = np.sum(np.diag(s_1))/size
manifold_value = np.full(size,s_1_value, dtype=np.float32)
# rbf = lambda x, sigma: math.exp((-predict)/(2*(math.pow(sigma,2))))
# vfunc = np.vectorize(rbf)
# W = vfunc(dm, sigma)
# np.fill_diagonal(W, 0)
# def calculate_S(W):
# d = np.sum(W, axis=1)
# D = np.sqrt(d*d[:, np.newaxis])
# return np.divide(W,D,where=D!=0)
# S = calculate_S(W)
# F = np.dot(S, predict)*alpha + (1-alpha)*predict
# n_iter = 400
# for t in range(n_iter):
# F = np.dot(S, F)*alpha + (1-alpha)*predict
# print('s_1_value',np.shape(manifold_value))
return manifold_value
def num_spanning_trees(A):
"""
Compute the number of spanning trees in an undirected graph
"""
from scipy.sparse import csgraph
L = csgraph.laplacian(A)
return np.linalg.det(L[1:, 1:])
def direct_compute_deepwalk_matrix(A, args, logger):
res = {}
try:
windows = args["prox_params"]['window']
b = args["prox_params"]['negative']
transform = args["prox_params"]['transform']
threshold = args["prox_params"]['threshold']
except KeyError:
raise MissingParamError
for window in windows:
n = A.shape[0]
vol = float(A.sum())
L, d_rt = csgraph.laplacian(A, normed=True, return_diag=True)
# X = D^{-1/2} A D^{-1/2}
X = sparse.identity(n) - L
S = np.zeros_like(X)
X_power = sparse.identity(n)
for i in range(window):
logger.info("Deep Walk %d-th power, %d/%d", i+1, i+1, window)
X_power = X_power.dot(X)
S += X_power
S *= vol / window / b
D_rt_inv = sparse.diags(d_rt ** -1)
M = D_rt_inv.dot(D_rt_inv.dot(S).T)
m = T.matrix()
if transform == 1:
logger.info("log transform")
res[window] = log_filter(M, threshold)
# res[window] = log_filter(M)
elif transform == 2:
res[window] = binary_filter(M, logger, threshold)
else:
logger.info("no transform")
res[window] = sparse.csr_matrix(M)
return res
def harmonic_function( W , fl ) : #calculating 'l' which indicates it is the number of labeled points l = len( fl ) print l #calculating 'n' total number of points n = len( W ) print n #calculating the laplacian W = np.matrix( W ) #G = np.arange(3) * np.arange(3)[:, np.newaxis] L = csgraph.laplacian( W , normed = False ) print len( L ), L[ 0 ] #calculating the harmonic function #fu = np.reshape( L , ( ) ) fl = np.matrix( fl ) fu = -( linalg.inv(L[(l):n,l:n]) ).dot( L[(l):n,0:l] ).dot( fl ) #compute the CMN solution #the unnormalized class proportion estimate from labeled data with laplace smoothing #q = sumColumn( fl ) + 1 #fu_CMN = fu * np.kron(numpy.ones((n-1,1)), q / sumColumn( fu ) ) #print fu return fu
def model_constrain(self, fixId, handleId, deformationMatrix):
#加入constrain
deformationMatrix_list = list(map(np.matrix, deformationMatrix))
self.deformedVerts = []
for vert in range(self.point_num):
if vert in fixId:
#固定点
self.deformedVerts.append((vert, self.vertex[vert]))
elif vert in handleId:
#先经过deformation matrix
deformedVert = np.append(self.vertex[vert], 1)
deformedVert = deformedVert.dot(
deformationMatrix_list[handleId.index(vert)])
deformedVert = np.delete(deformedVert, 3).flatten()
deformedVert = np.squeeze(np.asarray(deformedVert))
self.deformedVerts.append((vert, deformedVert))
#计算laplacian
process_num = len(self.deformedVerts)
self.laplacian = np.zeros(
(self.point_num + process_num, self.point_num + process_num),
dtype=np.float32)
self.laplacian[:self.point_num, :self.point_num] = csgraph.laplacian(
