# -*- coding: utf-8 -*-
"""Algorithms for spectral clustering"""
# Author: Gael Varoquaux gael.varoquaux@normalesup.org
# Brian Cheung
# Wei LI <kuantkid@gmail.com>
# License: BSD 3 clause
import warnings
import numpy as np
from ..base import BaseEstimator, ClusterMixin
from ..utils import check_random_state, as_float_array
from ..utils.validation import check_array
from ..metrics.pairwise import pairwise_kernels
from ..neighbors import kneighbors_graph
from ..manifold import spectral_embedding
from .k_means_ import k_means
def discretize(vectors, copy=True, max_svd_restarts=30, n_iter_max=20,
random_state=None):
"""Search for a partition matrix (clustering) which is closest to the
eigenvector embedding.
Parameters
----------
vectors : array-like, shape: (n_samples, n_clusters)
The embedding space of the samples.
copy : boolean, optional, default: True
Whether to copy vectors, or perform in-place normalization.
max_svd_restarts : int, optional, default: 30
Maximum number of attempts to restart SVD if convergence fails
n_iter_max : int, optional, default: 30
Maximum number of iterations to attempt in rotation and partition
matrix search if machine precision convergence is not reached
random_state : int, RandomState instance or None, optional, default: None
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`.
Returns
-------
labels : array of integers, shape: n_samples
The labels of the clusters.
References
----------
- Multiclass spectral clustering, 2003
Stella X. Yu, Jianbo Shi
http://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
Notes
-----
The eigenvector embedding is used to iteratively search for the
closest discrete partition. First, the eigenvector embedding is
normalized to the space of partition matrices. An optimal discrete
partition matrix closest to this normalized embedding multiplied by
an initial rotation is calculated. Fixing this discrete partition
matrix, an optimal rotation matrix is calculated. These two
calculations are performed until convergence. The discrete partition
matrix is returned as the clustering solution. Used in spectral
clustering, this method tends to be faster and more robust to random
initialization than k-means.
"""
from scipy.sparse import csc_matrix
from scipy.linalg import LinAlgError
random_state = check_random_state(random_state)
vectors = as_float_array(vectors, copy=copy)
eps = np.finfo(float).eps
n_samples, n_components = vectors.shape
# Normalize the eigenvectors to an equal length of a vector of ones.
# Reorient the eigenvectors to point in the negative direction with respect
# to the first element. This may have to do with constraining the
# eigenvectors to lie in a specific quadrant to make the discretization
# search easier.
norm_ones = np.sqrt(n_samples)
for i in range(vectors.shape[1]):
vectors[:, i] = (vectors[:, i] / np.linalg.norm(vectors[:, i])) \
* norm_ones
if vectors[0, i] != 0:
vectors[:, i] = -1 * vectors[:, i] * np.sign(vectors[0, i])
# Normalize the rows of the eigenvectors. Samples should lie on the unit
# hypersphere centered at the origin. This transforms the samples in the
# embedding space to the space of partition matrices.
vectors = vectors / np.sqrt((vectors ** 2).sum(axis=1))[:, np.newaxis]
svd_restarts = 0
has_converged = False
# If there is an exception we try to randomize and rerun SVD again
# do this max_svd_restarts times.
while (svd_restarts < max_svd_restarts) and not has_converged:
# Initialize first column of rotation matrix with a row of the
# eigenvectors
rotation = np.zeros((n_components, n_components))
rotation[:, 0] = vectors[random_state.randint(n_samples), :].T
# To initialize the rest of the rotation matrix, find the rows
# of the eigenvectors that are as orthogonal to each other as
# possible
c = np.zeros(n_samples)
for j in range(1, n_components):
# Accumulate c to ensure row is as orthogonal as possible to
# previous picks as well as current one
c += np.abs(np.dot(vectors, rotation[:, j - 1]))
rotation[:, j] = vectors[c.argmin(), :].T
last_objective_value = 0.0
n_iter = 0
while not has_converged:
n_iter += 1
t_discrete = np.dot(vectors, rotation)
labels = t_discrete.argmax(axis=1)
vectors_discrete = csc_matrix(
(np.ones(len(labels)), (np.arange(0, n_samples), labels)),
shape=(n_samples, n_components))
t_svd = vectors_discrete.T * vectors
try:
U, S, Vh = np.linalg.svd(t_svd)
svd_restarts += 1
except LinAlgError:
print("SVD did not converge, randomizing and trying again")
break
ncut_value = 2.0 * (n_samples - S.sum())
if ((abs(ncut_value - last_objective_value) < eps) or
(n_iter > n_iter_max)):
has_converged = True
else:
# otherwise calculate rotation and continue
last_objective_value = ncut_value
rotation = np.dot(Vh.T, U.T)
if not has_converged:
raise LinAlgError('SVD did not converge')
return labels
def spectral_clustering(affinity, n_clusters=8, n_components=None,
eigen_solver=None, random_state=None, n_init=10,
eigen_tol=0.0, assign_labels='kmeans'):
"""Apply clustering to a projection to the normalized laplacian.
