def spectral_embedding(adjacency, n_components=8, eigen_solver=None,
random_state=None, eigen_tol=0.0,
norm_laplacian=True, drop_first=True):
adjacency = check_symmetric(adjacency)
# eigen_solver = 'arpack'
# eigen_solver = 'amg'
norm_laplacian=False
random_state = check_random_state(random_state)
n_nodes = adjacency.shape[0]
if not _graph_is_connected(adjacency):
warnings.warn("Graph is not fully connected, spectral embedding"
" may not work as expected.")
laplacian, dd = csgraph_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)):
print("[INFILE] eigen_solver : ", eigen_solver, "norm_laplacian:", norm_laplacian)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
try:
laplacian *= -1
v0 = random_state.uniform(-1, 1, laplacian.shape[0])
lambdas, diffusion_map = eigsh(laplacian, k=n_components,
sigma=1.0, which='LM',
tol=eigen_tol, v0=v0)
embedding = diffusion_map.T[n_components::-1]
if norm_laplacian:
embedding = embedding / dd
except RuntimeError:
eigen_solver = "lobpcg"
laplacian *= -1
embedding = _deterministic_vector_sign_flip(embedding)
return embedding[:n_components].T
def spectral_embedding(adjacency, norm_laplacian=True):
adjacency = check_symmetric(adjacency)
n_nodes = adjacency.shape[0]
laplacian, dd = csgraph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
return np.linalg.pinv(laplacian.todense())
def transform(self):
# self.n_components = min(self.n_components, self.data.shape[1])
laplacian, dd = csgraph_laplacian(self.data,
normed=self.norm_laplacian,
return_diag=True)
laplacian = check_array(laplacian,
dtype=np.float64,
accept_sparse=True)
laplacian = _set_diag(laplacian, 1, self.norm_laplacian)
## Seed the global number generator because the pyamg
## interface apparently uses that...
## Also, see https://github.com/pyamg/pyamg/issues/139
np.random.seed(self.random_state.randint(2**31 - 1))
diag_shift = 1e-5 * sparse.eye(laplacian.shape[0])
laplacian += diag_shift
ml = smoothed_aggregation_solver(check_array(laplacian, "csr"))
laplacian -= diag_shift
M = ml.aspreconditioner()
X = self.random_state.rand(laplacian.shape[0], self.n_components + 1)
X[:, 0] = dd.ravel()
# laplacian *= -1
# v0 = self.random_state.uniform(-1, 1, laplacian.shape[0])
# eigvals, diffusion_map = eigsh(
# laplacian, k=self.n_components + 1, sigma=1.0, which="LM", tol=0.0, v0=v0
# )
# # eigsh needs reversing
# embedding = diffusion_map.T[::-1]
eigvals, diffusion_map = lobpcg(laplacian,
X,
M=M,
tol=1.0e-5,
largest=False)
embedding = diffusion_map.T
if self.norm_laplacian:
embedding = embedding / dd
if self.drop_first:
self.data_ = embedding[1:self.n_components].T
eigvals = eigvals[1:self.n_components]
else:
self.data_ = embedding[:self.n_components].T
self.eigvals_ = eigvals[::
-1] # reverse direction to have the largest first
def spectral_embedding(adjacency,
n_components=8,
eigen_solver=None,
random_state=None,
eigen_tol=0.0,
norm_laplacian=True,
drop_first=True):
"""Project the sample on the first eigenvectors of the graph Laplacian.
The adjacency matrix is used to compute a normalized graph Laplacian
whose spectrum (especially the eigenvectors associated to the
smallest eigenvalues) 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 eigenvector decomposition works as expected.
Note : Laplacian Eigenmaps is the actual algorithm implemented here.
Read more in the :ref:`User Guide <spectral_embedding>`.
Parameters
----------
adjacency : array-like or sparse matrix, shape: (n_samples, n_samples)
The adjacency 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, RandomState instance or None, optional, default: None
A pseudo random number generator used for the initialization of the
lobpcg eigenvectors decomposition. If int, random_state is the seed
used by the random number generator; If RandomState instance,
random_state is the random number generator; If None, the random number
generator is the RandomState instance used by `np.random`. Used when
``solver`` == 'amg'.
eigen_tol : float, optional, default=0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
norm_laplacian : bool, optional, default=True
If True, then compute normalized Laplacian.
