def _svd(self, array, n_components, n_discard):
"""Returns first `n_components` left and right singular
vectors u and v, discarding the first `n_discard`.
"""
if self.svd_method == "randomized":
kwargs = {}
if self.n_svd_vecs is not None:
kwargs["n_oversamples"] = self.n_svd_vecs
u, _, vt = randomized_svd(array, n_components, random_state=self.random_state, **kwargs)
elif self.svd_method == "arpack":
u, _, vt = svds(array, k=n_components, ncv=self.n_svd_vecs)
if np.any(np.isnan(vt)):
# some eigenvalues of A * A.T are negative, causing
# sqrt() to be np.nan. This causes some vectors in vt
# to be np.nan.
_, v = eigsh(safe_sparse_dot(array.T, array), ncv=self.n_svd_vecs)
vt = v.T
if np.any(np.isnan(u)):
_, u = eigsh(safe_sparse_dot(array, array.T), ncv=self.n_svd_vecs)
assert_all_finite(u)
assert_all_finite(vt)
u = u[:, n_discard:]
vt = vt[n_discard:]
return u, vt.T
def _svd(self, array, n_components, n_discard):
"""Returns first `n_components` left and right singular
vectors u and v, discarding the first `n_discard`.
"""
if self.svd_method == 'randomized':
kwargs = {}
if self.n_svd_vecs is not None:
kwargs['n_oversamples'] = self.n_svd_vecs
u, _, vt = randomized_svd(array,
n_components,
random_state=self.random_state,
**kwargs)
elif self.svd_method == 'arpack':
u, _, vt = svds(array, k=n_components, ncv=self.n_svd_vecs)
if np.any(np.isnan(vt)):
# some eigenvalues of A * A.T are negative, causing
# sqrt() to be np.nan. This causes some vectors in vt
# to be np.nan.
_, v = eigsh(safe_sparse_dot(array.T, array),
ncv=self.n_svd_vecs)
vt = v.T
if np.any(np.isnan(u)):
_, u = eigsh(safe_sparse_dot(array, array.T),
ncv=self.n_svd_vecs)
assert_all_finite(u)
assert_all_finite(vt)
u = u[:, n_discard:]
vt = vt[n_discard:]
return u, vt.T
def custom_svd(array, n_components, n_discard,n_svd_vecs): u, _, vt = svds(array, k=n_components, ncv=n_svd_vecs) if np.any(np.isnan(vt)): _, v = eigsh(safe_sparse_dot(array.T, array),ncv=n_svd_vecs) vt = v.T if np.any(np.isnan(u)): _, u = eigsh(safe_sparse_dot(array, array.T),ncv=n_svd_vecs) assert_all_finite(u) assert_all_finite(vt) u = u[:, n_discard:] vt = vt[n_discard:] return u, vt.T
def spectralcluster(A,
n_cluster,
n_neighbors=6,
random_state=None,
eigen_tol=0.0):
#maps = spectral_embedding(affinity, n_components=n_components,eigen_solver=eigen_solver,random_state=random_state,eigen_tol=eigen_tol, drop_first=False)
# dd is diag
laplacian, dd = graph_laplacian(A, normed=True, return_diag=True)
# set the diagonal of the laplacian matrix and convert it to a sparse format well suited for e # igenvalue decomposition
laplacian = _set_diag(laplacian, 1)
# diffusion_map is eigenvectors
# LM largest eigenvalues
laplacian *= -1
eigenvalues, eigenvectors = eigsh(laplacian,
k=n_cluster,
sigma=1.0,
which='LM',
tol=eigen_tol)
y = eigenvectors.T[n_cluster::-1] * dd
y = _deterministic_vector_sign_flip(y)[:n_cluster].T
random_state = check_random_state(random_state)
centroids, labels, _ = k_means(y, n_cluster, random_state=random_state)
return eigenvalues, y, centroids, labels
def my_uniteigenvector_zeroeigenvalue_cluster(k):
G = nx.read_gpickle('data/undirected(fortest).gpickle')
A = nx.adjacency_matrix(G, nodelist=G.nodes()[:-1], weight='weight')
#A=A.toarray()
#np.fill_diagonal(A,0.01) #add node with its own weight to itself
#Tri = np.diag(np.sum(A, axis=1))
#L = Tri - A
#Tri_1 = np.diag(np.reciprocal(np.sqrt(Tri).diagonal()))
#Ls = Tri_1.dot(L).dot(Tri_1)
Ls, dd = graph_laplacian(A,normed=True, return_diag=True)
eigenvalue_n, eigenvector_n = eigsh(Ls*(-1), k=k,
sigma=1.0, which='LM',
tol=0.0)
#for ic,vl in enumerate(eigenvalue_n):
# if abs(vl-0)<=1e-10:
# eigenvector_n[:, ic] = np.full(len(G.nodes()[:-1]),1.0 / math.sqrt(len(G.nodes()[:-1]))) # zero eigenvalue
eigenvector_n[:, -1] = np.full(len(G.nodes()[:-1]), 1.0 / math.sqrt(len(G.nodes()[:-1]))) # zero eigenvalue
for ir,n in enumerate(eigenvector_n):
eigenvector_n[ir]=n/float(np.linalg.norm(n)) # normalize to unitvector
_, labels, _ = k_means(eigenvector_n, k, random_state=None,
n_init=100)
return labels
def embed(self, laplacian, diagonal, k, tol=0):
k = k + 1
lambdas, diffusion_map = eigsh(-laplacian, k=k, which='SM', tol=tol)
embedding = diffusion_map.T[k::-1] * diagonal
if self.do_scale:
return scale(embedding[1:k].T, axis=1)
else:
return embedding[1:k].T
def runEmbed(data, n_components):
lambdas, vectors = eigsh(data, k=n_components)
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors/vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components + 1)] * lambdas[:n_components][None, :]
#embedding_sorted = np.argsort(embedding[:], axis=1)
return embedding
def runEmbed(data, n_components):
lambdas, vectors = eigsh(data, k=n_components)
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors / vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components + 1)] * lambdas[:n_components][None, :]
#embedding_sorted = np.argsort(embedding[:], axis=1)
return embedding
def seriation(self, A):
n_components = 2
eigen_tol = 0.00001
if sparse.issparse(A):
A = A.todense()
np.fill_diagonal(A, 0)
laplacian, dd = graph_laplacian(A, return_diag=True)
laplacian *= -1
lambdas, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which='LM',
tol=eigen_tol)
embedding = diffusion_map.T[n_components::-1] # * dd
sort_index = np.argsort(embedding[1])
return sort_index
def main(argv):
# Set defaults:
n_components_embedding = 25
comp_min = 2
comp_max = 20 + 1
varname = 'data'
filename = './test'
# Import files
f = h5py.File(('%s.mat' % filename), 'r')
dataCorr = np.array(f.get('%s' % varname))
# Prep matrix
K = (dataCorr + 1) / 2.