self.weight)
for i in range(process_num):
vert = self.deformedVerts[i][0]
newi = i + self.point_num
self.laplacian[newi, vert] = 1
self.laplacian[vert, newi] = 1
process_num = len(self.deformedVerts)
self.bArray = np.zeros((self.point_num + process_num, 3))
for i in range(process_num):
self.bArray[self.point_num + i] = self.deformedVerts[i][1]
self.compute_P()
def spectral_embedding(self, X, rad):
"""
Spectral Clustering
:param X: numpy.ndarray - input data matrix mxn , m data points with n dimensions
:param rad: float -radius for neighbor search
:return Y: numpy.ndarray - matrix m row, d attributes are reduced dimensional
"""
# Get the adjacency matrix/nearest neighbor graph; neighbors within the radius of 0.4
A = radius_neighbors_graph(
X.T,
rad,
mode="distance",
metric="minkowski",
p=2,
metric_params=None,
include_self=False,
)
A = A.toarray()
# Find the laplacian of the neighbour graph
# L = D - A ; where D is the diagonal degree matrix
L = csgraph.laplacian(A, normed=False)
# Embedd the data points i low dimension using the Eigen values/vectos
# of the laplacian graph to get the most optimal partition of the graph
eigval, eigvec = np.linalg.eig(L)
# the second smallest eigenvalue represents sparsest cut of the graph.
np.where(eigval == np.partition(eigval, 1)[1])
# Partition the graph using the smallest eigen value
y_spec = eigvec[:, 1].copy()
y_spec[y_spec < 0] = 0
y_spec[y_spec > 0] = 1
return y_spec
def compute_rfa(features,
k_neighbours=15,
distfn='sym',
connected=False,
sigma=1.0):
"""
Computes the target RFA similarity matrix. The RFA matrix of
similarities relates to the commute time between pairs of nodes, and it is
built on top of the Laplacian of a single connected component k-nearest
neighbour graph of the data.
"""
KNN = kneighbors_graph(features,
k_neighbours,
mode='distance',
include_self=False).toarray()
if 'sym' in distfn.lower():
KNN = np.maximum(KNN, KNN.T)
else:
KNN = np.minimum(KNN, KNN.T)
n_components, labels = csgraph.connected_components(KNN)
if connected and (n_components > 1):
from sklearn.metrics import pairwise_distances
distances = pairwise_distances(features, metric='euclidean')
KNN = connect_knn(KNN, distances, n_components, labels)
S = np.exp(-KNN / (sigma * features.size(1)))
S[KNN == 0] = 0
L = csgraph.laplacian(S, normed=False)
return torch.Tensor(np.linalg.inv(L + np.eye(L.shape[0])))
def HeatKernelSignature(adjacency_matrix, ntime_stamps=10):
"""
Compute node features as heat diffusion on graph
"""
# adjacency_matrix = np.array([[0,1,1,1,0,0,0,0,0,0,0,0],
# [1,0,1,1,0,0,0,0,0,0,0,0],
# [1,1,0,1,0,0,0,0,0,0,0,0],
# [1,1,1,0,1,1,0,0,0,0,0,0],
# [0,0,0,1,0,1,0,1,0,0,0,0],
# [0,0,0,1,1,0,1,0,0,1,1,1],
# [0,0,0,0,0,1,0,1,1,0,0,0],
# [0,0,0,0,1,0,1,0,1,0,0,0],
# [0,0,0,0,0,0,1,1,0,0,0,0],
# [0,0,0,0,0,1,0,0,0,0,1,1],
# [0,0,0,0,0,1,0,0,0,1,0,1],
# [0,0,0,0,0,1,0,0,0,1,1,0]])
HKS = np.zeros((len(adjacency_matrix), ntime_stamps))
graph_laplacian = csgraph.laplacian(adjacency_matrix)
[eigval, eigvec] = np.linalg.eigh(graph_laplacian)
time = np.logspace(math.log10(.5 / eigval[-1]),
math.log10(1 / eigval[1]),
num=ntime_stamps)
for t in range(ntime_stamps):
time
HKS[:, t] = np.diag(
np.matmul(
eigvec,
np.matmul(np.diagflat(np.exp(-time[t] * eigval)), eigvec.T)))
return HKS
def update_graph(self):