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 (Laplacian Eigenmaps) 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
----------
* https://en.wikipedia.org/wiki/LOBPCG
* Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method
Andrew V. Knyazev
https://doi.org/10.1137%2FS1064827500366124
"""
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 = csgraph_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:
# https://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 https://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
v0 = random_state.uniform(-1, 1, laplacian.shape[0])
lambdas, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which='LM',
tol=eigen_tol,
v0=v0)
embedding = diffusion_map.T[n_components::-1]
if norm_laplacian:
embedding = embedding / 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
if norm_laplacian:
embedding = embedding / 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]
if norm_laplacian:
embedding = embedding / 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]
if norm_laplacian:
embedding = embedding / 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
def spectral_embedding(adjacency, n_components=8, eigen_solver=None,
random_state=None, eigen_tol=0.0,
norm_laplacian=True, drop_first=True):
"""Project the sample on the first eigenvectors of the graph Laplacian.
The adjacency matrix is used to compute a normalized graph Laplacian
whose spectrum (especially the eigenvectors associated to the
smallest eigenvalues) 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 eigenvector decomposition works as expected.
Note : Laplacian Eigenmaps is the actual algorithm implemented here.
Read more in the :ref:`User Guide <spectral_embedding>`.
Parameters
----------
adjacency : array-like or sparse matrix, shape: (n_samples, n_samples)
The adjacency 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, RandomState instance or None, optional, default: None
A pseudo random number generator used for the initialization of the
lobpcg eigenvectors decomposition. If int, random_state is the seed
used by the random number generator; If RandomState instance,
random_state is the random number generator; If None, the random number
generator is the RandomState instance used by `np.random`. Used when
``solver`` == 'amg'.
eigen_tol : float, optional, default=0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
norm_laplacian : bool, optional, default=True
If True, then compute normalized Laplacian.
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 (Laplacian Eigenmaps) 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
----------
* https://en.wikipedia.org/wiki/LOBPCG
* Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method
Andrew V. Knyazev
https://doi.org/10.1137%2FS1064827500366124
"""
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 = csgraph_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:
# https://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 https://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
v0 = random_state.uniform(-1, 1, laplacian.shape[0])
lambdas, diffusion_map = eigsh(laplacian, k=n_components,
sigma=1.0, which='LM',
tol=eigen_tol, v0=v0)
embedding = diffusion_map.T[n_components::-1]
if norm_laplacian:
embedding = embedding / 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
if norm_laplacian:
embedding = embedding / 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]
if norm_laplacian:
embedding = embedding / 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]
if norm_laplacian:
embedding = embedding / 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
values_editions = []
liste_editions = []
for i in range(n):
for j in range(i):
if Adj[i][j] == 1:
liste_editions.append([i, j])
for K in range(2, Nmax):
try:
editions = []
#print('K = ',K)
n_components = K
norm_laplacian = True
laplacian, dd = csgraph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
n_nodes = adjacency.shape[0]
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()
_, diffusion_map = eigh(laplacian, check_finite=False)
embedding = diffusion_map.T[:n_components]
if norm_laplacian:
embedding = embedding / dd
else:
def spectral_embedding_sb(
adjacency,
n_components=8,
norm_laplacian=True,
drop_first=True,
):
"""Returns spectral embeddings.
Arguments
---------
adjacency : array-like or sparse graph
shape - (n_samples, n_samples)
The adjacency matrix of the graph to embed.
n_components : int
The dimension of the projection subspace.
norm_laplacian : bool
If True, then compute normalized Laplacian.
drop_first : bool
Whether to drop the first eigenvector.
Returns
-------
embedding : array
Spectral embeddings for each sample.
