"""Spectral Embedding"""

# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
#         Wei LI <kuantkid@gmail.com>
# License: BSD Style.

import warnings
import numpy as np

from scipy import sparse
from scipy.sparse.linalg import lobpcg
from scipy.sparse.linalg.eigen.lobpcg.lobpcg import symeig

from ..base import BaseEstimator, TransformerMixin
from ..externals import six
from ..utils import check_random_state
from ..utils.validation import atleast2d_or_csr
from ..utils.graph import graph_laplacian
from ..utils._csgraph import cs_graph_components
from ..utils.arpack import eigsh
from ..metrics.pairwise import rbf_kernel
from ..neighbors import kneighbors_graph


def _graph_connected_component(graph, node_id):
    """Find the largest graph connected components the 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 : array-like, shape: (n_samples,)
        An array of bool value indicates the indexes of the nodes
        belong to the largest connected components of the given query
        node
    """
    connected_components = np.zeros(shape=(graph.shape[0]), dtype=np.bool)
    connected_components[node_id] = True
    n_node = graph.shape[0]
    for i in range(n_node):
        last_num_component = connected_components.sum()
        _, node_to_add = np.where(graph[connected_components] != 0)
        connected_components[node_to_add] = True
        if last_num_component >= connected_components.sum():
            break
    return connected_components


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, _ = cs_graph_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):
    """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

    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):
        laplacian.flat[::n_nodes + 1] = value
    else:
        laplacian = laplacian.tocoo()
        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 comming 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,
                       mode=None):
    """Project the sample on the first eigen vectors of the graph Laplacian.

    The adjacency matrix is used to compute a normalized graph Laplacian
    whose spectrum (especially the eigen vectors associated to the
    smallest eigen values) 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 eigen vector decomposition works as expected.

    Parameters
    ----------
    adjacency : array-like or sparse matrix, shape: (n_samples, n_samples)
        The adjacency matrix of the graph to embed.

    n_components : integer, optional
        The dimension of the projection subspace.

    eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}
        The eigenvalue decomposition strategy to use. AMG requires pyamg
        to be installed. It can be faster on very large, sparse problems,
        but may also lead to instabilities.

    random_state : int seed, RandomState instance, or None (default)
        A pseudo random number generator used for the initialization of the
        lobpcg eigen vectors decomposition when eigen_solver == 'amg'.
        By default, arpack is used.

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

    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 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
    ----------
    * http://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
    """

    try:
        from pyamg import smoothed_aggregation_solver
    except ImportError:
        if eigen_solver == "amg" or mode == "amg":
            raise ValueError("The eigen_solver was set to 'amg', but pyamg is "
                             "not available.")

    if not mode is None:
        warnings.warn("'mode' was renamed to eigen_solver "
                      "and will be removed in 0.15.",
                      DeprecationWarning)
        eigen_solver = mode

    if eigen_solver is None:
        eigen_solver = 'arpack'
    elif not eigen_solver 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
    # Check that the matrices given is symmetric
    if ((not sparse.isspmatrix(adjacency) and
         not np.all((adjacency - adjacency.T) < 1e-10)) or
        (sparse.isspmatrix(adjacency) and
         (adjacency - adjacency.T).nnz > 0)):
        warnings.warn("Graph adjacency matrix should be symmetric. "
                      "Converted to be symmetric by average with its "
                      "transpose.")
    adjacency = .5 * (adjacency + adjacency.T)

    if not _graph_is_connected(adjacency):
        warnings.warn("Graph is not fully connected, spectral embedding"
                      " may not works as expected.")

    laplacian, dd = graph_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)

        # 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:
            lambdas, diffusion_map = eigsh(-laplacian, k=n_components,
                                           sigma=1.0, which='LM',
                                           tol=eigen_tol)
            embedding = diffusion_map.T[n_components::-1] * dd