"""Spectral Embedding."""

# Author: <NAME> <<EMAIL>>
#         <NAME> <<EMAIL>>
# 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
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.fixes import lobpcg
from ..metrics.pairwise import rbf_kernel
from ..neighbors import kneighbors_graph, NearestNeighbors
from ..utils.validation import _deprecate_positional_args
from ..utils.deprecation import deprecated


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


@_deprecate_positional_args
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, 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
        Determines the random number generator used for the initialization of
        the lobpcg eigenvectors decomposition when ``solver`` == 'amg'. Pass
        an int for reproducible results across multiple function calls.
        See :term: `Glossary <random_state>`.

    eigen_tol : float, default=0.0
        Stopping criterion for eigendecomposition of the Laplacian matrix
        when using arpack eigen_solver.

    norm_laplacian : bool, default=True
        If True, then compute 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

    * Toward the Optimal Preconditioned Eigensolver: Locally Optimal
      Block Preconditioned Conjugate Gradient Method
      <NAME>
      https://doi.org/10.1137%2FS1064827500366124
    """
    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
            laplacian *= -1
            v0 = _init_arpack_v0(laplacian.shape[0], random_state)
            _, 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

    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")
        # lobpcg needs double precision floats
        laplacian = check_array(laplacian, dtype=np.float64,
                                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()
        X = random_state.rand(laplacian.shape[0], n_components + 1)
        X[:, 0] = dd.ravel()
        _, diffusion_map = lobpcg(laplacian, X, M=M, tol=1.e-5,
                                  largest=False)
        embedding = diffusion_map.T
        if norm_laplacian:
            embedding = embedding / dd
        if embedding.shape[0] == 1:
            raise ValueError

    if 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()
            _, diffusion_map = eigh(laplacian, check_finite=False)
            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()
            _, 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


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
        Determines the random number generator used for the initialization of
        the lobpcg eigenvectors when ``solver`` == 'amg'.  Pass an int for
        reproducible results across multiple function calls.
        See :term: `Glossary <random_state>`.

    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.

    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_neighbors_ : int
        Number of nearest neighbors effectively used.

    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)

    References
    ----------

    - A Tutorial on Spectral Clustering, 2007
      <NAME>
      http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323

    - On Spectral Clustering: Analysis and an algorithm, 2001
      <NAME>, <NAME>, <NAME>
      http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8100

    - Normalized cuts and image segmentation, 2000
      <NAME>, <NAME>
      http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
    """
    @_deprecate_positional_args
    def __init__(self, n_components=2, *, affinity="nearest_neighbors",
                 gamma=None, random_state=None, eigen_solver=None,
                 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.n_neighbors = n_neighbors
        self.n_jobs = n_jobs

    def _more_tags(self):
        return {'pairwise': self.affinity in ["precomputed",
                                              "precomputed_nearest_neighbors"]}

    # TODO: Remove in 1.1
    # mypy error: Decorated property not supported
    @deprecated("Attribute _pairwise was deprecated in "  # type: ignore
                "version 0.24 and will be removed in 1.1 (renaming of 0.26).")
    @property
    def _pairwise(self):
        return 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

        Returns
        -------
        self : object
            Returns the instance itself.
        """

        X = self._validate_data(X, accept_sparse='csr', ensure_min_samples=2,
                                estimator=self)

        random_state = check_random_state(self.random_state)
        if isinstance(self.affinity, str):
            if self.affinity not in {"nearest_neighbors", "rbf", "precomputed",
                                     "precomputed_nearest_neighbors"}:
                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

    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

        Returns
        -------
        X_new : array-like of shape (n_samples, n_components)
        """
        self.fit(X)
        return self.embedding_

"""Spectral Embedding."""

# Author: <NAME> <<EMAIL>>
#         <NAME> <<EMAIL>>
# 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
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.fixes import lobpcg
from ..metrics.pairwise import rbf_kernel
from ..neighbors import kneighbors_graph, NearestNeighbors
from ..utils.validation import _deprecate_positional_args
from ..utils.deprecation import deprecated


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


@_deprecate_positional_args
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, 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
        Determines the random number generator used for the initialization of
        the lobpcg eigenvectors decomposition when ``solver`` == 'amg'. Pass
        an int for reproducible results across multiple function calls.
        See :term: `Glossary <random_state>`.

    eigen_tol : float, default=0.0
        Stopping criterion for eigendecomposition of the Laplacian matrix
        when using arpack eigen_solver.

    norm_laplacian : bool, default=True
        If True, then compute 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

    * Toward the Optimal Preconditioned Eigensolver: Locally Optimal
      Block Preconditioned Conjugate Gradient Method
      <NAME>
      https://doi.org/10.1137%2FS1064827500366124
    """
    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
            laplacian *= -1
            v0 = _init_arpack_v0(laplacian.shape[0], random_state)
            _, 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

    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")
        # lobpcg needs double precision floats
        laplacian = check_array(laplacian, dtype=np.float64,
                                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()
        X = random_state.rand(laplacian.shape[0], n_components + 1)
        X[:, 0] = dd.ravel()
        _, diffusion_map = lobpcg(laplacian, X, M=M, tol=1.e-5,
                                  largest=False)
        embedding = diffusion_map.T
        if norm_laplacian:
            embedding = embedding / dd
        if embedding.shape[0] == 1:
            raise ValueError

    if 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()
            _, diffusion_map = eigh(laplacian, check_finite=False)
            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()
            _, 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


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
        Determines the random number generator used for the initialization of
        the lobpcg eigenvectors when ``solver`` == 'amg'.  Pass an int for
        reproducible results across multiple function calls.
        See :term: `Glossary <random_state>`.