Example
-------
>>> import numpy as np
>>> from speechbrain.processing import diarization as diar
>>> affinity = np.array([[1, 1, 1, 0.5, 0, 0, 0, 0, 0, 0.5],
... [1, 1, 1, 0, 0, 0, 0, 0, 0, 0],
... [1, 1, 1, 0, 0, 0, 0, 0, 0, 0],
... [0.5, 0, 0, 1, 1, 1, 0, 0, 0, 0],
... [0, 0, 0, 1, 1, 1, 0, 0, 0, 0],
... [0, 0, 0, 1, 1, 1, 0, 0, 0, 0],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
... [0, 0, 0, 0, 0, 0, 1, 1, 1, 1],
... [0.5, 0, 0, 0, 0, 0, 1, 1, 1, 1]])
>>> embs = diar.spectral_embedding_sb(affinity, 3)
>>> # Notice similar embeddings
>>> print(np.around(embs , decimals=3))
[[ 0.075 0.244 0.285]
[ 0.083 0.356 -0.203]
[ 0.083 0.356 -0.203]
[ 0.26 -0.149 0.154]
[ 0.29 -0.218 -0.11 ]
[ 0.29 -0.218 -0.11 ]
[-0.198 -0.084 -0.122]
[-0.198 -0.084 -0.122]
[-0.198 -0.084 -0.122]
[-0.167 -0.044 0.316]]
"""
# 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 = csgraph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
laplacian *= -1
vals, diffusion_map = eigsh(
laplacian,
k=n_components,
sigma=1.0,
which="LM",
)
embedding = diffusion_map.T[n_components::-1]
if norm_laplacian:
embedding = embedding / dd
embedding = _deterministic_vector_sign_flip(embedding)
if drop_first:
return embedding[1:n_components].T
else:
return embedding[:n_components].T
def kind_joint(K, n_clusters, init, maxit, disp, tol, norm_laplacian):
if maxit <= 0:
raise ValueError('Number of iterations should be a positive number,'
' got %d instead' % maxit)
if tol <= 0:
raise ValueError('The tolerance should be a positive number,'
' got %d instead' % tol)
if K.shape[0] != K.shape[1]:
warnings.warn('Input is not an affinity matrix. Kernelize using KNN'
'graph now')
X = kneighbors_graph(K, n_neighbors=5)
else:
X = (K + K.T) / 2
# set initial V
V = spectral_embedding(X, n_components=n_clusters,
drop_first=False, norm_laplacian=norm_laplacian)
# set initial idx
n = X.shape[0]
if hasattr(init, '__array__'):
idx = np.array(init).reshape(max(init.shape()), )
if idx.shape[0] != n:
raise ValueError('The init should be the same as the total'
'observations, got %d instead.' % idx.shape[0])
else:
km = KindAP(n_clusters=n_clusters)
idx = km.fit_predict_L(V)
# set rho
rho = 1 / n
# set history info
hist = [0 for i in range(maxit)]
itr = 0
for itr in range(maxit):
Vp, idxp = V, idx
laplacian, dd = csgraph_laplacian(X, normed=norm_laplacian,
return_diag=True)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
laplacian *= -1
v0 = np.random.uniform(-1, 1, laplacian.shape[0])
H = sparse.csc_matrix((np.ones(n), (np.arange(n), idx)), shape=(n, n_clusters))
lambdas, diffusion_map = eigsh(laplacian + rho * sparse.csc_matrix.dot(H, H.T),
k=n_clusters, sigma=1.0, which='LM', v0=v0)
embedding = diffusion_map.T[n_clusters::-1]
V = _deterministic_vector_sign_flip(embedding)
# if norm_laplacian:
# V = embedding / dd
obj = rho * np.sum(sparse.csc_matrix.dot(V, H) ** 2) + np.trace(
np.dot(sparse.csc_matrix.dot(V, laplacian), V.T))
hist[itr] = 0.5 * obj
V = V.T
ki = KindAP(n_clusters=n_clusters)
idx = ki.fit_predict_L(V)
# stopping criteria
idxchg = sum(idx != idxp)
Vrel = norm(V - Vp, 'fro') / norm(Vp, 'fro')
if disp:
print('iter: %3d, Obj: %6.2e, Vrel: %6.2e, idxchg: %6d' % (itr, obj, Vrel, idxchg))
if not idxchg or Vrel < tol:
break
return idx, V, hist[:min(maxit, itr + 1)]
def predict_k(self, 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)
normed_laplacian, dd = csgraph_laplacian(affinity_matrix, normed=True,
return_diag=True)
laplacian = _set_diag(normed_laplacian, 1, True)
"""
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
test = []
x_axis = []
for i in range(1, eigenvalues.size):
# if len(x_axis) <= 20:
x_axis.append(i)
test.append(eigenvalues[i])
gap = eigenvalues[i] - eigenvalues[i - 1]
if gap > max_gap:
max_gap = gap
gap_pre_index = i - 1
k = gap_pre_index + 1
plot(x_axis, test, '*')
return k
def spectral_embedding(
adjacency,
*,
n_components=8,
eigen_solver=None,
random_state=None,
eigen_tol="auto",
norm_laplacian=True,
drop_first=True,
):
"""Project the sample on the first eigenvectors of the graph Laplacian.
The adjacency matrix is used to compute a normalized graph Laplacian
whose spectrum (especially the eigenvectors associated to the
smallest eigenvalues) 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 eigenvector decomposition works as expected.
Note : Laplacian Eigenmaps is the actual algorithm implemented here.
Read more in the :ref:`User Guide <spectral_embedding>`.