v = np.sqrt(np.sum(K, axis=1))
A = K / (v[:, None] * v[None, :])
del K
A = np.squeeze(A * [A > 0])
# Run embedding
lambdas, vectors = eigsh(A, k=n_components_embedding)
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors / vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding +
1)] * lambdas[:n_components_embedding][None, :]
# Run kmeans clustering
def kmeans(embedding, n_components):
est = KMeans(n_clusters=n_components,
n_jobs=-1,
init='k-means++',
n_init=300)
est.fit_transform(embedding)
labels = est.labels_
data = labels.astype(np.float)
return data
results = list()
for n_components in xrange(comp_min, comp_max):
results.append(kmeans(embedding, n_components))
savemat(('%s_results.mat' % filename), {'results': results})
def runFiedler(conn):
# https://github.com/margulies/topography
# prep for embedding
K = (conn + 1) / 2.
v = np.sqrt(np.sum(K, axis=1))
A = K/(v[:, None] * v[None, :])
del K
A = np.squeeze(A * [A > 0])
# diffusion embedding
n_components_embedding = 2
lambdas, vectors = eigsh(A, k=n_components_embedding+1)
del A
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors/vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding + 1 + 1)] * lambdas[:n_components_embedding+1][None, :]
return embedding
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 = graph_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 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 _fit_transform(self, K):
""" Fit's using kernel K"""
# center kernel
K = self._centerer.fit_transform(K)
if self.n_components is None:
n_components = K.shape[0]
else:
n_components = min(K.shape[0], self.n_components)
# compute eigenvectors
if self.eigen_solver == 'auto':
if K.shape[0] > 200 and n_components < 10:
eigen_solver = 'arpack'
else:
eigen_solver = 'dense'
else:
eigen_solver = self.eigen_solver
if eigen_solver == 'dense':
self.lambdas_, self.alphas_ = linalg.eigh(
K, eigvals=(K.shape[0] - n_components, K.shape[0] - 1))
self.evals_, self.evecs_ = linalg.eigh(K)
elif eigen_solver == 'arpack':
self.lambdas_, self.alphas_ = eigsh(K,
n_components,
which="LA",
tol=self.tol,
maxiter=self.max_iter)
# sort eigenvectors in descending order
indices = self.lambdas_.argsort()[::-1]
self.lambdas_ = self.lambdas_[indices]
self.alphas_ = self.alphas_[:, indices]
# remove eigenvectors with a zero eigenvalue
if self.remove_zero_eig or self.n_components is None:
self.alphas_ = self.alphas_[:, self.lambdas_ > 0]
self.lambdas_ = self.lambdas_[self.lambdas_ > 0]
return K
def main(argv):
# Set defaults:
n_components_embedding = 25
comp_min = 2
comp_max = 20 + 1
varname = 'data'
filename = './test'
# Import files
f = h5py.File(('%s.mat' % filename),'r')
dataCorr = np.array(f.get('%s' % varname))
# Prep matrix
K = (dataCorr + 1) / 2.
v = np.sqrt(np.sum(K, axis=1))
A = K/(v[:, None] * v[None, :])
del K
A = np.squeeze(A * [A > 0])
# Run embedding
lambdas, vectors = eigsh(A, k=n_components_embedding)
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors/vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding + 1)] * lambdas[:n_components_embedding][None, :]
# Run kmeans clustering
def kmeans(embedding, n_components):
est = KMeans(n_clusters=n_components, n_jobs=-1, init='k-means++', n_init=300)
est.fit_transform(embedding)
labels = est.labels_
data = labels.astype(np.float)
return data
results = list()
for n_components in xrange(comp_min,comp_max):
results.append(kmeans(embedding, n_components))
savemat(('%s_results.mat' % filename), {'results':results})
def DoFiedler(conn):
# prep for embedding
K = (conn + 1) / 2.
