t_sne.py

# Author: Alexander Fabisch  -- <afabisch@informatik.uni-bremen.de>
# Author: Christopher Moody <chrisemoody@gmail.com>
# Author: Nick Travers <nickt@squareup.com>
# License: BSD 3 clause (C) 2014

# This is the exact and Barnes-Hut t-SNE implementation. There are other
# modifications of the algorithm:
# * Fast Optimization for t-SNE:
#   http://cseweb.ucsd.edu/~lvdmaaten/workshops/nips2010/papers/vandermaaten.pdf

from time import time
import numpy as np
from scipy import linalg
import scipy.sparse as sp
from scipy.spatial.distance import pdist
from scipy.spatial.distance import squareform
from scipy.sparse import csr_matrix
from ..neighbors import NearestNeighbors
from ..base import BaseEstimator
from ..utils import check_array
from ..utils import check_random_state
from ..decomposition import PCA
from ..metrics.pairwise import pairwise_distances
from . import _utils
from . import _barnes_hut_tsne
from ..externals.six import string_types
from ..utils import deprecated


MACHINE_EPSILON = np.finfo(np.double).eps


def _joint_probabilities(distances, desired_perplexity, verbose):
    """Compute joint probabilities p_ij from distances.

    Parameters
    ----------
    distances : array, shape (n_samples * (n_samples-1) / 2,)
        Distances of samples are stored as condensed matrices, i.e.
        we omit the diagonal and duplicate entries and store everything
        in a one-dimensional array.

    desired_perplexity : float
        Desired perplexity of the joint probability distributions.

    verbose : int
        Verbosity level.

    Returns
    -------
    P : array, shape (n_samples * (n_samples-1) / 2,)
        Condensed joint probability matrix.
    """
    # Compute conditional probabilities such that they approximately match
    # the desired perplexity
    distances = distances.astype(np.float32, copy=False)
    conditional_P = _utils._binary_search_perplexity(
        distances, None, desired_perplexity, verbose)
    P = conditional_P + conditional_P.T
    sum_P = np.maximum(np.sum(P), MACHINE_EPSILON)
    P = np.maximum(squareform(P) / sum_P, MACHINE_EPSILON)
    return P


def _joint_probabilities_nn(distances, neighbors, desired_perplexity, verbose):
    """Compute joint probabilities p_ij from distances using just nearest
    neighbors.

    This method is approximately equal to _joint_probabilities. The latter
    is O(N), but limiting the joint probability to nearest neighbors improves
    this substantially to O(uN).

    Parameters
    ----------
    distances : array, shape (n_samples, k)
        Distances of samples to its k nearest neighbors.

    neighbors : array, shape (n_samples, k)
        Indices of the k nearest-neighbors for each samples.

    desired_perplexity : float
        Desired perplexity of the joint probability distributions.

    verbose : int
        Verbosity level.

    Returns
    -------
    P : csr sparse matrix, shape (n_samples, n_samples)
        Condensed joint probability matrix with only nearest neighbors.
    """
    t0 = time()
    # Compute conditional probabilities such that they approximately match
    # the desired perplexity
    n_samples, k = neighbors.shape
    distances = distances.astype(np.float32, copy=False)
    neighbors = neighbors.astype(np.int64, copy=False)
    conditional_P = _utils._binary_search_perplexity(
        distances, neighbors, desired_perplexity, verbose)
    assert np.all(np.isfinite(conditional_P)), \
        "All probabilities should be finite"

    # Symmetrize the joint probability distribution using sparse operations
    P = csr_matrix((conditional_P.ravel(), neighbors.ravel(),
                    range(0, n_samples * k + 1, k)),
                   shape=(n_samples, n_samples))
    P = P + P.T

    # Normalize the joint probability distribution
    sum_P = np.maximum(P.sum(), MACHINE_EPSILON)
    P /= sum_P

    assert np.all(np.abs(P.data) <= 1.0)
    if verbose >= 2:
        duration = time() - t0
        print("[t-SNE] Computed conditional probabilities in {:.3f}s"
              .format(duration))
    return P


def _kl_divergence(params, P, degrees_of_freedom, n_samples, n_components,
                   skip_num_points=0):
    """t-SNE objective function: gradient of the KL divergence
    of p_ijs and q_ijs and the absolute error.