adj = rbf_kernel(self.user_feature_matrix)
self.L = csgraph.laplacian(adj, normed=True)
A_t_1 = self.A.copy()
self.A = np.kron(self.L + 0.01 * np.identity(self.user_num),
np.identity(self.dimension))
self.cov += self.alpha * (self.A - A_t_1)
def compute(self, data, target, class_samples):
"""
Compute the difference matrix and the eigenvalues
Args:
data: data samples, ndarray (n_samples, n_features)
target: target samples, ndarray (n_samples)
class_samples : class samples, ndarray (n_class, M, n_features)
"""
# Compute E_{p(x\mid C_i)} [p(x\mid C_j)]
self.S, self.similarity_arrays = compute_expectation_with_monte_carlo(
data,
target,
class_samples,
class_indices=self.class_indices,
n_class=self.n_class,
k_nearest=self.k_nearest,
distance=self.distance,
)
# Compute the D matrix
self.W = np.empty([self.n_class, self.n_class])
for i, j in product(range(self.n_class), range(self.n_class)):
self.W[i, j] = 1 - scipy.spatial.distance.braycurtis(
self.S[i], self.S[j])
self.difference = 1 - self.W
# Get the Laplacian and its eigen values
self.L_mat, dd = laplacian(self.W, False, True)
self.evals, self.evecs = np.linalg.eigh(self.L_mat)
self.csg = self._csg_from_evals(self.evals)
def fit(
self,
affinity_matrix,
labels,
n_labeled,
n_eig_vecs=10,
classifier_type='knn', # ['knn', 'svc']
n_neighbors=5 # Only necessary if classifier_type == 'knn'
):
if issparse(affinity_matrix):
affinity_matrix = affinity_matrix.todense()
# Symmetrize the adjacency matrix
self.affinity_matrix = np.maximum(affinity_matrix, affinity_matrix.T)
laplace = laplacian(self.affinity_matrix, normed=True)
# eigsh gets eigenvectors and values from the graph laplacian. We get an extra
# eigenvector because the first vector corresponds to eigenvalue 0
_, eig_vecs = eigsh(laplace,
k=n_eig_vecs + 1,
which='SM',
maxiter=5000)
# Don't consider the eigenvector corresponding to eigenvalue 0
self.embeddings = eig_vecs[:, 1:]
embeddings_labeled = self.embeddings[:n_labeled, :]
self.classifier = neighbors.KNeighborsClassifier(
n_neighbors=n_neighbors)
self.classifier.fit(embeddings_labeled, labels[:n_labeled])
def _check_laplacian_dtype_none(A, desired_L, desired_d, normed,
use_out_degree, copy, dtype, arr_type):
mat = arr_type(A, dtype=dtype)
L, d = csgraph.laplacian(
mat,
normed=normed,
return_diag=True,
use_out_degree=use_out_degree,
copy=copy,
dtype=None,
)
if normed and check_int_type(mat):
assert L.dtype == np.float64
assert d.dtype == np.float64
_assert_allclose_sparse(L, desired_L, atol=1e-12)
_assert_allclose_sparse(d, desired_d, atol=1e-12)
else:
assert L.dtype == dtype
assert d.dtype == dtype
desired_L = np.asarray(desired_L).astype(dtype)
desired_d = np.asarray(desired_d).astype(dtype)
_assert_allclose_sparse(L, desired_L, atol=1e-12)
_assert_allclose_sparse(d, desired_d, atol=1e-12)
if not copy:
if not (normed and check_int_type(mat)):
if type(mat) is np.ndarray:
assert_allclose(L, mat)
elif mat.format == "coo":
_assert_allclose_sparse(L, mat)
def learn_knn_graph(signals, node_num, k=5):
#print('Learning KNN Graph')
adj=rbf_kernel(signals.T)
np.fill_diagonal(adj,0)
knn_adj=filter_graph_to_knn(adj, node_num, k=k)
knn_lap=csgraph.laplacian(knn_adj, normed=False)
return knn_adj, knn_lap
def _build_A(self):
r"""
Builds the coefficient matrix based on throat conductance values.