v = np.sqrt(np.sum(K, axis=1))
A = K / (v[:, None] * v[None, :])
del K
A = np.squeeze(A * [A > 0])
# diffusion embedding
n_components_embedding = 2
lambdas, vectors = eigsh(A, k=n_components_embedding + 1)
del A
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors / vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding + 1 +
1)] * lambdas[:n_components_embedding + 1][None, :]
return embedding
def _fit_transform(self, K):
""" Fit's using kernel K"""
# center kernel
K = self._centerer.fit_transform(K)
if self.n_components is None:
n_components = K.shape[0]
else:
n_components = min(K.shape[0], self.n_components)
# compute eigenvectors
if self.eigen_solver == 'auto':
if K.shape[0] > 200 and n_components < 10:
eigen_solver = 'arpack'
else:
eigen_solver = 'dense'
else:
eigen_solver = self.eigen_solver
if eigen_solver == 'dense':
self.lambdas_, self.alphas_ = linalg.eigh(
K, eigvals=(K.shape[0] - n_components, K.shape[0] - 1))
self.evals_, self.evecs_ = linalg.eigh(K)
elif eigen_solver == 'arpack':
self.lambdas_, self.alphas_ = eigsh(K, n_components,
which="LA",
tol=self.tol,
maxiter=self.max_iter)
# sort eigenvectors in descending order
indices = self.lambdas_.argsort()[::-1]
self.lambdas_ = self.lambdas_[indices]
self.alphas_ = self.alphas_[:, indices]
# remove eigenvectors with a zero eigenvalue
if self.remove_zero_eig or self.n_components is None:
self.alphas_ = self.alphas_[:, self.lambdas_ > 0]
self.lambdas_ = self.lambdas_[self.lambdas_ > 0]
return K
def predict_k(affinity_matrix): normed_laplacian, dd = graph_laplacian(affinity_matrix, normed=True, return_diag=True) laplacian = _set_diag(normed_laplacian, 1,norm_laplacian=True) n_components = affinity_matrix.shape[0] - 1 eigenvalues, eigenvectors = eigsh(-laplacian, k=n_components, which="LM", sigma=1.0, maxiter=5000) eigenvalues = -eigenvalues[::-1] # Reverse and sign inversion. max_gap = 0 gap_pre_index = 0 for i in range(1, eigenvalues.size): gap = eigenvalues[i] - eigenvalues[i - 1] if gap > max_gap: max_gap = gap gap_pre_index = i - 1 k = gap_pre_index + 1 return k
def spectralcluster(A, n_cluster, n_neighbors=6, random_state=None, eigen_tol=0.0):
#maps = spectral_embedding(affinity, n_components=n_components,eigen_solver=eigen_solver,random_state=random_state,eigen_tol=eigen_tol, drop_first=False)
# dd is diag
laplacian, dd = graph_laplacian(A, normed=True, return_diag=True)
# set the diagonal of the laplacian matrix and convert it to a sparse format well suited for e # igenvalue decomposition
laplacian = _set_diag(laplacian, 1)
# diffusion_map is eigenvectors
# LM largest eigenvalues
laplacian *= -1
eigenvalues, eigenvectors = eigsh(laplacian, k=n_cluster,
sigma=1.0, which='LM',
tol=eigen_tol)
y = eigenvectors.T[n_cluster::-1] * dd
y = _deterministic_vector_sign_flip(y)[:n_cluster].T
random_state = check_random_state(random_state)
centroids, labels, _ = k_means(y, n_cluster, random_state=random_state)
return eigenvalues, y, centroids, labels
def DoFiedler(conn):
# prep for embedding
# K : matrix of similarities / Kernel matrix / Gram matrix
# make conn non-negative, -1<since conn<1
K = (conn + 1) / 2.
# axis=1 meaning operating over rows, "row sum's of K"
v = np.sqrt(np.sum(K, axis=1))
# make a random walk on data, D is diagonal matrix
D = v[:, None] * v[None, :]
# row-normalization of K gives transition matrix A => A = D^-1 * K
A = K/D
del K
A = np.squeeze(A * [A > 0])
n_components_embedding = 5
lambdas, vectors = eigsh(A, k=n_components_embedding)
del A
# sorting eigenvalues and -vectors in descending order
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors/vectors[:, 0][:, None]
# begin from second largest eigenvalue and corr. eigenvector
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding + 1)] * lambdas[:n_components_embedding][None, :]
return embedding
fullsize = len(dataAll)
del dataAll
# correlate
dataCorr = np.corrcoef(np.transpose(np.array(dataNorm)))
del dataNorm
dataCorr[np.isnan(dataCorr)] = 0
# prep for embedding
K = (dataCorr + 1) / 2.
del dataCorr
v = np.sqrt(np.sum(K, axis=1))
A = K/(v[:, None] * v[None, :])
del K
A = np.squeeze(A * [A > 0])
# diffusion embedding
lambdas, vectors = eigsh(A, k=n_components_embedding+1)
del A
lambdas = lambdas[::-1]
vectors = vectors[:, ::-1]
psi = vectors/vectors[:, 0][:, None]
lambdas = lambdas[1:] / (1 - lambdas[1:])
embedding = psi[:, 1:(n_components_embedding + 1 + 1)] * lambdas[:n_components_embedding+1][None, :]
# kmeans clustering
results = []
for n_components in xrange(comp_min, comp_max+1):
est = KMeans(n_clusters=n_components, n_jobs=-1, init='k-means++', n_init=300)
est.fit_transform(embedding)
labels = est.labels_
clust = labels.astype(np.float)
# reinsert zeros:
padded = np.zeros(fullsize)
def spectral_embedding(self,
adjacency,
n_components=8,
eigen_solver=None,
random_state=None,
eigen_tol=0.0,
drop_first=True):
"""
see original at https://github.com/scikit-learn/scikit-learn/blob/14031f6/sklearn/manifold/spectral_embedding_.py#L133
custermize1: return lambdas with the embedded matrix.
custermize2: norm_laplacian is always True
"""
norm_laplacian = True
adjacency = check_symmetric(adjacency)
try:
from pyamg import smoothed_aggregation_solver
except ImportError:
if eigen_solver == "amg":
raise ValueError(
"The eigen_solver was set to 'amg', but pyamg is "
"not available.")
if eigen_solver is None:
eigen_solver = 'arpack'
elif eigen_solver not in ('arpack', 'lobpcg', 'amg'):
raise ValueError("Unknown value for eigen_solver: '%s'."
