"""Spectral Embedding"""
# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
# Wei LI <kuantkid@gmail.com>
# License: BSD 3 clause
import warnings
import numpy as np
from scipy import sparse
from scipy.linalg import eigh
from scipy.sparse.linalg import eigsh, lobpcg
from scipy.sparse.csgraph import connected_components
from ..base import BaseEstimator
from ..externals import six
from ..utils import check_random_state, check_array, check_symmetric
from ..utils.extmath import _deterministic_vector_sign_flip
from ..metrics.pairwise import rbf_kernel
from ..neighbors import kneighbors_graph
def _graph_connected_component(graph, node_id):
"""Find the largest graph connected components that contains one
given node
Parameters
----------
graph : array-like, shape: (n_samples, n_samples)
adjacency matrix of the graph, non-zero weight means an edge
between the nodes
node_id : int
The index of the query node of the graph
Returns
-------
connected_components_matrix : array-like, shape: (n_samples,)
An array of bool value indicating the indexes of the nodes
belonging to the largest connected components of the given query
node
"""
n_node = graph.shape[0]
if sparse.issparse(graph):
# speed up row-wise access to boolean connection mask
graph = graph.tocsr()
connected_nodes = np.zeros(n_node, dtype=np.bool)
nodes_to_explore = np.zeros(n_node, dtype=np.bool)
nodes_to_explore[node_id] = True
for _ in range(n_node):
last_num_component = connected_nodes.sum()
np.logical_or(connected_nodes, nodes_to_explore, out=connected_nodes)
if last_num_component >= connected_nodes.sum():
break
indices = np.where(nodes_to_explore)[0]
nodes_to_explore.fill(False)
for i in indices:
if sparse.issparse(graph):
neighbors = graph[i].toarray().ravel()
else:
neighbors = graph[i]
np.logical_or(nodes_to_explore, neighbors, out=nodes_to_explore)
return connected_nodes
def _graph_is_connected(graph):
""" Return whether the graph is connected (True) or Not (False)
Parameters
----------
graph : array-like or sparse matrix, shape: (n_samples, n_samples)
adjacency matrix of the graph, non-zero weight means an edge
between the nodes
Returns
-------
is_connected : bool
True means the graph is fully connected and False means not
"""
if sparse.isspmatrix(graph):
# sparse graph, find all the connected components
n_connected_components, _ = connected_components(graph)
return n_connected_components == 1
else:
# dense graph, find all connected components start from node 0
return _graph_connected_component(graph, 0).sum() == graph.shape[0]
def _set_diag(laplacian, value, norm_laplacian):
"""Set the diagonal of the laplacian matrix and convert it to a
sparse format well suited for eigenvalue decomposition
Parameters
----------
laplacian : array or sparse matrix
The graph laplacian
value : float
The value of the diagonal
norm_laplacian : bool
Whether the value of the diagonal should be changed or not
Returns
-------
laplacian : array or sparse matrix
An array of matrix in a form that is well suited to fast
eigenvalue decomposition, depending on the band width of the
matrix.
"""
n_nodes = laplacian.shape[0]
# We need all entries in the diagonal to values
if not sparse.isspmatrix(laplacian):
if norm_laplacian:
laplacian.flat[::n_nodes + 1] = value
else:
laplacian = laplacian.tocoo()
if norm_laplacian:
diag_idx = (laplacian.row == laplacian.col)
laplacian.data[diag_idx] = value
# If the matrix has a small number of diagonals (as in the
# case of structured matrices coming from images), the
# dia format might be best suited for matvec products:
n_diags = np.unique(laplacian.row - laplacian.col).size
if n_diags <= 7:
# 3 or less outer diagonals on each side
laplacian = laplacian.todia()
else:
# csr has the fastest matvec and is thus best suited to
# arpack
laplacian = laplacian.tocsr()
return laplacian
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
http://dx.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 = sparse.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:
# 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
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] * 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
else:
return embedding[:n_components].T
class SpectralEmbedding(BaseEstimator):
"""Spectral embedding for non-linear dimensionality reduction.
Forms an affinity matrix given by the specified function and
applies spectral decomposition to the corresponding graph laplacian.
The resulting transformation is given by the value of the
eigenvectors for each data point.
Note : Laplacian Eigenmaps is the actual algorithm implemented here.
Read more in the :ref:`User Guide <spectral_embedding>`.