    Parameters
    ----------
    params : array, shape (n_params,)
        Unraveled embedding.

    P : array, shape (n_samples * (n_samples-1) / 2,)
        Condensed joint probability matrix.

    degrees_of_freedom : float
        Degrees of freedom of the Student's-t distribution.

    n_samples : int
        Number of samples.

    n_components : int
        Dimension of the embedded space.

    skip_num_points : int (optional, default:0)
        This does not compute the gradient for points with indices below
        `skip_num_points`. This is useful when computing transforms of new
        data where you'd like to keep the old data fixed.

    Returns
    -------
    kl_divergence : float
        Kullback-Leibler divergence of p_ij and q_ij.

    grad : array, shape (n_params,)
        Unraveled gradient of the Kullback-Leibler divergence with respect to
        the embedding.
    """
    X_embedded = params.reshape(n_samples, n_components)

    # Q is a heavy-tailed distribution: Student's t-distribution
    dist = pdist(X_embedded, "sqeuclidean")
    dist /= degrees_of_freedom
    dist += 1.
    dist **= (degrees_of_freedom + 1.0) / -2.0
    Q = np.maximum(dist / (2.0 * np.sum(dist)), MACHINE_EPSILON)

    # Optimization trick below: np.dot(x, y) is faster than
    # np.sum(x * y) because it calls BLAS

    # Objective: C (Kullback-Leibler divergence of P and Q)
    kl_divergence = 2.0 * np.dot(P, np.log(np.maximum(P, MACHINE_EPSILON) / Q))

    # Gradient: dC/dY
    # pdist always returns double precision distances. Thus we need to take
    grad = np.ndarray((n_samples, n_components), dtype=params.dtype)
    PQd = squareform((P - Q) * dist)
    for i in range(skip_num_points, n_samples):
        grad[i] = np.dot(np.ravel(PQd[i], order='K'),
                         X_embedded[i] - X_embedded)
    grad = grad.ravel()
    c = 2.0 * (degrees_of_freedom + 1.0) / degrees_of_freedom
    grad *= c

    return kl_divergence, grad


def _kl_divergence_bh(params, P, degrees_of_freedom, n_samples, n_components,
                      angle=0.5, skip_num_points=0, verbose=False):
    """t-SNE objective function: KL divergence of p_ijs and q_ijs.

    Uses Barnes-Hut tree methods to calculate the gradient that
    runs in O(NlogN) instead of O(N^2)

    Parameters
    ----------
    params : array, shape (n_params,)
        Unraveled embedding.

    P : csr sparse matrix, shape (n_samples, n_sample)
        Sparse approximate joint probability matrix, computed only for the
        k nearest-neighbors and symmetrized.

    degrees_of_freedom : float
        Degrees of freedom of the Student's-t distribution.

    n_samples : int
        Number of samples.

    n_components : int
        Dimension of the embedded space.

    angle : float (default: 0.5)
        This is the trade-off between speed and accuracy for Barnes-Hut T-SNE.
        'angle' is the angular size (referred to as theta in [3]) of a distant
        node as measured from a point. If this size is below 'angle' then it is
        used as a summary node of all points contained within it.
        This method is not very sensitive to changes in this parameter
        in the range of 0.2 - 0.8. Angle less than 0.2 has quickly increasing
        computation time and angle greater 0.8 has quickly increasing error.

    skip_num_points : int (optional, default:0)
        This does not compute the gradient for points with indices below
        `skip_num_points`. This is useful when computing transforms of new
        data where you'd like to keep the old data fixed.

    verbose : int
        Verbosity level.

