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Inzynierka/Lib/site-packages/sklearn/manifold/_spectral_embedding.py

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"""Spectral Embedding."""
# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
# Wei LI <kuantkid@gmail.com>
# License: BSD 3 clause
from numbers import Integral, Real
import warnings
import numpy as np
from scipy import sparse
from scipy.linalg import eigh
from scipy.sparse.linalg import eigsh
from scipy.sparse.csgraph import connected_components
from scipy.sparse.csgraph import laplacian as csgraph_laplacian
from ..base import BaseEstimator
from ..utils import (
check_array,
check_random_state,
check_symmetric,
)
from ..utils._arpack import _init_arpack_v0
from ..utils.extmath import _deterministic_vector_sign_flip
from ..utils._param_validation import Interval, StrOptions
from ..utils.fixes import lobpcg
from ..metrics.pairwise import rbf_kernel
from ..neighbors import kneighbors_graph, NearestNeighbors
def _graph_connected_component(graph, node_id):
"""Find the largest graph connected components that contains one
given node.
Parameters
----------
graph : array-like of 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 of 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=bool)
nodes_to_explore = np.zeros(n_node, dtype=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, sparse matrix} of 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 : {ndarray, 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, 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="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
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 : int, default=2
The dimension of the projected subspace.
affinity : {'nearest_neighbors', 'rbf', 'precomputed', \
'precomputed_nearest_neighbors'} or callable, \
default='nearest_neighbors'
How to construct the affinity matrix.
- 'nearest_neighbors' : construct the affinity matrix by computing a
graph of nearest neighbors.
- 'rbf' : construct the affinity matrix by computing a radial basis
function (RBF) kernel.
- 'precomputed' : interpret ``X`` as a precomputed affinity matrix.
- 'precomputed_nearest_neighbors' : interpret ``X`` as a sparse graph
of precomputed nearest neighbors, and constructs the affinity matrix
by selecting the ``n_neighbors`` nearest neighbors.
- 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, default=None
Kernel coefficient for rbf kernel. If None, gamma will be set to
1/n_features.
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_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.
If None, then ``'arpack'`` is used.
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="lobpcg"` or `eigen_solver="amg"`
values of `tol<1e-5` may lead to convergence issues and should be
avoided.
.. versionadded:: 1.2
n_neighbors : int, default=None
Number of nearest neighbors for nearest_neighbors graph building.
If None, n_neighbors will be set to max(n_samples/10, 1).
n_jobs : int, default=None
The number of parallel jobs to run.
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
Attributes
----------
embedding_ : ndarray of shape (n_samples, n_components)
Spectral embedding of the training matrix.
affinity_matrix_ : ndarray of shape (n_samples, n_samples)
Affinity_matrix constructed from samples or precomputed.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
n_neighbors_ : int
Number of nearest neighbors effectively used.
See Also
--------
Isomap : Non-linear dimensionality reduction through Isometric Mapping.
References
----------
- :doi:`A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg
<10.1007/s11222-007-9033-z>`
- `On Spectral Clustering: Analysis and an algorithm, 2001
Andrew Y. Ng, Michael I. Jordan, Yair Weiss
<https://citeseerx.ist.psu.edu/doc_view/pid/796c5d6336fc52aa84db575fb821c78918b65f58>`_
- :doi:`Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik
<10.1109/34.868688>`
Examples
--------
>>> from sklearn.datasets import load_digits
>>> from sklearn.manifold import SpectralEmbedding
>>> X, _ = load_digits(return_X_y=True)
>>> X.shape
(1797, 64)
>>> embedding = SpectralEmbedding(n_components=2)
>>> X_transformed = embedding.fit_transform(X[:100])
>>> X_transformed.shape
(100, 2)
"""
_parameter_constraints: dict = {
"n_components": [Interval(Integral, 1, None, closed="left")],
"affinity": [
StrOptions(
{
"nearest_neighbors",
"rbf",
"precomputed",
"precomputed_nearest_neighbors",
},
),
callable,
],
"gamma": [Interval(Real, 0, None, closed="left"), None],
"random_state": ["random_state"],
"eigen_solver": [StrOptions({"arpack", "lobpcg", "amg"}), None],
"eigen_tol": [Interval(Real, 0, None, closed="left"), StrOptions({"auto"})],
"n_neighbors": [Interval(Integral, 1, None, closed="left"), None],
"n_jobs": [None, Integral],
}
def __init__(
self,
n_components=2,
*,
affinity="nearest_neighbors",
gamma=None,
random_state=None,
eigen_solver=None,
eigen_tol="auto",
n_neighbors=None,
n_jobs=None,
):
self.n_components = n_components
self.affinity = affinity
self.gamma = gamma
self.random_state = random_state
self.eigen_solver = eigen_solver
self.eigen_tol = eigen_tol
self.n_neighbors = n_neighbors
self.n_jobs = n_jobs
def _more_tags(self):
return {
"pairwise": self.affinity
in ["precomputed", "precomputed_nearest_neighbors"]
}
def _get_affinity_matrix(self, X, Y=None):
"""Calculate the affinity matrix from data
Parameters
----------
X : array-like of 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 of shape (n_samples, n_samples),
Interpret X as precomputed adjacency graph computed from
samples.
Y: Ignored
Returns
-------
affinity_matrix of shape (n_samples, n_samples)
"""
if self.affinity == "precomputed":
self.affinity_matrix_ = X
return self.affinity_matrix_
if self.affinity == "precomputed_nearest_neighbors":
estimator = NearestNeighbors(
n_neighbors=self.n_neighbors, n_jobs=self.n_jobs, metric="precomputed"
).fit(X)
connectivity = estimator.kneighbors_graph(X=X, mode="connectivity")
self.affinity_matrix_ = 0.5 * (connectivity + connectivity.T)
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_
def fit(self, X, y=None):
"""Fit the model from data in X.
Parameters
----------
X : {array-like, sparse matrix} of 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, sparse matrix}, shape (n_samples, n_samples),
Interpret X as precomputed adjacency graph computed from
samples.
y : Ignored
Not used, present for API consistency by convention.
Returns
-------
self : object
Returns the instance itself.
"""
self._validate_params()
X = self._validate_data(X, accept_sparse="csr", ensure_min_samples=2)
random_state = check_random_state(self.random_state)
affinity_matrix = self._get_affinity_matrix(X)
self.embedding_ = spectral_embedding(
affinity_matrix,
n_components=self.n_components,
eigen_solver=self.eigen_solver,
eigen_tol=self.eigen_tol,
random_state=random_state,
)
return self
def fit_transform(self, X, y=None):
"""Fit the model from data in X and transform X.
Parameters
----------
X : {array-like, sparse matrix} of 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, sparse matrix} of shape (n_samples, n_samples),
Interpret X as precomputed adjacency graph computed from
samples.
y : Ignored
Not used, present for API consistency by convention.
Returns
-------
X_new : array-like of shape (n_samples, n_components)
Spectral embedding of the training matrix.
"""
self.fit(X)
return self.embedding_
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