"Should be 'amg', 'arpack', or 'lobpcg'" %
eigen_solver)
random_state = check_random_state(random_state)
n_nodes = adjacency.shape[0]
# Whether to drop the first eigenvector
if drop_first:
n_components = n_components + 1
if not _graph_is_connected(adjacency):
warnings.warn("Graph is not fully connected, spectral embedding"
" may not work as expected.")
laplacian, dd = graph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
if (eigen_solver == 'arpack' or eigen_solver != 'lobpcg' and
(not sparse.isspmatrix(laplacian) or n_nodes < 5 * n_components)):
# lobpcg used with eigen_solver='amg' has bugs for low number of nodes
# for details see the source code in scipy:
# https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen
# /lobpcg/lobpcg.py#L237
# or matlab:
# http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m
laplacian = _set_diag(laplacian, 1, norm_laplacian)
# Here we'll use shift-invert mode for fast eigenvalues
# (see http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
# for a short explanation of what this means)
# Because the normalized Laplacian has eigenvalues between 0 and 2,
# I - L has eigenvalues between -1 and 1. ARPACK is most efficient
# when finding eigenvalues of largest magnitude (keyword which='LM')
# and when these eigenvalues are very large compared to the rest.
# For very large, very sparse graphs, I - L can have many, many
# eigenvalues very near 1.0. This leads to slow convergence. So
# instead, we'll use ARPACK's shift-invert mode, asking for the
# eigenvalues near 1.0. This effectively spreads-out the spectrum
# near 1.0 and leads to much faster convergence: potentially an
# orders-of-magnitude speedup over simply using keyword which='LA'
# in standard mode.
try:
# We are computing the opposite of the laplacian inplace so as
# to spare a memory allocation of a possibly very large array
laplacian *= -1
lambdas, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which='LM',
tol=eigen_tol)
embedding = diffusion_map.T[n_components::-1] * dd
except RuntimeError:
# When submatrices are exactly singular, an LU decomposition
# in arpack fails. We fallback to lobpcg
eigen_solver = "lobpcg"
# Revert the laplacian to its opposite to have lobpcg work
laplacian *= -1
if eigen_solver == 'amg':
# Use AMG to get a preconditioner and speed up the eigenvalue
# problem.
if not sparse.issparse(laplacian):
warnings.warn("AMG works better for sparse matrices")
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
ml = smoothed_aggregation_solver(check_array(laplacian, 'csr'))
M = ml.aspreconditioner()
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
M=M,
tol=1.e-12,
largest=False)
embedding = diffusion_map.T * dd
if embedding.shape[0] == 1:
raise ValueError
elif eigen_solver == "lobpcg":
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
if n_nodes < 5 * n_components + 1:
# see note above under arpack why lobpcg has problems with small
# number of nodes
# lobpcg will fallback to eigh, so we short circuit it
if sparse.isspmatrix(laplacian):
laplacian = laplacian.toarray()
lambdas, diffusion_map = eigh(laplacian)
embedding = diffusion_map.T[:n_components] * dd
else:
laplacian = _set_diag(laplacian, 1, norm_laplacian)
# We increase the number of eigenvectors requested, as lobpcg
# doesn't behave well in low dimension
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
tol=1e-15,
largest=False,
maxiter=2000)
embedding = diffusion_map.T[:n_components] * dd
if embedding.shape[0] == 1:
raise ValueError
embedding = _deterministic_vector_sign_flip(embedding)
if drop_first:
return embedding[1:n_components].T, lambdas
else:
return embedding[:n_components].T, lambdas
def null_space(M, k, k_skip=1, eigen_solver='dense', tol=1E-6, max_iter=100,
random_state=None):
"""
Find the null space of a matrix M.
Parameters
----------
M : {array, matrix, sparse matrix, LinearOperator}
Input covariance matrix: should be symmetric positive semi-definite
k : integer
Number of eigenvalues/vectors to return
k_skip : integer, optional
Number of low eigenvalues to skip.
eigen_solver : string, {'auto', 'arpack', 'dense'}
auto : algorithm will attempt to choose the best method for input data
arpack : use arnoldi iteration in shift-invert mode.
For this method, M may be a dense matrix, sparse matrix,
or general linear operator.
Warning: ARPACK can be unstable for some problems. It is
best to try several random seeds in order to check results.
dense : use standard dense matrix operations for the eigenvalue
decomposition. For this method, M must be an array
or matrix type. This method should be avoided for
large problems.
tol : float, optional
Tolerance for 'arpack' method.
Not used if eigen_solver=='dense'.
max_iter : maximum number of iterations for 'arpack' method
not used if eigen_solver=='dense'
random_state: numpy.RandomState or int, optional
The generator or seed used to determine the starting vector for arpack
iterations. Defaults to numpy.random.
Returns
-------
embedding_vectors : array[float, float], shape=(n_components, n_samples)
Eigenvectors used for embedding
eigenvectors : array[float, float], shape=(n_features, n_samples)
All eigenvectors, in descending order by eigenvalues. Eigenvectors are
stored in columns. The vector corresponding to evals[i] is stored in evecs[:,i]
eigenvalues : array[float], shape=(n_features)
All eigenvalues, in descending order
"""
if eigen_solver == 'auto':
if M.shape[0] > 200 and k + k_skip < 10:
eigen_solver = 'arpack'
else:
eigen_solver = 'dense'
if eigen_solver == 'arpack':
random_state = check_random_state(random_state)
v0 = random_state.rand(M.shape[0])
try:
eigen_values, eigen_vectors = eigsh(M, k + k_skip, sigma=0.0,
tol=tol, maxiter=max_iter,
v0=v0)
except RuntimeError as msg:
raise ValueError("Error in determining null-space with ARPACK. "
"Error message: '%s'. "
"Note that method='arpack' can fail when the "
"weight matrix is singular or otherwise "
"ill-behaved. method='dense' is recommended. "
"See online documentation for more information."