    Returns
    -------
    kl_divergence : float
        Kullback-Leibler divergence of p_ij and q_ij.

    grad : array, shape (n_params,)
        Unraveled gradient of the Kullback-Leibler divergence with respect to
        the embedding.
    """
    params = params.astype(np.float32, copy=False)
    X_embedded = params.reshape(n_samples, n_components)

    val_P = P.data.astype(np.float32, copy=False)
    neighbors = P.indices.astype(np.int64, copy=False)
    indptr = P.indptr.astype(np.int64, copy=False)

    grad = np.zeros(X_embedded.shape, dtype=np.float32)
    error = _barnes_hut_tsne.gradient(val_P, X_embedded, neighbors, indptr,
                                      grad, angle, n_components, verbose,
                                      dof=degrees_of_freedom)
    c = 2.0 * (degrees_of_freedom + 1.0) / degrees_of_freedom
    grad = grad.ravel()
    grad *= c

    return error, grad


def _gradient_descent(objective, p0, it, n_iter,
                      n_iter_check=1, n_iter_without_progress=300,
                      momentum=0.8, learning_rate=200.0, min_gain=0.01,
                      min_grad_norm=1e-7, verbose=0, args=None, kwargs=None):
    """Batch gradient descent with momentum and individual gains.

    Parameters
    ----------
    objective : function or callable
        Should return a tuple of cost and gradient for a given parameter
        vector. When expensive to compute, the cost can optionally
        be None and can be computed every n_iter_check steps using
        the objective_error function.

    p0 : array-like, shape (n_params,)
        Initial parameter vector.

    it : int
        Current number of iterations (this function will be called more than
        once during the optimization).

    n_iter : int
        Maximum number of gradient descent iterations.

    n_iter_check : int
        Number of iterations before evaluating the global error. If the error
        is sufficiently low, we abort the optimization.

    n_iter_without_progress : int, optional (default: 300)
        Maximum number of iterations without progress before we abort the
        optimization.

    momentum : float, within (0.0, 1.0), optional (default: 0.8)
        The momentum generates a weight for previous gradients that decays
        exponentially.

    learning_rate : float, optional (default: 200.0)
        The learning rate for t-SNE is usually in the range [10.0, 1000.0]. If
        the learning rate is too high, the data may look like a 'ball' with any
        point approximately equidistant from its nearest neighbours. If the
        learning rate is too low, most points may look compressed in a dense
        cloud with few outliers.

    min_gain : float, optional (default: 0.01)
        Minimum individual gain for each parameter.

    min_grad_norm : float, optional (default: 1e-7)
        If the gradient norm is below this threshold, the optimization will
        be aborted.

    verbose : int, optional (default: 0)
        Verbosity level.

    args : sequence
        Arguments to pass to objective function.

    kwargs : dict
        Keyword arguments to pass to objective function.

    Returns
    -------
    p : array, shape (n_params,)
        Optimum parameters.

    error : float
        Optimum.

    i : int
        Last iteration.
    """
    if args is None:
        args = []
    if kwargs is None:
        kwargs = {}

    p = p0.copy().ravel()
    update = np.zeros_like(p)
    gains = np.ones_like(p)
    error = np.finfo(np.float).max
    best_error = np.finfo(np.float).max
    best_iter = i = it

    tic = time()
    for i in range(it, n_iter):
        error, grad = objective(p, *args, **kwargs)
        grad_norm = linalg.norm(grad)

        inc = update * grad < 0.0
        dec = np.invert(inc)
        gains[inc] += 0.2
        gains[dec] *= 0.8
        np.clip(gains, min_gain, np.inf, out=gains)
        grad *= gains
        update = momentum * update - learning_rate * grad
        p += update

        if (i + 1) % n_iter_check == 0:
            toc = time()
            duration = toc - tic
            tic = toc

            if verbose >= 2:
                print("[t-SNE] Iteration %d: error = %.7f,"
                      " gradient norm = %.7f"
                      " (%s iterations in %0.3fs)"
                      % (i + 1, error, grad_norm, n_iter_check, duration))

            if error < best_error:
                best_error = error
                best_iter = i
            elif i - best_iter > n_iter_without_progress:
                if verbose >= 2:
                    print("[t-SNE] Iteration %d: did not make any progress "
                          "during the last %d episodes. Finished."
                          % (i + 1, n_iter_without_progress))
                break
            if grad_norm <= min_grad_norm:
                if verbose >= 2:
                    print("[t-SNE] Iteration %d: gradient norm %f. Finished."
                          % (i + 1, grad_norm))
                break

    return p, error, i


def trustworthiness(X, X_embedded, n_neighbors=5, precomputed=False):
    """Expresses to what extent the local structure is retained.