Parameters
----------
adjacency : {array-like, sparse graph} of shape (n_samples, n_samples)
The adjacency matrix of the graph to embed.
n_components : int, default=8
The dimension of the projection subspace.
eigen_solver : {'arpack', 'lobpcg', '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. If None, then ``'arpack'`` is
used.
random_state : int, RandomState instance or None, default=None
A pseudo random number generator used for the initialization
of the lobpcg eigen vectors decomposition when `eigen_solver ==
'amg'`, and for the K-Means initialization. Use an int to make
the results deterministic across calls (See
:term:`Glossary <random_state>`).
.. note::
When using `eigen_solver == 'amg'`,
it is necessary to also fix the global numpy seed with
`np.random.seed(int)` to get deterministic results. See
https://github.com/pyamg/pyamg/issues/139 for further
information.
eigen_tol : float, default="auto"
Stopping criterion for eigendecomposition of the Laplacian matrix.
If `eigen_tol="auto"` then the passed tolerance will depend on the
`eigen_solver`:
- If `eigen_solver="arpack"`, then `eigen_tol=0.0`;
- If `eigen_solver="lobpcg"` or `eigen_solver="amg"`, then
`eigen_tol=None` which configures the underlying `lobpcg` solver to
automatically resolve the value according to their heuristics. See,
:func:`scipy.sparse.linalg.lobpcg` for details.
Note that when using `eigen_solver="amg"` values of `tol<1e-5` may lead
to convergence issues and should be avoided.
.. versionadded:: 1.2
Added 'auto' option.
norm_laplacian : bool, default=True
If True, then compute symmetric normalized Laplacian.
drop_first : bool, 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 : ndarray of shape (n_samples, n_components)
The reduced samples.
Notes
-----
Spectral Embedding (Laplacian Eigenmaps) 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
----------
* https://en.wikipedia.org/wiki/LOBPCG
* :doi:`"Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method",
Andrew V. Knyazev
<10.1137/S1064827500366124>`
"""
adjacency = check_symmetric(adjacency)
try:
from pyamg import smoothed_aggregation_solver
except ImportError as e:
if eigen_solver == "amg":
raise ValueError(
"The eigen_solver was set to 'amg', but pyamg is not available."
) from e
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 = csgraph_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:
# https://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 https://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
tol = 0 if eigen_tol == "auto" else eigen_tol
laplacian *= -1
v0 = _init_arpack_v0(laplacian.shape[0], random_state)
_, diffusion_map = eigsh(laplacian,
k=n_components,
sigma=1.0,
which="LM",
tol=tol,
v0=v0)
embedding = diffusion_map.T[n_components::-1]
if norm_laplacian:
# recover u = D^-1/2 x from the eigenvector output x
embedding = embedding / 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
elif 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 = check_array(laplacian,
dtype=[np.float64, np.float32],
accept_sparse=True)
laplacian = _set_diag(laplacian, 1, norm_laplacian)
# The Laplacian matrix is always singular, having at least one zero
# eigenvalue, corresponding to the trivial eigenvector, which is a
# constant. Using a singular matrix for preconditioning may result in
# random failures in LOBPCG and is not supported by the existing
# theory:
# see https://doi.org/10.1007/s10208-015-9297-1
# Shift the Laplacian so its diagononal is not all ones. The shift
# does change the eigenpairs however, so we'll feed the shifted
# matrix to the solver and afterward set it back to the original.
diag_shift = 1e-5 * sparse.eye(laplacian.shape[0])
laplacian += diag_shift
ml = smoothed_aggregation_solver(
check_array(laplacian, accept_sparse="csr"))
laplacian -= diag_shift
M = ml.aspreconditioner()
# Create initial approximation X to eigenvectors
X = random_state.standard_normal(size=(laplacian.shape[0],
n_components + 1))
X[:, 0] = dd.ravel()
X = X.astype(laplacian.dtype)
tol = None if eigen_tol == "auto" else eigen_tol
_, diffusion_map = lobpcg(laplacian, X, M=M, tol=tol, largest=False)
embedding = diffusion_map.T
if norm_laplacian:
# recover u = D^-1/2 x from the eigenvector output x
embedding = embedding / dd
if embedding.shape[0] == 1:
raise ValueError
if eigen_solver == "lobpcg":
laplacian = check_array(laplacian,
dtype=[np.float64, np.float32],
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()
_, diffusion_map = eigh(laplacian, check_finite=False)
embedding = diffusion_map.T[:n_components]
if norm_laplacian:
# recover u = D^-1/2 x from the eigenvector output x
embedding = embedding / 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 and create initial
# approximation X to eigenvectors
X = random_state.standard_normal(size=(laplacian.shape[0],
n_components + 1))
X[:, 0] = dd.ravel()
X = X.astype(laplacian.dtype)
tol = None if eigen_tol == "auto" else eigen_tol
_, diffusion_map = lobpcg(laplacian,
X,
tol=tol,
largest=False,
maxiter=2000)
embedding = diffusion_map.T[:n_components]
if norm_laplacian:
# recover u = D^-1/2 x from the eigenvector output x
embedding = embedding / 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
def spectral_embedding(adjacency,
n_components=8,
random_state=np.random.RandomState(),
eigen_tol=0.0,
norm_laplacian=True,
drop_first=True):
n_nodes = adjacency.shape[0]
# Whether to drop the first eigenvector
if drop_first:
n_components = n_components + 1
print('solving eigenvectors...')