% msg)
return eigen_vectors[:, k_skip:], eigen_values[k_skip:]
elif eigen_solver == 'dense':
if hasattr(M, 'toarray'):
M = M.toarray()
eigen_values, eigen_vectors = eigh(
M, eigvals=(k_skip, k + k_skip - 1))
index = np.argsort(np.abs(eigen_values))
evals, evecs = eigh(M, overwrite_a=True)
order = np.argsort(evals)[::-1]
evals = evals[order]
evecs = evecs[:, order]
return eigen_vectors[:, index], evals, evecs
else:
raise ValueError("Unrecognized eigen_solver '%s'" % eigen_solver)
def predict_k(affinity_matrix):
"""
Predict number of clusters based on the eigengap.
Parameters
----------
affinity_matrix : array-like or sparse matrix, shape: (n_samples, n_samples)
adjacency matrix.
Each element of this matrix contains a measure of similarity between two of the data points.
Returns
----------
k : integer
estimated number of cluster.
Note
---------
If graph is not fully connected, zero component as single cluster.
References
----------
A Tutorial on Spectral Clustering, 2007
Luxburg, Ulrike
http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/Luxburg07_tutorial_4488%5b0%5d.pdf
"""
"""
If normed=True, L = D^(-1/2) * (D - A) * D^(-1/2) else L = D - A.
normed=True is recommended.
"""
normed_laplacian, dd = graph_laplacian(affinity_matrix, normed=True, return_diag=True)
laplacian = _set_diag(normed_laplacian, 1)
"""
n_components size is N - 1.
Setting N - 1 may lead to slow execution time...
"""
n_components = affinity_matrix.shape[0] - 1
"""
shift-invert mode
The shift-invert mode provides more than just a fast way to obtain a few small eigenvalues.
http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
The normalized Laplacian has eigenvalues between 0 and 2.
I - L has eigenvalues between -1 and 1.
"""
eigenvalues, eigenvectors = eigsh(-laplacian, k=n_components, which="LM", sigma=1.0, maxiter=5000)
eigenvalues = -eigenvalues[::-1] # Reverse and sign inversion.
max_gap = 0
gap_pre_index = 0
for i in range(1, eigenvalues.size):
gap = eigenvalues[i] - eigenvalues[i - 1]
if gap > max_gap:
max_gap = gap
gap_pre_index = i - 1
k = gap_pre_index + 1
return k
cat_tuple = (cat, matrix[cat].mean())
cat_perc.append(cat_tuple)
# sort category percentages
cat_perc = sorted(cat_perc, key=lambda x: x[1])
graph = cosine_similarity(matrix) # use cosine similarity, as in Noulas et al.
# https://github.com/mingmingyang/auto_spectral_clustering/blob/master/autosp.py
# how to calculate spectral clusters
norm_laplacian, dd = graph_laplacian(graph, normed=True, return_diag=True)
laplacian = _set_diag(norm_laplacian, 1, norm_laplacian=True)
n_components = graph.shape[0] - 1
eigenvalues, eigenvectors = eigsh(-laplacian,
k=n_components,
which="LM",
sigma=1.0,
maxiter=5000)
eigenvalues = -eigenvalues[::-1]
max_gap = 0
gap_pre_index = 0
for i in range(1, eigenvalues.size):
gap = eigenvalues[i] - eigenvalues[i - 1]
if gap > max_gap:
max_gap = gap
gap_pre_index = i - 1
k = gap_pre_index + 1
print k
def predict_k(affinity_matrix):
"""
Predict number of clusters based on the eigengap.
Parameters
----------
affinity_matrix : array-like or sparse matrix, shape: (n_samples, n_samples)
adjacency matrix.
Each element of this matrix contains a measure of similarity between two of the data points.
Returns
----------
k : integer
estimated number of cluster.
Note
---------
If graph is not fully connected, zero component as single cluster.
References
----------
A Tutorial on Spectral Clustering, 2007
Luxburg, Ulrike
http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/Luxburg07_tutorial_4488%5b0%5d.pdf
"""
"""
If normed=True, L = D^(-1/2) * (D - A) * D^(-1/2) else L = D - A.
normed=True is recommended.
"""
normed_laplacian, dd = graph_laplacian(affinity_matrix,
normed=True,
return_diag=True)
laplacian = _set_diag(normed_laplacian, 1)
"""
n_components size is N - 1.
Setting N - 1 may lead to slow execution time...
"""
n_components = affinity_matrix.shape[0] - 1
"""
shift-invert mode
The shift-invert mode provides more than just a fast way to obtain a few small eigenvalues.
http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
The normalized Laplacian has eigenvalues between 0 and 2.
I - L has eigenvalues between -1 and 1.
"""
eigenvalues, eigenvectors = eigsh(-laplacian,
k=n_components,
which="LM",
sigma=1.0,
maxiter=5000)
eigenvalues = -eigenvalues[::-1] # Reverse and sign inversion.
max_gap = 0
gap_pre_index = 0
for i in range(1, eigenvalues.size):
gap = eigenvalues[i] - eigenvalues[i - 1]
if gap > max_gap:
max_gap = gap
gap_pre_index = i - 1
k = gap_pre_index + 1
return k
def spectral_embedding(laplacian,
n_components=8,
eigen_solver=None,
random_state=None,
eigen_tol=1e-20,
drop_first=False):
"""
----------------------------------------------------------------
*****!!!sklearn function variation for spectral embeding!!!*****
----------------------------------------------------------------
Project the sample on the first eigenvectors of the graph Laplacian.