Notes
-----
The conductance to use is specified in stored in the algorithm's
settings under ``alg.settings['conductance']``.
In subclasses, conductance is set by default. For instance, in
``FickianDiffusion``, it is set to
``throat.diffusive_conductance``, although it can be changed.
"""
gvals = self.settings['conductance']
# FIXME: this needs to be properly addressed (see issue #1548)
try:
if gvals in self._get_iterative_props():
self.settings._update({"cache_A": False, "cache_b": False})
except AttributeError:
pass
if not self.settings['cache_A']:
self._pure_A = None
if self._pure_A is None:
phase = self.project[self.settings.phase]
g = phase[gvals]
am = self.network.create_adjacency_matrix(weights=g, fmt='coo')
self._pure_A = spgr.laplacian(am).astype(float)
self.A = self._pure_A.copy()
def spectral_decomposition(matrix):
'''
Perform eigendecomposition on a weighted graph matrix,
by converting
Parameters
----------
matrix: 2-d numpy array
Weighted graph adjacency matrix.
Returns
---------
eigen_vals: numpy array
Vector of the graph laplacian eigenvalues.
Sorted from largest to smallest.
eigen_vecs: 2-d numpy array
Matrix of the graph laplacian eigenvectors.
Sorted from largest to smallest.
'''
affinity = compute_similarity(matrix)
laplacian = csgraph.laplacian(affinity, normed=True)
eigen_vals, eigen_vecs = eigh(laplacian)
eigen_vals, eigen_vecs = np.flip(eigen_vals), np.flip(eigen_vecs)
return eigen_vals, eigen_vecs
def fit(self, X):
"""
Learn the model using the training data.
:param X: Training data.
"""
print('Just fitting')
W = (self.alpha * self.W_s) + (self.gamma * self.W_F)
L, diag_p = csgraph.laplacian(W, normed=self.normed, return_diag=True)
# - Formulate the eigenproblem.
lhs_matrix = (X.T.dot(L.dot(X)))
rhs_matrix = None
# - Solve the problem
eigval, eigvec = eigh(a=lhs_matrix,
b=rhs_matrix,
overwrite_a=True,
overwrite_b=True,
check_finite=True)
eigval = np.real(eigval)
# - Select eigenvectors based on eigenvalues
# -- get indices of k smallest eigen values
k_eig_ixs = np.argpartition(eigval, self.k)[:self.k]
# -- columns of eigvec are the eigen vectors corresponding to the eigen values
# --- select column vectors corresponding to k largest eigen values
self.V = eigvec[:, k_eig_ixs]
def initializeGW_clustering(Gepsilon, relationFileName, newW):
G = newW
n = newW.shape[0]
L = csgraph.laplacian(G, normed = False)
I = np.identity(n)
GW = I + Gepsilon*L # W is a double stochastic matrix
print GW
return GW.T
def _prepare_inputs(self, X, W):
self.X = X
# set up prior M
if self.params['use_cov']:
self.M = np.cov(X.T)
else:
self.M = np.identity(X.shape[1])
L = laplacian(W, normed=False)
self.loss_matrix = self.X.T.dot(L.dot(self.X))
def _check_symmetric_graph_laplacian(mat, normed):
if not hasattr(mat, "shape"):
mat = eval(mat, dict(np=np, sparse=sparse))
if sparse.issparse(mat):
sp_mat = mat
mat = sp_mat.todense()
else:
sp_mat = sparse.csr_matrix(mat)
laplacian = csgraph.laplacian(mat, normed=normed)
n_nodes = mat.shape[0]
if not normed:
assert_array_almost_equal(laplacian.sum(axis=0), np.zeros(n_nodes))
assert_array_almost_equal(laplacian.T, laplacian)
assert_array_almost_equal(laplacian, csgraph.laplacian(sp_mat, normed=normed).todense())
assert_array_almost_equal(laplacian, _explicit_laplacian(mat, normed=normed))
def test_graph_laplacian():
for mat in (np.arange(10) * np.arange(10)[:, np.newaxis],
np.ones((7, 7)),
np.eye(19),
np.vander(np.arange(4)) + np.vander(np.arange(4)).T,
):