    The trustworthiness is within [0, 1]. It is defined as

    .. math::

        T(k) = 1 - \frac{2}{nk (2n - 3k - 1)} \sum^n_{i=1}
            \sum_{j \in U^{(k)}_i} (r(i, j) - k)

    where :math:`r(i, j)` is the rank of the embedded datapoint j
    according to the pairwise distances between the embedded datapoints,
    :math:`U^{(k)}_i` is the set of points that are in the k nearest
    neighbors in the embedded space but not in the original space.

    * "Neighborhood Preservation in Nonlinear Projection Methods: An
      Experimental Study"
      J. Venna, S. Kaski
    * "Learning a Parametric Embedding by Preserving Local Structure"
      L.J.P. van der Maaten

    Parameters
    ----------
    X : array, shape (n_samples, n_features) or (n_samples, n_samples)
        If the metric is 'precomputed' X must be a square distance
        matrix. Otherwise it contains a sample per row.

    X_embedded : array, shape (n_samples, n_components)
        Embedding of the training data in low-dimensional space.

    n_neighbors : int, optional (default: 5)
        Number of neighbors k that will be considered.

    precomputed : bool, optional (default: False)
        Set this flag if X is a precomputed square distance matrix.

    Returns
    -------
    trustworthiness : float
        Trustworthiness of the low-dimensional embedding.
    """
    if precomputed:
        dist_X = X
    else:
        dist_X = pairwise_distances(X, squared=True)
    dist_X_embedded = pairwise_distances(X_embedded, squared=True)
    ind_X = np.argsort(dist_X, axis=1)
    ind_X_embedded = np.argsort(dist_X_embedded, axis=1)[:, 1:n_neighbors + 1]

    n_samples = X.shape[0]
    t = 0.0
    ranks = np.zeros(n_neighbors)
    for i in range(n_samples):
        for j in range(n_neighbors):
            ranks[j] = np.where(ind_X[i] == ind_X_embedded[i, j])[0][0]
        ranks -= n_neighbors
        t += np.sum(ranks[ranks > 0])
    t = 1.0 - t * (2.0 / (n_samples * n_neighbors *
                          (2.0 * n_samples - 3.0 * n_neighbors - 1.0)))
    return t


class TSNE(BaseEstimator):
    """t-distributed Stochastic Neighbor Embedding.

    t-SNE [1] is a tool to visualize high-dimensional data. It converts
    similarities between data points to joint probabilities and tries
    to minimize the Kullback-Leibler divergence between the joint
    probabilities of the low-dimensional embedding and the
    high-dimensional data. t-SNE has a cost function that is not convex,
    i.e. with different initializations we can get different results.

    It is highly recommended to use another dimensionality reduction
    method (e.g. PCA for dense data or TruncatedSVD for sparse data)
    to reduce the number of dimensions to a reasonable amount (e.g. 50)
    if the number of features is very high. This will suppress some
    noise and speed up the computation of pairwise distances between
    samples. For more tips see Laurens van der Maaten's FAQ [2].

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

    Parameters
    ----------
    n_components : int, optional (default: 2)
        Dimension of the embedded space.

    perplexity : float, optional (default: 30)
        The perplexity is related to the number of nearest neighbors that
        is used in other manifold learning algorithms. Larger datasets
        usually require a larger perplexity. Consider selecting a value
        between 5 and 50. The choice is not extremely critical since t-SNE
        is quite insensitive to this parameter.

    early_exaggeration : float, optional (default: 12.0)
        Controls how tight natural clusters in the original space are in
        the embedded space and how much space will be between them. For
        larger values, the space between natural clusters will be larger
        in the embedded space. Again, the choice of this parameter is not
        very critical. If the cost function increases during initial
        optimization, the early exaggeration factor or the learning rate
        might be too high.

    learning_rate : float, optional (default: 200.0)
        The learning rate for t-SNE is usually in the range [10.0, 1000.0]. If
        the learning rate is too high, the data may look like a 'ball' with any
        point approximately equidistant from its nearest neighbours. If the
        learning rate is too low, most points may look compressed in a dense
        cloud with few outliers. If the cost function gets stuck in a bad local
        minimum increasing the learning rate may help.

    n_iter : int, optional (default: 1000)
        Maximum number of iterations for the optimization. Should be at
        least 250.

    n_iter_without_progress : int, optional (default: 300)
        Maximum number of iterations without progress before we abort the
        optimization, used after 250 initial iterations with early
        exaggeration. Note that progress is only checked every 50 iterations so
        this value is rounded to the next multiple of 50.