This embedding can also 'work' even if the ``adjacency`` variable is
not strictly the adjacency matrix of a graph but more generally
an affinity or similarity matrix between samples (for instance the
heat kernel of a euclidean distance matrix or a k-NN matrix).
However care must taken to always make the affinity matrix symmetric
so that the eigenvector decomposition works as expected.
Read more in the :ref:`User Guide <spectral_embedding>`.
Parameters
----------
laplacian : array-like or sparse matrix, shape: (n_samples, n_samples)
The laplacian matrix of the graph to embed.
n_components : integer, optional, default 8
The dimension of the projection subspace.
eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}, default None
The eigenvalue decomposition strategy to use. AMG requires pyamg
to be installed. It can be faster on very large, sparse problems,
but may also lead to instabilities.
random_state : int seed, RandomState instance, or None (default)
A pseudo random number generator used for the initialization of the
lobpcg eigenvectors decomposition when eigen_solver == 'amg'.
By default, arpack is used.
eigen_tol : float, optional, default=0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
drop_first : bool, optional, default=True
Whether to drop the first eigenvector. For spectral embedding, this
should be True as the first eigenvector should be constant vector for
connected graph, but for spectral clustering, this should be kept as
False to retain the first eigenvector.
Returns
-------
embedding : array, shape=(n_samples, n_components)
The reduced samples.
Notes
-----
Spectral embedding is most useful when the graph has one connected
component. If there graph has many components, the first few eigenvectors
will simply uncover the connected components of the graph.
References
----------
* http://en.wikipedia.org/wiki/LOBPCG
* Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method
Andrew V. Knyazev
http://dx.doi.org/10.1137%2FS1064827500366124
"""
try:
from pyamg import smoothed_aggregation_solver
except ImportError:
if eigen_solver == "amg":
raise ValueError("The eigen_solver was set to 'amg', but pyamg is "
"not available.")
if eigen_solver is None:
eigen_solver = 'arpack'
elif eigen_solver not in ('arpack', 'lobpcg', 'amg'):
raise ValueError("Unknown value for eigen_solver: '%s'."
"Should be 'amg', 'arpack', or 'lobpcg'" %
eigen_solver)
random_state = check_random_state(random_state)
n_nodes = laplacian.shape[0]
# Whether to drop the first eigenvector
if drop_first:
n_components = n_components + 1
dd = laplacian.diagonal()
if (eigen_solver == 'arpack' or eigen_solver != 'lobpcg' and
(not sparse.isspmatrix(laplacian) or n_nodes < 5 * n_components)):
# lobpcg used with eigen_solver='amg' has bugs for low number of nodes
# for details see the source code in scipy:
# https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen
# /lobpcg/lobpcg.py#L237
# or matlab:
# http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m
laplacian = _set_diag(laplacian, 1)
# Here we'll use shift-invert mode for fast eigenvalues
# (see http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
# for a short explanation of what this means)
# Because the normalized Laplacian has eigenvalues between 0 and 2,
# I - L has eigenvalues between -1 and 1. ARPACK is most efficient
# when finding eigenvalues of largest magnitude (keyword which='LM')
# and when these eigenvalues are very large compared to the rest.
# For very large, very sparse graphs, I - L can have many, many
# eigenvalues very near 1.0. This leads to slow convergence. So
# instead, we'll use ARPACK's shift-invert mode, asking for the
# eigenvalues near 1.0. This effectively spreads-out the spectrum
# near 1.0 and leads to much faster convergence: potentially an
# orders-of-magnitude speedup over simply using keyword which='LA'
# in standard mode.
try:
# We are computing the opposite of the laplacian inplace so as
# to spare a memory allocation of a possibly very large array
laplacian *= -1
lambdas, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which='LM',
tol=eigen_tol)
embedding = diffusion_map.T[n_components::-1] * dd
except RuntimeError:
# When submatrices are exactly singular, an LU decomposition
# in arpack fails. We fallback to lobpcg
eigen_solver = "lobpcg"
# Revert the laplacian to its opposite to have lobpcg work
laplacian *= -1
if eigen_solver == 'amg':
# Use AMG to get a preconditioner and speed up the eigenvalue
# problem.
if not sparse.issparse(laplacian):
warnings.warn("AMG works better for sparse matrices")
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
laplacian = _set_diag(laplacian, 1)
ml = smoothed_aggregation_solver(check_array(laplacian, 'csr'))
M = ml.aspreconditioner()
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
M=M,
tol=1.e-12,
largest=False)
embedding = diffusion_map.T * dd
if embedding.shape[0] == 1:
raise ValueError
elif eigen_solver == "lobpcg":
# lobpcg needs double precision floats
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
if n_nodes < 5 * n_components + 1:
# see note above under arpack why lobpcg has problems with small
# number of nodes
# lobpcg will fallback to eigh, so we short circuit it
if sparse.isspmatrix(laplacian):
laplacian = laplacian.toarray()
lambdas, diffusion_map = eigh(laplacian)
embedding = diffusion_map.T[:n_components] * dd
else:
laplacian = _set_diag(laplacian, 1)
# We increase the number of eigenvectors requested, as lobpcg
# doesn't behave well in low dimension
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian,
X,
tol=1e-15,
largest=False,
maxiter=2000)
embedding = diffusion_map.T[:n_components] * dd
if embedding.shape[0] == 1:
raise ValueError
embedding = _deterministic_vector_sign_flip(embedding)
if drop_first:
return embedding[1:n_components].T
else:
return embedding[:n_components].T
from sklearn.utils.arpack import eigsh
app = service.prodbox.CinemaService()
X = app.getWeightedSearchFeatures(15)
graph = kneighbors_graph(X, 10)
lap = graph_laplacian(graph, True)
from sklearn.decomposition import TruncatedSVD
svd = TruncatedSVD(n_components=30, algorithm="arpack")
lap = spectral_embedding_._set_diag(lap, 1)
svd.fit(-lap)
eigenvalues = np.diag(svd.components_ * (-lap).todense() * svd.components_.T)
eigenvalues2, _ = eigsh(-lap, k=30, which='LM', sigma=1)
print(eigenvalues)
print(eigenvalues2)
se = SpectralEmbedding(n_components=30,
eigen_solver='arpack',
affinity="nearest_neighbors")
se.fit(X)
app.quit()
# TODO : check budget distribution, draw budget conditionnaly
out = connected_components(graph)
def spectral_embedding(adjacency, n_components=8, eigen_solver=None,
random_state=None, eigen_tol=0.0,
norm_laplacian=True, drop_first=True,
mode=None):
"""Project the sample on the first eigen vectors of the graph Laplacian.
