"""Isomap for manifold learning"""

# Author: Jake Vanderplas  -- <vanderplas@astro.washington.edu>
# License: BSD 3 clause (C) 2011

import numpy as np
from ..base import BaseEstimator, TransformerMixin
from ..neighbors import NearestNeighbors, kneighbors_graph
from ..utils.validation import check_is_fitted
from ..utils.validation import _deprecate_positional_args
from ..utils.graph import graph_shortest_path
from ..decomposition import KernelPCA
from ..preprocessing import KernelCenterer


class Isomap(TransformerMixin, BaseEstimator):
    """Isomap Embedding

    Non-linear dimensionality reduction through Isometric Mapping

    Read more in the :ref:`User Guide <isomap>`.

    Parameters
    ----------
    n_neighbors : int, default=5
        number of neighbors to consider for each point.

    n_components : int, default=2
        number of coordinates for the manifold

    eigen_solver : {'auto', 'arpack', 'dense'}, default='auto'
        'auto' : Attempt to choose the most efficient solver
        for the given problem.

        'arpack' : Use Arnoldi decomposition to find the eigenvalues
        and eigenvectors.

        'dense' : Use a direct solver (i.e. LAPACK)
        for the eigenvalue decomposition.

    tol : float, default=0
        Convergence tolerance passed to arpack or lobpcg.
        not used if eigen_solver == 'dense'.

    max_iter : int, default=None
        Maximum number of iterations for the arpack solver.
        not used if eigen_solver == 'dense'.

    path_method : {'auto', 'FW', 'D'}, default='auto'
        Method to use in finding shortest path.

        'auto' : attempt to choose the best algorithm automatically.

        'FW' : Floyd-Warshall algorithm.

        'D' : Dijkstra's algorithm.

    neighbors_algorithm : {'auto', 'brute', 'kd_tree', 'ball_tree'}, \
                          default='auto'
        Algorithm to use for nearest neighbors search,
        passed to neighbors.NearestNeighbors instance.

    n_jobs : int or None, 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.

    metric : string, or callable, default="minkowski"
        The metric to use when calculating distance between instances in a
        feature array. If metric is a string or callable, it must be one of
        the options allowed by :func:`sklearn.metrics.pairwise_distances` for
        its metric parameter.
        If metric is "precomputed", X is assumed to be a distance matrix and
        must be square. X may be a :term:`Glossary <sparse graph>`.

        .. versionadded:: 0.22

    p : int, default=2
        Parameter for the Minkowski metric from
        sklearn.metrics.pairwise.pairwise_distances. When p = 1, this is
        equivalent to using manhattan_distance (l1), and euclidean_distance
        (l2) for p = 2. For arbitrary p, minkowski_distance (l_p) is used.

        .. versionadded:: 0.22

    metric_params : dict, default=None
        Additional keyword arguments for the metric function.

        .. versionadded:: 0.22

    Attributes
    ----------
    embedding_ : array-like, shape (n_samples, n_components)
        Stores the embedding vectors.

    kernel_pca_ : object
        :class:`~sklearn.decomposition.KernelPCA` object used to implement the
        embedding.

    nbrs_ : sklearn.neighbors.NearestNeighbors instance
        Stores nearest neighbors instance, including BallTree or KDtree
        if applicable.

    dist_matrix_ : array-like, shape (n_samples, n_samples)
        Stores the geodesic distance matrix of training data.

    Examples
    --------
    >>> from sklearn.datasets import load_digits
    >>> from sklearn.manifold import Isomap
    >>> X, _ = load_digits(return_X_y=True)
    >>> X.shape
    (1797, 64)
    >>> embedding = Isomap(n_components=2)
    >>> X_transformed = embedding.fit_transform(X[:100])
    >>> X_transformed.shape
    (100, 2)