sp_mat = sparse.csr_matrix(mat)
for normed in (True, False):
laplacian = csgraph.laplacian(mat, normed=normed)
n_nodes = mat.shape[0]
if not normed:
np.testing.assert_array_almost_equal(laplacian.sum(axis=0),
np.zeros(n_nodes))
np.testing.assert_array_almost_equal(laplacian.T,
laplacian)
np.testing.assert_array_almost_equal(\
laplacian,
csgraph.laplacian(sp_mat, normed=normed).todense())
def generate_laplacian(df):
G = np.zeros((sum(df.shape),sum(df.shape)))
### node indexed from 0, row first, columns second
for i in range(0,df.shape[0]):
for j in range(0,df.shape[1]):
if df.iat[i,j] > 0:
connect(G,(i, df.shape[0]+j,1))
G = csgraph.laplacian(G)
return G
def __init__(self, X, W, use_cov=True):
self.X = X
# set up prior M
if use_cov:
self.M = np.cov(X.T)
else:
self.M = np.identity(X.shape[1])
L = laplacian(W, normed=False)
self.loss_matrix = self.X.T.dot(L).dot(self.X)
def _prepare_inputs(self, X, W):
self.X_ = X = check_array(X)
W = check_array(W, accept_sparse=True)
# set up prior M
if self.use_cov:
self.M_ = pinvh(np.cov(X, rowvar = False))
else:
self.M_ = np.identity(X.shape[1])
L = laplacian(W, normed=False)
return X.T.dot(L.dot(X))
def compute_clusters(graph): lap_data = csgraph.laplacian(asarray(graph.get_adjacency(attribute="weight").data), normed=False) lap_cov = covariate_laplacian(graph, 'department') lap_total = 1*lap_data + 10000000000*lap_cov + 1000*eye(len(graph.vs)) + 1000000 [evals, evecs] = eig(lap_total) coords = array((log(evecs[2]/sqrt(evals[2])), log(evecs[1]/sqrt(evals[1])))) coords = transpose(abs(coords)) return coords
def _compute_reg_neighbors(n_ch_x, n_delays, reg_type, method='direct',
normed=False):
"""Compute regularization parameter from neighbors."""
from scipy.sparse.csgraph import laplacian
known_types = ('ridge', 'laplacian')
if isinstance(reg_type, str):
reg_type = (reg_type,) * 2
if len(reg_type) != 2:
raise ValueError('reg_type must have two elements, got %s'
% (len(reg_type),))
for r in reg_type:
if r not in known_types:
raise ValueError('reg_type entries must be one of %s, got %s'
% (known_types, r))
reg_time = (reg_type[0] == 'laplacian' and n_delays > 1)
reg_chs = (reg_type[1] == 'laplacian' and n_ch_x > 1)
if not reg_time and not reg_chs:
return np.eye(n_ch_x * n_delays)
# regularize time
if reg_time:
reg = np.eye(n_delays)
stride = n_delays + 1
reg.flat[1::stride] += -1
reg.flat[n_delays::stride] += -1
reg.flat[n_delays + 1:-n_delays - 1:stride] += 1
args = [reg] * n_ch_x
reg = linalg.block_diag(*args)
else:
reg = np.zeros((n_delays * n_ch_x,) * 2)
# regularize features
if reg_chs:
block = n_delays * n_delays
row_offset = block * n_ch_x
stride = n_delays * n_ch_x + 1
reg.flat[n_delays:-row_offset:stride] += -1
reg.flat[n_delays + row_offset::stride] += 1
reg.flat[row_offset:-n_delays:stride] += -1
reg.flat[:-(n_delays + row_offset):stride] += 1
assert np.array_equal(reg[::-1, ::-1], reg)
if method == 'direct':
if normed:
norm = np.sqrt(np.diag(reg))
reg /= norm
reg /= norm[:, np.newaxis]
return reg
else:
# Use csgraph. Note that our -1's above are really the neighbors!
# If we ever want to allow arbitrary adjacency matrices, this is how
# we'd want to do it.
reg = laplacian(-reg, normed=normed)
return reg
def load_dataset(self, interactions, user_count=150, item_count=200, split=0.5):
# interactions = [us,is,rs]
user_range = len(interactions[0])
user_idx = interactions[0]
item_idx = interactions[1]
ratings = interactions[2]
print("Initialising model variables ...")