MMP:TO CHANGE THIS
The adjacency matrix is used to compute a normalized graph Laplacian
whose spectrum (especially the eigen vectors associated to the
smallest eigen values) has an interpretation in terms of minimal
number of cuts necessary to split the graph into comparably sized
components.
This embedding can also 'work' even if the ``adjacency`` variable is
not strictly the adjacency matrix of a graph but more generally
an affinity or similarity matrix between samples (for instance the
heat kernel of a euclidean distance matrix or a k-NN matrix).
However care must taken to always make the affinity matrix symmetric
so that the eigen vector decomposition works as expected.
Parameters
----------
adjacency : array-like or sparse matrix, shape: (n_samples, n_samples)
The adjacency matrix of the graph to embed.
n_components : integer, optional
The dimension of the projection subspace.
eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}
The eigenvalue decomposition strategy to use. AMG requires pyamg
to be installed. It can be faster on very large, sparse problems,
but may also lead to instabilities.
random_state : int seed, RandomState instance, or None (default)
A pseudo random number generator used for the initialization of the
lobpcg eigen vectors decomposition when eigen_solver == 'amg'.
By default, arpack is used.
eigen_tol : float, optional, default=0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
drop_first : bool, optional, default=True
Whether to drop the first eigenvector. For spectral embedding, this
should be True as the first eigenvector should be constant vector for
connected graph, but for spectral clustering, this should be kept as
False to retain the first eigenvector.
Returns
-------
embedding : array, shape=(n_samples, n_components)
The reduced samples.
Notes
-----
Spectral embedding is most useful when the graph has one connected
component. If there graph has many components, the first few eigenvectors
will simply uncover the connected components of the graph.
References
----------
* http://en.wikipedia.org/wiki/LOBPCG
* Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method
Andrew V. Knyazev
http://dx.doi.org/10.1137%2FS1064827500366124
"""
try:
from pyamg import smoothed_aggregation_solver
except ImportError:
if eigen_solver == "amg" or mode == "amg":
raise ValueError("The eigen_solver was set to 'amg', but pyamg is "
"not available.")
if not mode is None:
warnings.warn("'mode' was renamed to eigen_solver "
"and will be removed in 0.15.",
DeprecationWarning)
eigen_solver = mode
if eigen_solver is None:
eigen_solver = 'arpack'
elif not eigen_solver in ('arpack', 'lobpcg', 'amg'):
raise ValueError("Unknown value for eigen_solver: '%s'."
"Should be 'amg', 'arpack', or 'lobpcg'"
% eigen_solver)
random_state = check_random_state(random_state)
n_nodes = adjacency.shape[0]
# Whether to drop the first eigenvector
if drop_first:
n_components = n_components + 1
# Check that the matrices given is symmetric
if ((not sparse.isspmatrix(adjacency) and
not np.all((adjacency - adjacency.T) < 1e-10)) or
(sparse.isspmatrix(adjacency) and
not np.all((adjacency - adjacency.T).data < 1e-10))):
warnings.warn("Graph adjacency matrix should be symmetric. "
"Converted to be symmetric by average with its "
"transpose.")
adjacency = .5 * (adjacency + adjacency.T)
if not _graph_is_connected(adjacency):
warnings.warn("Graph is not fully connected, spectral embedding"
" may not work as expected.")
laplacian, dd = graph_laplacian(adjacency,
normed=norm_laplacian, return_diag=True)
if (eigen_solver == 'arpack'
or eigen_solver != 'lobpcg' and
(not sparse.isspmatrix(laplacian)
or n_nodes < 5 * n_components)):
# lobpcg used with eigen_solver='amg' has bugs for low number of nodes
# for details see the source code in scipy:
# https://github.com/scipy/scipy/blob/v0.11.0/scipy/sparse/linalg/eigen
# /lobpcg/lobpcg.py#L237
# or matlab:
# http://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m
laplacian = _set_diag(laplacian, 1)