    References
    ----------

    .. [1] Tenenbaum, J.B.; De Silva, V.; & Langford, J.C. A global geometric
           framework for nonlinear dimensionality reduction. Science 290 (5500)
    """
    @_deprecate_positional_args
    def __init__(self, *, n_neighbors=5, n_components=2, eigen_solver='auto',
                 tol=0, max_iter=None, path_method='auto',
                 neighbors_algorithm='auto', n_jobs=None, metric='minkowski',
                 p=2, metric_params=None):
        self.n_neighbors = n_neighbors
        self.n_components = n_components
        self.eigen_solver = eigen_solver
        self.tol = tol
        self.max_iter = max_iter
        self.path_method = path_method
        self.neighbors_algorithm = neighbors_algorithm
        self.n_jobs = n_jobs
        self.metric = metric
        self.p = p
        self.metric_params = metric_params

    def _fit_transform(self, X):
        self.nbrs_ = NearestNeighbors(n_neighbors=self.n_neighbors,
                                      algorithm=self.neighbors_algorithm,
                                      metric=self.metric, p=self.p,
                                      metric_params=self.metric_params,
                                      n_jobs=self.n_jobs)
        self.nbrs_.fit(X)
        self.n_features_in_ = self.nbrs_.n_features_in_

        self.kernel_pca_ = KernelPCA(n_components=self.n_components,
                                     kernel="precomputed",
                                     eigen_solver=self.eigen_solver,
                                     tol=self.tol, max_iter=self.max_iter,
                                     n_jobs=self.n_jobs)

        kng = kneighbors_graph(self.nbrs_, self.n_neighbors,
                               metric=self.metric, p=self.p,
                               metric_params=self.metric_params,
                               mode='distance', n_jobs=self.n_jobs)

        self.dist_matrix_ = graph_shortest_path(kng,
                                                method=self.path_method,
                                                directed=False)
        G = self.dist_matrix_ ** 2
        G *= -0.5

        self.embedding_ = self.kernel_pca_.fit_transform(G)

    def reconstruction_error(self):
        """Compute the reconstruction error for the embedding.

        Returns
        -------
        reconstruction_error : float

        Notes
        -----
        The cost function of an isomap embedding is

        ``E = frobenius_norm[K(D) - K(D_fit)] / n_samples``

        Where D is the matrix of distances for the input data X,
        D_fit is the matrix of distances for the output embedding X_fit,
        and K is the isomap kernel:

        ``K(D) = -0.5 * (I - 1/n_samples) * D^2 * (I - 1/n_samples)``
        """
        G = -0.5 * self.dist_matrix_ ** 2
        G_center = KernelCenterer().fit_transform(G)
        evals = self.kernel_pca_.lambdas_
        return np.sqrt(np.sum(G_center ** 2) - np.sum(evals ** 2)) / G.shape[0]

    def fit(self, X, y=None):
        """Compute the embedding vectors for data X

        Parameters
        ----------
        X : {array-like, sparse graph, BallTree, KDTree, NearestNeighbors}
            Sample data, shape = (n_samples, n_features), in the form of a
            numpy array, sparse graph, precomputed tree, or NearestNeighbors
            object.

        y : Ignored

        Returns
        -------
        self : returns an instance of self.
        """
        self._fit_transform(X)
        return self

    def fit_transform(self, X, y=None):
        """Fit the model from data in X and transform X.

        Parameters
        ----------
        X : {array-like, sparse graph, BallTree, KDTree}
            Training vector, where n_samples in the number of samples
            and n_features is the number of features.

        y : Ignored

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

    def transform(self, X):
        """Transform X.

        This is implemented by linking the points X into the graph of geodesic
        distances of the training data. First the `n_neighbors` nearest
        neighbors of X are found in the training data, and from these the
        shortest geodesic distances from each point in X to each point in
        the training data are computed in order to construct the kernel.
        The embedding of X is the projection of this kernel onto the
        embedding vectors of the training set.

        Parameters
        ----------
        X : array-like, shape (n_queries, n_features)
            If neighbors_algorithm='precomputed', X is assumed to be a
            distance matrix or a sparse graph of shape
            (n_queries, n_samples_fit).

        Returns
        -------
        X_new : array-like, shape (n_queries, n_components)
        """
        check_is_fitted(self)
        distances, indices = self.nbrs_.kneighbors(X, return_distance=True)

        # Create the graph of shortest distances from X to
        # training data via the nearest neighbors of X.
        # This can be done as a single array operation, but it potentially
        # takes a lot of memory.  To avoid that, use a loop:

        n_samples_fit = self.nbrs_.n_samples_fit_
        n_queries = distances.shape[0]
        G_X = np.zeros((n_queries, n_samples_fit))
        for i in range(n_queries):
            G_X[i] = np.min(self.dist_matrix_[indices[i]] +
                            distances[i][:, None], 0)

        G_X **= 2
        G_X *= -0.5

        return self.kernel_pca_.transform(G_X)