In practice Spectral Clustering is very useful when the structure of
the individual clusters is highly non-convex or more generally when
a measure of the center and spread of the cluster is not a suitable
description of the complete cluster. For instance when clusters are
nested circles on the 2D plan.
If affinity is the adjacency matrix of a graph, this method can be
used to find normalized graph cuts.
Read more in the :ref:`User Guide <spectral_clustering>`.
Parameters
-----------
affinity : array-like or sparse matrix, shape: (n_samples, n_samples)
The affinity matrix describing the relationship of the samples to
embed. **Must be symmetric**.
Possible examples:
- adjacency matrix of a graph,
- heat kernel of the pairwise distance matrix of the samples,
- symmetric k-nearest neighbours connectivity matrix of the samples.
n_clusters : integer, optional
Number of clusters to extract.
n_components : integer, optional, default is n_clusters
Number of eigen vectors to use for the spectral embedding
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, RandomState instance or None, optional, default: None
A pseudo random number generator used for the initialization of the
lobpcg eigen vectors decomposition when eigen_solver == 'amg' and by
the K-Means initialization. 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`.
n_init : int, optional, default: 10
Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of
n_init consecutive runs in terms of inertia.
eigen_tol : float, optional, default: 0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
assign_labels : {'kmeans', 'discretize'}, default: 'kmeans'
The strategy to use to assign labels in the embedding
space. There are two ways to assign labels after the laplacian
embedding. k-means can be applied and is a popular choice. But it can
also be sensitive to initialization. Discretization is another
approach which is less sensitive to random initialization. See
the 'Multiclass spectral clustering' paper referenced below for
more details on the discretization approach.
Returns
-------
labels : array of integers, shape: n_samples
The labels of the clusters.
References
----------
- Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
- A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
- Multiclass spectral clustering, 2003
Stella X. Yu, Jianbo Shi
http://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
Notes
------
The graph should contain only one connect component, elsewhere
the results make little sense.
This algorithm solves the normalized cut for k=2: it is a
normalized spectral clustering.
"""
if assign_labels not in ('kmeans', 'discretize'):
raise ValueError("The 'assign_labels' parameter should be "
"'kmeans' or 'discretize', but '%s' was given"
% assign_labels)
random_state = check_random_state(random_state)
n_components = n_clusters if n_components is None else n_components
maps = spectral_embedding(affinity, n_components=n_components,
eigen_solver=eigen_solver,
random_state=random_state,
eigen_tol=eigen_tol, drop_first=False)
if assign_labels == 'kmeans':
_, labels, _ = k_means(maps, n_clusters, random_state=random_state,
n_init=n_init)
else:
labels = discretize(maps, random_state=random_state)
return labels
class SpectralClustering(BaseEstimator, ClusterMixin):
"""Apply clustering to a projection to the normalized laplacian.
In practice Spectral Clustering is very useful when the structure of
the individual clusters is highly non-convex or more generally when
a measure of the center and spread of the cluster is not a suitable
description of the complete cluster. For instance when clusters are
nested circles on the 2D plan.
If affinity is the adjacency matrix of a graph, this method can be
used to find normalized graph cuts.
When calling ``fit``, an affinity matrix is constructed using either
kernel function such the Gaussian (aka RBF) kernel of the euclidean
distanced ``d(X, X)``::
np.exp(-gamma * d(X,X) ** 2)
or a k-nearest neighbors connectivity matrix.
Alternatively, using ``precomputed``, a user-provided affinity
matrix can be used.
Read more in the :ref:`User Guide <spectral_clustering>`.
Parameters
-----------
n_clusters : 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, RandomState instance or None, optional, default: None
A pseudo random number generator used for the initialization of the
lobpcg eigen vectors decomposition when eigen_solver == 'amg' and by
the K-Means initialization. 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`.
n_init : int, optional, default: 10
Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of
n_init consecutive runs in terms of inertia.
gamma : float, default=1.0
Kernel coefficient for rbf, poly, sigmoid, laplacian and chi2 kernels.
Ignored for ``affinity='nearest_neighbors'``.
affinity : string, array-like or callable, default 'rbf'
If a string, this may be one of 'nearest_neighbors', 'precomputed',
'rbf' or one of the kernels supported by
`sklearn.metrics.pairwise_kernels`.