        .. versionadded:: 0.17
           parameter *n_iter_without_progress* to control stopping criteria.

    min_grad_norm : float, optional (default: 1e-7)
        If the gradient norm is below this threshold, the optimization will
        be stopped.

    metric : string or callable, optional
        The metric to use when calculating distance between instances in a
        feature array. If metric is a string, it must be one of the options
        allowed by scipy.spatial.distance.pdist for its metric parameter, or
        a metric listed in pairwise.PAIRWISE_DISTANCE_FUNCTIONS.
        If metric is "precomputed", X is assumed to be a distance matrix.
        Alternatively, if metric is a callable function, it is called on each
        pair of instances (rows) and the resulting value recorded. The callable
        should take two arrays from X as input and return a value indicating
        the distance between them. The default is "euclidean" which is
        interpreted as squared euclidean distance.

    init : string or numpy array, optional (default: "random")
        Initialization of embedding. Possible options are 'random', 'pca',
        and a numpy array of shape (n_samples, n_components).
        PCA initialization cannot be used with precomputed distances and is
        usually more globally stable than random initialization.

    verbose : int, optional (default: 0)
        Verbosity level.

    random_state : int, RandomState instance or None, optional (default: None)
        If int, random_state is the seed used by the random number generator;
        If RandomState instance, random_state is the random number generator;
        If None, the random number generator is the RandomState instance used
        by `np.random`.  Note that different initializations might result in
        different local minima of the cost function.

    method : string (default: 'barnes_hut')
        By default the gradient calculation algorithm uses Barnes-Hut
        approximation running in O(NlogN) time. method='exact'
        will run on the slower, but exact, algorithm in O(N^2) time. The
        exact algorithm should be used when nearest-neighbor errors need
        to be better than 3%. However, the exact method cannot scale to
        millions of examples.

        .. versionadded:: 0.17
           Approximate optimization *method* via the Barnes-Hut.

    angle : float (default: 0.5)
        Only used if method='barnes_hut'
        This is the trade-off between speed and accuracy for Barnes-Hut T-SNE.
        'angle' is the angular size (referred to as theta in [3]) of a distant
        node as measured from a point. If this size is below 'angle' then it is
        used as a summary node of all points contained within it.
        This method is not very sensitive to changes in this parameter
        in the range of 0.2 - 0.8. Angle less than 0.2 has quickly increasing
        computation time and angle greater 0.8 has quickly increasing error.

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

    kl_divergence_ : float
        Kullback-Leibler divergence after optimization.

    n_iter_ : int
        Number of iterations run.

    Examples
    --------

    >>> import numpy as np
    >>> from sklearn.manifold import TSNE
    >>> X = np.array([[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 1]])
    >>> X_embedded = TSNE(n_components=2).fit_transform(X)
    >>> X_embedded.shape
    (4, 2)

    References
    ----------

    [1] van der Maaten, L.J.P.; Hinton, G.E. Visualizing High-Dimensional Data
        Using t-SNE. Journal of Machine Learning Research 9:2579-2605, 2008.