# Here we'll use shift-invert mode for fast eigenvalues
# (see http://docs.scipy.org/doc/scipy/reference/tutorial/arpack.html
# for a short explanation of what this means)
# Because the normalized Laplacian has eigenvalues between 0 and 2,
# I - L has eigenvalues between -1 and 1. ARPACK is most efficient
# when finding eigenvalues of largest magnitude (keyword which='LM')
# and when these eigenvalues are very large compared to the rest.
# For very large, very sparse graphs, I - L can have many, many
# eigenvalues very near 1.0. This leads to slow convergence. So
# instead, we'll use ARPACK's shift-invert mode, asking for the
# eigenvalues near 1.0. This effectively spreads-out the spectrum
# near 1.0 and leads to much faster convergence: potentially an
# orders-of-magnitude speedup over simply using keyword which='LA'
# in standard mode.
try:
lambdas, diffusion_map = eigsh(-laplacian, k=n_components,
sigma=1.0, which='LM',
tol=eigen_tol)
embedding = diffusion_map.T[n_components::-1] * dd
except RuntimeError:
# When submatrices are exactly singular, an LU decomposition
# in arpack fails. We fallback to lobpcg
eigen_solver = "lobpcg"
if eigen_solver == 'amg':
# Use AMG to get a preconditioner and speed up the eigenvalue
# problem.
if not sparse.issparse(laplacian):
warnings.warn("AMG works better for sparse matrices")
laplacian = laplacian.astype(np.float) # lobpcg needs native floats
laplacian = _set_diag(laplacian, 1)
ml = smoothed_aggregation_solver(atleast2d_or_csr(laplacian))
M = ml.aspreconditioner()
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian, X, M=M, tol=1.e-12,
largest=False)
embedding = diffusion_map.T * dd
if embedding.shape[0] == 1:
raise ValueError
elif eigen_solver == "lobpcg":
laplacian = laplacian.astype(np.float) # lobpcg needs native floats
if n_nodes < 5 * n_components + 1:
# see note above under arpack why lobpcg has problems with small
# number of nodes
# lobpcg will fallback to symeig, so we short circuit it
if sparse.isspmatrix(laplacian):
laplacian = laplacian.todense()
lambdas, diffusion_map = symeig(laplacian)
embedding = diffusion_map.T[:n_components] * dd
else:
# lobpcg needs native floats
laplacian = laplacian.astype(np.float)
laplacian = _set_diag(laplacian, 1)
# We increase the number of eigenvectors requested, as lobpcg
# doesn't behave well in low dimension
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
lambdas, diffusion_map = lobpcg(laplacian, X, tol=1e-15,
largest=False, maxiter=2000)
embedding = diffusion_map.T[:n_components] * dd
if embedding.shape[0] == 1:
raise ValueError
if drop_first:
return embedding[1:n_components].T
else:
return embedding[:n_components].T
import numpy as np from sklearn.utils.arpack import eigsh app = service.prodbox.CinemaService() X = app.getWeightedSearchFeatures(15) graph = kneighbors_graph(X, 10) lap = graph_laplacian(graph, True) from sklearn.decomposition import TruncatedSVD svd = TruncatedSVD(n_components = 30, algorithm="arpack") lap = spectral_embedding_._set_diag(lap, 1) svd.fit(-lap) eigenvalues = np.diag(svd.components_ * (-lap).todense() * svd.components_.T) eigenvalues2, _ = eigsh(-lap, k=30, which='LM', sigma=1) print(eigenvalues) print(eigenvalues2) se = SpectralEmbedding(n_components = 30, eigen_solver='arpack', affinity="nearest_neighbors") se.fit(X) app.quit() # TODO : check budget distribution, draw budget conditionnaly out = connected_components(graph)
def null_space(M, k, k_skip=1, eigen_solver='arpack', tol=1E-6, max_iter=100,
random_state=None):
"""
Find the null space of a matrix M.
Parameters
----------
M : {array, matrix, sparse matrix, LinearOperator}
Input covariance matrix: should be symmetric positive semi-definite
k : integer
Number of eigenvalues/vectors to return
k_skip : integer, optional
Number of low eigenvalues to skip.
eigen_solver : string, {'auto', 'arpack', 'dense'}
auto : algorithm will attempt to choose the best method for input data
arpack : use arnoldi iteration in shift-invert mode.
For this method, M may be a dense matrix, sparse matrix,
or general linear operator.
Warning: ARPACK can be unstable for some problems. It is
best to try several random seeds in order to check results.
dense : use standard dense matrix operations for the eigenvalue
decomposition. For this method, M must be an array
or matrix type. This method should be avoided for
large problems.
tol : float, optional
Tolerance for 'arpack' method.
Not used if eigen_solver=='dense'.
max_iter : maximum number of iterations for 'arpack' method
not used if eigen_solver=='dense'
random_state: numpy.RandomState or int, optional
The generator or seed used to determine the starting vector for arpack
iterations. Defaults to numpy.random.
"""
if eigen_solver == 'auto':
if M.shape[0] > 200 and k + k_skip < 10:
eigen_solver = 'arpack'
else:
eigen_solver = 'dense'
if eigen_solver == 'arpack':
random_state = check_random_state(random_state)
# initialize with [-1,1] as in ARPACK
v0 = random_state.uniform(-1, 1, M.shape[0])
try:
eigen_values, eigen_vectors = eigsh(M, k + k_skip, sigma=0.0,
tol=tol, maxiter=max_iter,
v0=v0)
except RuntimeError as msg:
raise ValueError("Error in determining null-space with ARPACK. "
"Error message: '%s'. "
"Note that method='arpack' can fail when the "
"weight matrix is singular or otherwise "
"ill-behaved. method='dense' is recommended. "
"See online documentation for more information."
% msg)
return eigen_vectors[:, k_skip:], np.sum(eigen_values[k_skip:])
elif eigen_solver == 'dense':
if hasattr(M, 'toarray'):
M = M.toarray()
eigen_values, eigen_vectors = eigh(
M, eigvals=(k_skip, k + k_skip - 1), overwrite_a=True)
index = np.argsort(np.abs(eigen_values))
return eigen_vectors[:, index], np.sum(eigen_values)
else:
raise ValueError("Unrecognized eigen_solver '%s'" % eigen_solver)