uig = UserItemGraph(user_count, item_count)
print("Building dataset ...")
for j in range(user_range):
u = user_idx[j]-1; i=item_idx[j]-1;
uig.M[u,i] = ratings[j]
uig.O[u,i] = 1
print("Computing Leave-one-out test split ...")
for u in user_idx:
heldout = np.random.choice(list(uig.O[u-1, :].nonzero())[0],1)
uig.Otest[u-1, heldout] = 1
uig.Otraining = uig.O - uig.Otest
print("Building user interaction matrix ...")
# User interactions
Wrow = np.zeros((user_count, user_count), dtype=np.int)
Wrow = interaction_matrix(Wrow, uig.O, 1)
print("Building item interaction matrix ...")
# Item interactions
Wcol = np.zeros((item_count, item_count), dtype=np.int)
Wcol = interaction_matrix(Wcol, uig.O, 0)
print("Computing Laplacian of interactions ...")
uig.Lrow = csgraph.laplacian(Wrow, normed=True)
uig.Lcol = csgraph.laplacian(Wcol, normed=True)
self.graph = uig
def initializeGW( Gepsilon ,n, relationFileName):
W = np.identity(n)
with open(relationFileName) as f:
for line in f:
line = line.split('\t')
if line[0] != 'userID':
if int(line[0])<=n and int(line[1]) <=n:
W[int(line[0])][int(line[1])] +=1
G = W
L = csgraph.laplacian(G, normed = False)
I = np.identity(n)
GW = I + Gepsilon*L # W is a double stochastic matrix
print GW
return GW.T
def kwok_lau(graph, v, k, epsilon, path_length_scaler, volbd_scaler):
num_vertices = graph.shape[0]
volbd = k * volbd_scaler
vertices = list(range(num_vertices))
print("N: {0}\nVolume bound: {1}\n".format(num_vertices, volbd))
p = [np.zeros(num_vertices, int)]
p[0][v] = 1
last = p[-1]
# length of walk to compute
# need to get W!
I = mlib.identity(num_vertices, int)
# assuming symmetric here
L, D_vector = csglib.laplacian(graph, return_diag=True)
D = mlib.diags(D_vector, (0), format='csc')
lazy_walk = 0.5 * (I + lalib.inv(D) * graph)
num_iterations = math.ceil(num_vertices ** 2 * math.log(num_vertices, 2))
for t in range(1, num_iterations):
p.append(last * lazy_walk)
last = p[-1]
# value function for sorting:
sortkey = lambda t: (lambda vertex: p[t][vertex] / D_vector[vertex])
# initialize set now
S = dict()
S[0,1] = p[0][v]
outset = S[0,1]
# when S has one element, the conductance is 1
# conductance is <= 1
outcond = 2
for t in range(1, num_iterations):
# so#rt all at once here?
p[t] = p[t-1] * lazy_walk
for j in range(1, num_vertices):