    [2] van der Maaten, L.J.P. t-Distributed Stochastic Neighbor Embedding
        http://homepage.tudelft.nl/19j49/t-SNE.html

    [3] L.J.P. van der Maaten. Accelerating t-SNE using Tree-Based Algorithms.
        Journal of Machine Learning Research 15(Oct):3221-3245, 2014.
        http://lvdmaaten.github.io/publications/papers/JMLR_2014.pdf
    """
    # Control the number of exploration iterations with early_exaggeration on
    _EXPLORATION_N_ITER = 250

    # Control the number of iterations between progress checks
    _N_ITER_CHECK = 50

    def __init__(self, n_components=2, perplexity=30.0,
                 early_exaggeration=12.0, learning_rate=200.0, n_iter=1000,
                 n_iter_without_progress=300, min_grad_norm=1e-7,
                 metric="euclidean", init="random", verbose=0,
                 random_state=None, method='barnes_hut', angle=0.5):
        self.n_components = n_components
        self.perplexity = perplexity
        self.early_exaggeration = early_exaggeration
        self.learning_rate = learning_rate
        self.n_iter = n_iter
        self.n_iter_without_progress = n_iter_without_progress
        self.min_grad_norm = min_grad_norm
        self.metric = metric
        self.init = init
        self.verbose = verbose
        self.random_state = random_state
        self.method = method
        self.angle = angle

    def _fit(self, X, skip_num_points=0):
        """Fit the model using X as training data.

        Note that sparse arrays can only be handled by method='exact'.
        It is recommended that you convert your sparse array to dense
        (e.g. `X.toarray()`) if it fits in memory, or otherwise using a
        dimensionality reduction technique (e.g. TruncatedSVD).

        Parameters
        ----------
        X : array, shape (n_samples, n_features) or (n_samples, n_samples)
            If the metric is 'precomputed' X must be a square distance
            matrix. Otherwise it contains a sample per row. Note that this
            when method='barnes_hut', X cannot be a sparse array and if need be
            will be converted to a 32 bit float array. Method='exact' allows
            sparse arrays and 64bit floating point inputs.

        skip_num_points : int (optional, default:0)
            This does not compute the gradient for points with indices below
            `skip_num_points`. This is useful when computing transforms of new
            data where you'd like to keep the old data fixed.
        """
        if self.method not in ['barnes_hut', 'exact']:
            raise ValueError("'method' must be 'barnes_hut' or 'exact'")
        if self.angle < 0.0 or self.angle > 1.0:
            raise ValueError("'angle' must be between 0.0 - 1.0")
        if self.metric == "precomputed":
            if isinstance(self.init, string_types) and self.init == 'pca':
                raise ValueError("The parameter init=\"pca\" cannot be "
                                 "used with metric=\"precomputed\".")
            if X.shape[0] != X.shape[1]:
                raise ValueError("X should be a square distance matrix")
            if np.any(X < 0):
                raise ValueError("All distances should be positive, the "
                                 "precomputed distances given as X is not "
                                 "correct")
        if self.method == 'barnes_hut' and sp.issparse(X):
            raise TypeError('A sparse matrix was passed, but dense '
                            'data is required for method="barnes_hut". Use '
                            'X.toarray() to convert to a dense numpy array if '
                            'the array is small enough for it to fit in '
                            'memory. Otherwise consider dimensionality '
                            'reduction techniques (e.g. TruncatedSVD)')
        else:
            X = check_array(X, accept_sparse=['csr', 'csc', 'coo'],
                            dtype=[np.float32, np.float64])
        if self.method == 'barnes_hut' and self.n_components > 3:
            raise ValueError("'n_components' should be inferior to 4 for the "
                             "barnes_hut algorithm as it relies on "
                             "quad-tree or oct-tree.")
        random_state = check_random_state(self.random_state)

        if self.early_exaggeration < 1.0:
            raise ValueError("early_exaggeration must be at least 1, but is {}"
                             .format(self.early_exaggeration))

        if self.n_iter < 250:
            raise ValueError("n_iter should be at least 250")

        n_samples = X.shape[0]

        neighbors_nn = None
        if self.method == "exact":
            # Retrieve the distance matrix, either using the precomputed one or
            # computing it.
            if self.metric == "precomputed":
                distances = X
            else:
                if self.verbose:
                    print("[t-SNE] Computing pairwise distances...")

                if self.metric == "euclidean":
                    distances = pairwise_distances(X, metric=self.metric,
                                                   squared=True)
                else:
                    distances = pairwise_distances(X, metric=self.metric)

                if np.any(distances < 0):
                    raise ValueError("All distances should be positive, the "
                                     "metric given is not correct")