laplacian, dd = csgraph_laplacian(adjacency,
normed=norm_laplacian,
return_diag=True)
laplacian = check_array(laplacian, dtype=np.float64, accept_sparse=True)
laplacian = set_diag(laplacian, 1, norm_laplacian)
diag_shift = 1e-5 * sparse.eye(laplacian.shape[0])
laplacian += diag_shift
ml = smoothed_aggregation_solver(check_array(laplacian, 'csr'))
laplacian -= diag_shift
M = ml.aspreconditioner()
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
for attempt_num in range(1, 4):
try:
_, diffusion_map = lobpcg(laplacian,
X,
M=M,
tol=1.e-5,
largest=False)
continue
except:
print(
'LOBPCG eigensolver failed, attempting to recondition on different eigenvector approximation (attempt {}/3)'
.format(attempt_num))
X = random_state.rand(laplacian.shape[0], n_components + 1)
X[:, 0] = dd.ravel()
embedding = diffusion_map.T
if norm_laplacian:
embedding = embedding / dd
if embedding.shape[0] == 1:
raise ValueError
# laplacian = _set_diag(laplacian, 1, norm_laplacian)
# laplacian *= -1
# v0 = random_state.uniform(-1, 1, laplacian.shape[0])
# lambdas, diffusion_map = eigsh(laplacian, k=n_components,
# sigma=1.0, which='LM',
# tol=eigen_tol, v0=v0)
# embedding = diffusion_map.T[n_components::-1]
# if norm_laplacian:
# embedding = embedding / dd
embedding = deterministic_vector_sign_flip(embedding)
if drop_first:
return embedding[1:n_components].T
else:
return embedding[:n_components].T
def generate_A_spec_cluster(num_unknowns,
add_diag=False,
num_clusters=2,
unit_std=False,
dim=2,
dist='gauss',
gamma=None,
distance=False,
return_x=False,
n_neighbors=10):
"""
Similar params to https://scikit-learn.org/stable/auto_examples/cluster/plot_cluster_comparison.html
With spectral clustering
"""
centers = num_clusters
if num_clusters == 2 and not unit_std:
cluster_std = [
1.0, 2.5
] # looks good, sometimes graph is connected, sometimes not
size_factor = 1
else:
cluster_std = 1.0
size_factor = num_unknowns / 1000
center_box = [-10 * size_factor, 10 * size_factor]
norm_laplacian = True
if dist == 'gauss':
X, y = datasets.make_blobs(n_samples=num_unknowns,
n_features=dim,
centers=centers,
cluster_std=cluster_std,
center_box=center_box)
elif dist == 'moons':
X, y = datasets.make_moons(n_samples=num_unknowns, noise=.05)
elif dist == 'circles':
X, y = datasets.make_circles(n_samples=num_unknowns,
noise=.05,
factor=.5)
elif dist == 'random':
X = np.random.rand(num_unknowns, dim)
X = StandardScaler().fit_transform(X)
if distance:
mode = 'distance'
else:
mode = 'connectivity'
connectivity = kneighbors_graph(X,
n_neighbors=n_neighbors,
mode=mode,
include_self=True)
if gamma is not None:
np.exp(-(gamma * connectivity.data)**2, out=connectivity.data)
affinity_matrix = 0.5 * (connectivity + connectivity.T)
laplacian, dd = csgraph_laplacian(affinity_matrix,
normed=norm_laplacian,
return_diag=True)
# set diagonal to 1 if normed
if norm_laplacian:
diag_idx = (laplacian.row == laplacian.col)
laplacian.data[diag_idx] = 1
if add_diag:
small_diag = scipy.sparse.diags(
np.random.uniform(0, 0.02, num_unknowns))
laplacian += small_diag
if return_x:
return X, laplacian
else:
return laplacian