# compute new S[t,j]
# don't want to include the entire graph, that's dumb...
# should also put another bound in here for later
S[t,j] = computeS(sortkey(t), vertices, j, num_vertices)
# find smallest S_{t,j}
currcond = conductance(S[t,j], L, D)
if (currcond < outcond and volume(S[t,j], D) <= volbd):
outset = S[t,j]
outcond = currcond
return outset
def constructLaplacianMatrix(self, W, Gepsilon): G = W.copy() #Convert adjacency matrix of weighted graph to adjacency matrix of unweighted graph for i in self.users: for j in self.users: if G[i.id][j.id] > 0: G[i.id][j.id] = 1 L = csgraph.laplacian(G, normed = False) print L I = np.identity(n = G.shape[0]) GW = I + Gepsilon*L # W is a double stochastic matrix print 'GW', GW return GW.T
def initializeGW(W, epsilon):
n = len(W)
#print 'W', W
G = np.zeros(shape = (n, n))
for i in range(n):
for j in range(n):
if W[i][j] > 0:
G[i][j] = 1
L = csgraph.laplacian(G, normed = False)
I = np.identity(n)
GW = I + epsilon*L
print GW
#showheatmap(GW)
return GW
def _build_graph(self):
"""Graph matrix for Label Spreading computes the graph laplacian"""
# compute affinity matrix (or gram matrix)
if self.kernel == 'knn':
self.nn_fit = None
n_samples = self.X_.shape[0]
affinity_matrix = self._get_kernel(self.X_)
laplacian = csgraph.laplacian(affinity_matrix, normed=True)
laplacian = -laplacian
if sparse.isspmatrix(laplacian):
diag_mask = (laplacian.row == laplacian.col)
laplacian.data[diag_mask] = 0.0
else:
laplacian.flat[::n_samples + 1] = 0.0 # set diag to 0.0
return laplacian
def initializeGW(self, Gepsilon): n = len(self.users) a = np.ones(n-1) b =np.ones(n); c = np.ones(n-1) k1 = -1 k2 = 0 k3 = 1 A = np.diag(a, k1) + np.diag(b, k2) + np.diag(c, k3) G = A L = csgraph.laplacian(G, normed = False) I = np.identity(n) GW = I + Gepsilon*L # W is a double stochastic matrix print GW return GW.T
def fit(self, X):
'''Obtain the top-k eigensystem of the graph Laplacian
The eigen solver adopts shift-invert mode as described in
http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
'''
nbrs = NearestNeighbors(n_neighbors=self.n_nbrs).fit(X)
# NOTE W is a dense graph thus may lead to memory leak
W = nbrs.kneighbors_graph(X).toarray()
W_sym = np.maximum(W, W.T)
L = csr_matrix(csgraph.laplacian(W_sym, normed=True))
[Sigma, U] = eigsh(L, self.n_clusters+1, sigma=0, which='LM')
# remove the trivial (smallest) eigenvalues & vectors
self.Sigma, self.U = Sigma[1:], U[:,1:]
def test_arpack_eigsh_initialization():
# Non-regression test that shows null-space computation is better with
# initialization of eigsh from [-1,1] instead of [0,1]
random_state = check_random_state(42)
A = random_state.rand(50, 50)
A = np.dot(A.T, A) # create s.p.d. matrix
A = laplacian(A) + 1e-7 * np.identity(A.shape[0])
k = 5
# Test if eigsh is working correctly
# New initialization [-1,1] (as in original ARPACK)
# Was [0,1] before, with which this test could fail
v0 = random_state.uniform(-1, 1, A.shape[0])
w, _ = eigsh(A, k=k, sigma=0.0, v0=v0)
# Eigenvalues of s.p.d. matrix should be nonnegative, w[0] is smallest
assert_greater_equal(w[0], 0)
def initializeGW_label(Gepsilon ,n, relationFileName, label, diagnol):
W = np.identity(n)
with open(relationFileName) as f:
for line in f:
line = line.split('\t')
if line[0] != 'userID' and label[int(line[0])]!=10000 and label[int(line[1])]!=10000: #10000 means not top 100 user.
W[label[int(line[0])]][label[int(line[1])]] += 1
# don't need it
'''
if diagnol=='1' or diagnol=='0':
for i in range(n):
W[i][i] = int(diagnol)
'''
G = W
L = csgraph.laplacian(G, normed = False)
I = np.identity(n)
GW = I + Gepsilon*L # W is a double stochastic matrix
print GW
return GW.T
def test_spectral_embedding_unnormalized():
# Test that spectral_embedding is also processing unnormalized laplacian
# correctly
random_state = np.random.RandomState(36)
data = random_state.randn(10, 30)
sims = rbf_kernel(data)
n_components = 8
embedding_1 = spectral_embedding(sims,
norm_laplacian=False,
n_components=n_components,
drop_first=False)
# Verify using manual computation with dense eigh
laplacian, dd = csgraph.laplacian(sims, normed=False,
return_diag=True)
_, diffusion_map = eigh(laplacian)
embedding_2 = diffusion_map.T[:n_components]
embedding_2 = _deterministic_vector_sign_flip(embedding_2).T
assert_array_almost_equal(embedding_1, embedding_2)
def laplacian_eigs(csr_matrix):
"""
Computes the eigenvalues and eigenvectors of the laplacian matrix.