Parameters
-----------
n_components : integer, default: 2
The dimension of the projected subspace.
affinity : string or callable, default : "nearest_neighbors"
How to construct the affinity matrix.
- 'nearest_neighbors' : construct affinity matrix by knn graph
- 'rbf' : construct affinity matrix by rbf kernel
- 'precomputed' : interpret X as precomputed affinity matrix
- callable : use passed in function as affinity
the function takes in data matrix (n_samples, n_features)
and return affinity matrix (n_samples, n_samples).
gamma : float, optional, default : 1/n_features
Kernel coefficient for rbf kernel.
random_state : int, RandomState instance or None, optional, default: None
A pseudo random number generator used for the initialization of the
lobpcg eigenvectors. 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_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.
n_neighbors : int, default : max(n_samples/10 , 1)
Number of nearest neighbors for nearest_neighbors graph building.
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
----------
embedding_ : array, shape = (n_samples, n_components)
Spectral embedding of the training matrix.
affinity_matrix_ : array, shape = (n_samples, n_samples)
Affinity_matrix constructed from samples or precomputed.
References
----------
- A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323
- On Spectral Clustering: Analysis and an algorithm, 2001
Andrew Y. Ng, Michael I. Jordan, Yair Weiss
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8100
- Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
"""
def __init__(self, n_components=2, affinity="nearest_neighbors",
gamma=None, random_state=None, eigen_solver=None,
n_neighbors=None, n_jobs=1):
self.n_components = n_components
self.affinity = affinity
self.gamma = gamma
self.random_state = random_state
self.eigen_solver = eigen_solver
self.n_neighbors = n_neighbors
self.n_jobs = n_jobs
@property
def _pairwise(self):
return self.affinity == "precomputed"
def _get_affinity_matrix(self, X, Y=None):
"""Calculate the affinity matrix from data
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training vector, where n_samples is the number of samples
and n_features is the number of features.
If affinity is "precomputed"
X : array-like, shape (n_samples, n_samples),
Interpret X as precomputed adjacency graph computed from
samples.
Y: Ignored.
Returns
-------
affinity_matrix, shape (n_samples, n_samples)
"""
if self.affinity == 'precomputed':
self.affinity_matrix_ = X
return self.affinity_matrix_
if self.affinity == 'nearest_neighbors':
if sparse.issparse(X):
warnings.warn("Nearest neighbors affinity currently does "
"not support sparse input, falling back to "
"rbf affinity")
self.affinity = "rbf"
else:
self.n_neighbors_ = (self.n_neighbors
if self.n_neighbors is not None
else max(int(X.shape[0] / 10), 1))
self.affinity_matrix_ = kneighbors_graph(X, self.n_neighbors_,
include_self=True,
n_jobs=self.n_jobs)
# currently only symmetric affinity_matrix supported
self.affinity_matrix_ = 0.5 * (self.affinity_matrix_ +
self.affinity_matrix_.T)
return self.affinity_matrix_
if self.affinity == 'rbf':
self.gamma_ = (self.gamma
if self.gamma is not None else 1.0 / X.shape[1])
self.affinity_matrix_ = rbf_kernel(X, gamma=self.gamma_)
return self.affinity_matrix_
self.affinity_matrix_ = self.affinity(X)
return self.affinity_matrix_
[docs] def fit(self, X, y=None):
"""Fit the model from data in X.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Training vector, where n_samples is the number of samples
and n_features is the number of features.
If affinity is "precomputed"
X : array-like, shape (n_samples, n_samples),
Interpret X as precomputed adjacency graph computed from
samples.
Y: Ignored.
Returns
-------
self : object
Returns the instance itself.
"""
X = check_array(X, ensure_min_samples=2, estimator=self)
random_state = check_random_state(self.random_state)
if isinstance(self.affinity, six.string_types):
if self.affinity not in set(("nearest_neighbors", "rbf",
"precomputed")):
raise ValueError(("%s is not a valid affinity. Expected "
"'precomputed', 'rbf', 'nearest_neighbors' "
"or a callable.") % self.affinity)
elif not callable(self.affinity):
raise ValueError(("'affinity' is expected to be an affinity "
"name or a callable. Got: %s") % self.affinity)
affinity_matrix = self._get_affinity_matrix(X)
self.embedding_ = spectral_embedding(affinity_matrix,
n_components=self.n_components,
eigen_solver=self.eigen_solver,
random_state=random_state)
return self