Only kernels that produce similarity scores (non-negative values that
increase with similarity) should be used. This property is not checked
by the clustering algorithm.
n_neighbors : integer
Number of neighbors to use when constructing the affinity matrix using
the nearest neighbors method. Ignored for ``affinity='rbf'``.
eigen_tol : float, optional, default: 0.0
Stopping criterion for eigendecomposition of the Laplacian matrix
when using arpack eigen_solver.
assign_labels : {'kmeans', 'discretize'}, default: 'kmeans'
The strategy to use to assign labels in the embedding
space. There are two ways to assign labels after the laplacian
embedding. k-means can be applied and is a popular choice. But it can
also be sensitive to initialization. Discretization is another approach
which is less sensitive to random initialization.
degree : float, default=3
Degree of the polynomial kernel. Ignored by other kernels.
coef0 : float, default=1
Zero coefficient for polynomial and sigmoid kernels.
Ignored by other kernels.
kernel_params : dictionary of string to any, optional
Parameters (keyword arguments) and values for kernel passed as
callable object. Ignored by other kernels.
n_jobs : int, optional (default = 1)
The number of parallel jobs to run.
If ``-1``, then the number of jobs is set to the number of CPU cores.
Attributes
----------
affinity_matrix_ : array-like, shape (n_samples, n_samples)
Affinity matrix used for clustering. Available only if after calling
``fit``.
labels_ :
Labels of each point
Notes
-----
If you have an affinity matrix, such as a distance matrix,
for which 0 means identical elements, and high values means
very dissimilar elements, it can be transformed in a
similarity matrix that is well suited for the algorithm by
applying the Gaussian (RBF, heat) kernel::
np.exp(- dist_matrix ** 2 / (2. * delta ** 2))
Where ``delta`` is a free parameter representing the width of the Gaussian
kernel.
Another alternative is to take a symmetric version of the k
nearest neighbors connectivity matrix of the points.
If the pyamg package is installed, it is used: this greatly
speeds up computation.
References
----------
- Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
- A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
- Multiclass spectral clustering, 2003
Stella X. Yu, Jianbo Shi
http://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf
"""
def __init__(self, n_clusters=8, eigen_solver=None, random_state=None,
n_init=10, gamma=1., affinity='rbf', n_neighbors=10,
eigen_tol=0.0, assign_labels='kmeans', degree=3, coef0=1,
kernel_params=None, n_jobs=1):
self.n_clusters = n_clusters
self.eigen_solver = eigen_solver
self.random_state = random_state
self.n_init = n_init
self.gamma = gamma
self.affinity = affinity
self.n_neighbors = n_neighbors
self.eigen_tol = eigen_tol
self.assign_labels = assign_labels
self.degree = degree
self.coef0 = coef0
self.kernel_params = kernel_params
self.n_jobs = n_jobs
[docs] def fit(self, X, y=None):
"""Creates an affinity matrix for X using the selected affinity,
then applies spectral clustering to this affinity matrix.
Parameters
----------
X : array-like or sparse matrix, shape (n_samples, n_features)
OR, if affinity==`precomputed`, a precomputed affinity
matrix of shape (n_samples, n_samples)
y : Ignored
"""
X = check_array(X, accept_sparse=['csr', 'csc', 'coo'],
dtype=np.float64)
if X.shape[0] == X.shape[1] and self.affinity != "precomputed":
warnings.warn("The spectral clustering API has changed. ``fit``"
"now constructs an affinity matrix from data. To use"
" a custom affinity matrix, "
"set ``affinity=precomputed``.")
if self.affinity == 'nearest_neighbors':
connectivity = kneighbors_graph(X, n_neighbors=self.n_neighbors, include_self=True,
n_jobs=self.n_jobs)
self.affinity_matrix_ = 0.5 * (connectivity + connectivity.T)
elif self.affinity == 'precomputed':
self.affinity_matrix_ = X
else:
params = self.kernel_params
if params is None:
params = {}
if not callable(self.affinity):
params['gamma'] = self.gamma
params['degree'] = self.degree
params['coef0'] = self.coef0
self.affinity_matrix_ = pairwise_kernels(X, metric=self.affinity,
filter_params=True,
**params)
random_state = check_random_state(self.random_state)
self.labels_ = spectral_clustering(self.affinity_matrix_,
n_clusters=self.n_clusters,
eigen_solver=self.eigen_solver,
random_state=random_state,
n_init=self.n_init,
eigen_tol=self.eigen_tol,
assign_labels=self.assign_labels)
return self
@property
def _pairwise(self):
return self.affinity == "precomputed"