    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.

    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_neighbors_ : int
        Number of nearest neighbors effectively used.

    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)

    References
    ----------

    - A Tutorial on Spectral Clustering, 2007
      <NAME>
      http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323

    - On Spectral Clustering: Analysis and an algorithm, 2001
      <NAME>, <NAME>, <NAME>
      http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8100

    - Normalized cuts and image segmentation, 2000
      <NAME>, <NAME>
      http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324
    """
    @_deprecate_positional_args
    def __init__(self, n_components=2, *, affinity="nearest_neighbors",
                 gamma=None, random_state=None, eigen_solver=None,
                 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.n_neighbors = n_neighbors
        self.n_jobs = n_jobs

    def _more_tags(self):
        return {'pairwise': self.affinity in ["precomputed",
                                              "precomputed_nearest_neighbors"]}

    # TODO: Remove in 1.1
    # mypy error: Decorated property not supported
    @deprecated("Attribute _pairwise was deprecated in "  # type: ignore
                "version 0.24 and will be removed in 1.1 (renaming of 0.26).")
    @property
    def _pairwise(self):
        return 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

        Returns
        -------
        self : object
            Returns the instance itself.
        """

        X = self._validate_data(X, accept_sparse='csr', ensure_min_samples=2,
                                estimator=self)

        random_state = check_random_state(self.random_state)
        if isinstance(self.affinity, str):
            if self.affinity not in {"nearest_neighbors", "rbf", "precomputed",
                                     "precomputed_nearest_neighbors"}:
                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

    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

        Returns
        -------
        X_new : array-like of shape (n_samples, n_components)
        """
        self.fit(X)
        return self.embedding_

Spectral Embedding. # Author: <NAME> <<EMAIL>> #         <NAME> <<EMAIL>> # License: BSD 3 clause 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. # speed up row-wise access to boolean connection mask 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. # sparse graph, find all the connected components # dense graph, find all connected components start from node 0 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. # We need all entries in the diagonal to values # 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: # 3 or less outer diagonals on each side # csr has the fastest matvec and is thus best suited to # arpack 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         Determines the random number generator used for the initialization of         the lobpcg eigenvectors decomposition when ``solver`` == 'amg'. Pass         an int for reproducible results across multiple function calls.         See :term: `Glossary <random_state>`.      eigen_tol : float, default=0.0         Stopping criterion for eigendecomposition of the Laplacian matrix         when using arpack eigen_solver.      norm_laplacian : bool, default=True         If True, then compute 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      * Toward the Optimal Preconditioned Eigensolver: Locally Optimal       Block Preconditioned Conjugate Gradient Method       <NAME>       https://doi.org/10.1137%2FS1064827500366124 # Whether to drop the first eigenvector # 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 # 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. # We are computing the opposite of the laplacian inplace so as # to spare a memory allocation of a possibly very large array # When submatrices are exactly singular, an LU decomposition # in arpack fails. We fallback to lobpcg # Revert the laplacian to its opposite to have lobpcg work # Use AMG to get a preconditioner and speed up the eigenvalue # problem. # lobpcg needs double precision floats # 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. # lobpcg needs double precision floats # see note above under arpack why lobpcg has problems with small # number of nodes # lobpcg will fallback to eigh, so we short circuit it # We increase the number of eigenvectors requested, as lobpcg # doesn't behave well in low dimension 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         Determines the random number generator used for the initialization of         the lobpcg eigenvectors when ``solver`` == 'amg'.  Pass an int for         reproducible results across multiple function calls.         See :term: `Glossary <random_state>`.      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.      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_neighbors_ : int         Number of nearest neighbors effectively used.      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)      References     ----------      - A Tutorial on Spectral Clustering, 2007       <NAME>       http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.9323      - On Spectral Clustering: Analysis and an algorithm, 2001       <NAME>, <NAME>, <NAME>       http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.8100      - Normalized cuts and image segmentation, 2000       <NAME>, <NAME>       http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.160.2324 # TODO: Remove in 1.1 # mypy error: Decorated property not supported # type: ignore 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) # currently only symmetric affinity_matrix supported 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          Returns         -------         self : object             Returns the instance itself. 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          Returns         -------         X_new : array-like of shape (n_samples, n_components)