Parameters
----------
csr_matrix: scipy csr_matrix
The adjacency matrix of the graph.
Returns
-------
eig_vals : (M,) double or complex ndarray
The eigenvalues, each repeated according to its multiplicity.
eig_vecs : (M, M) double or complex ndarray
The normalized left eigenvector corresponding to the eigenvalue
``w[i]`` is the column v[:,i].
"""
stderr.write('Computing Eigenvectors...')
lap = csgraph.laplacian(csr_matrix, normed=False)
eig_vals, eig_vecs = linalg.eigh(lap.todense(), type=3)
stderr.write('\rEigenvector decomposition complete.\n')
return eig_vals, eig_vecs
def low_rank_align(X, Y, Cxy, d=None, mu=0.8):
"""Input: data matrices X,Y, correspondence matrix Cxy,
embedding dimension d, and correspondence weight mu
Output: embedded X and embedded Y
"""
nx, dx = X.shape
ny, dy = Y.shape
assert Cxy.shape==(nx,ny), \
'Correspondence matrix must be shape num_X_samples X num_Y_samples.'
C = np.fliplr(block_diag(np.fliplr(Cxy),np.fliplr(Cxy.T)))
if d is None:
d = min(dx,dy)
Rx = low_rank_repr(X,d)
Ry = low_rank_repr(Y,d)
R = block_diag(Rx,Ry)
tmp = np.eye(R.shape[0]) - R
M = tmp.T.dot(tmp)
L = laplacian(C)
eigen_prob = (1-mu)*M + 2*mu*L
_,F = eigh(eigen_prob,eigvals=(1,d),overwrite_a=True)
Xembed = F[:nx]
Yembed = F[nx:]
return Xembed, Yembed
def initializeGW(FeatureVectors, Gepsilon): n = len(FeatureVectors) W = np.zeros(shape = (n, n)) for i in range(n): sSim = 0 for j in range(n): sim = np.dot(FeatureVectors[i],FeatureVectors[j]) print 'sim',sim if i == j: sim += 1 W[i][j] = sim sSim += sim W[i] /= sSim for a in range(n): print '%.3f' % W[i][a], print '' G = W L = csgraph.laplacian(G, normed = False) I = np.identity(n) GW = I + Gepsilon*L # W is a double stochastic matrix print GW return GW.T
def _build_A(self, force=False):
r"""
Builds the coefficient matrix based on conductances between pores.
The conductance to use is specified in the algorithm's ``settings``
under ``conductance``. In subclasses (e.g. ``FickianDiffusion``)
this is set by default, though it can be overwritten.
Parameters
----------
force : Boolean (default is ``False``)
If set to ``True`` then the A matrix is built from new. If
``False`` (the default), a cached version of A is returned. The
cached version is *clean* in the sense that no boundary conditions
or sources terms have been added to it.
"""
if force:
self._pure_A = None
if self._pure_A is None:
network = self.project.network
phase = self.project.phases()[self.settings['phase']]
g = phase[self.settings['conductance']]
am = network.create_adjacency_matrix(weights=g, fmt='coo')
self._pure_A = spgr.laplacian(am).astype(float)
self.A = self._pure_A.copy()
def formulate_graph_laplacian(self,X):
"""
Input:
X --> The feature set of essays, of size N * D
Output:
L --> Laplacian Matrix of the graph formed by the essay set X.
We form the graph W from X by computing the similarity
between x and y for all x and y belonging to X
Then we find the degree matrix D of W, which is a diagonal
matrix.
Laplacian L is defined as
L = D - W
Normalized Laplacian is defined as
l_norm = D^(-1/2)*L*D^(-1/2)
Normalized Laplacian are known to work marginally better
than in graph diffusion.
D --> The degree matrix D of W
Note:
None
"""
W = self.similarity_measure(X)
W = self.sparsify(W,100,0)
D = self.calculate_degree_matrix(W)
return csgraph.laplacian(W, normed=True), D