            # compute the joint probability distribution for the input space
            P = _joint_probabilities(distances, self.perplexity, self.verbose)
            assert np.all(np.isfinite(P)), "All probabilities should be finite"
            assert np.all(P >= 0), "All probabilities should be non-negative"
            assert np.all(P <= 1), ("All probabilities should be less "
                                    "or then equal to one")

        else:
            # Cpmpute the number of nearest neighbors to find.
            # LvdM uses 3 * perplexity as the number of neighbors.
            # In the event that we have very small # of points
            # set the neighbors to n - 1.
            k = min(n_samples - 1, int(3. * self.perplexity + 1))

            if self.verbose:
                print("[t-SNE] Computing {} nearest neighbors...".format(k))

            # Find the nearest neighbors for every point
            knn = NearestNeighbors(algorithm='auto', n_neighbors=k,
                                   metric=self.metric)
            t0 = time()
            knn.fit(X)
            duration = time() - t0
            if self.verbose:
                print("[t-SNE] Indexed {} samples in {:.3f}s...".format(
                    n_samples, duration))

            t0 = time()
            distances_nn, neighbors_nn = knn.kneighbors(
                None, n_neighbors=k)
            duration = time() - t0
            if self.verbose:
                print("[t-SNE] Computed neighbors for {} samples in {:.3f}s..."
                      .format(n_samples, duration))

            # Free the memory used by the ball_tree
            del knn

            if self.metric == "euclidean":
                # knn return the euclidean distance but we need it squared
                # to be consistent with the 'exact' method. Note that the
                # the method was derived using the euclidean method as in the
                # input space. Not sure of the implication of using a different
                # metric.
                distances_nn **= 2

            # compute the joint probability distribution for the input space
            P = _joint_probabilities_nn(distances_nn, neighbors_nn,
                                        self.perplexity, self.verbose)

        if isinstance(self.init, np.ndarray):
            X_embedded = self.init
        elif self.init == 'pca':
            pca = PCA(n_components=self.n_components, svd_solver='randomized',
                      random_state=random_state)
            X_embedded = pca.fit_transform(X).astype(np.float32, copy=False)
        elif self.init == 'random':
            # The embedding is initialized with iid samples from Gaussians with
            # standard deviation 1e-4.
            X_embedded = 1e-4 * random_state.randn(
                n_samples, self.n_components).astype(np.float32)
        else:
            raise ValueError("'init' must be 'pca', 'random', or "
                             "a numpy array")

        # Degrees of freedom of the Student's t-distribution. The suggestion
        # degrees_of_freedom = n_components - 1 comes from
        # "Learning a Parametric Embedding by Preserving Local Structure"
        # Laurens van der Maaten, 2009.
        degrees_of_freedom = max(self.n_components - 1.0, 1)

        return self._tsne(P, degrees_of_freedom, n_samples,
                          X_embedded=X_embedded,
                          neighbors=neighbors_nn,
                          skip_num_points=skip_num_points)

    @property
    @deprecated("Attribute n_iter_final was deprecated in version 0.19 and "
                "will be removed in 0.21. Use ``n_iter_`` instead")
    def n_iter_final(self):
        return self.n_iter_

    def _tsne(self, P, degrees_of_freedom, n_samples, X_embedded,
              neighbors=None, skip_num_points=0):
        """Runs t-SNE."""
        # t-SNE minimizes the Kullback-Leiber divergence of the Gaussians P
        # and the Student's t-distributions Q. The optimization algorithm that
        # we use is batch gradient descent with two stages:
        # * initial optimization with early exaggeration and momentum at 0.5
        # * final optimization with momentum at 0.8
        params = X_embedded.ravel()

        opt_args = {
            "it": 0,
            "n_iter_check": self._N_ITER_CHECK,
            "min_grad_norm": self.min_grad_norm,
            "learning_rate": self.learning_rate,
            "verbose": self.verbose,
            "kwargs": dict(skip_num_points=skip_num_points),
            "args": [P, degrees_of_freedom, n_samples, self.n_components],
            "n_iter_without_progress": self._EXPLORATION_N_ITER,
            "n_iter": self._EXPLORATION_N_ITER,
            "momentum": 0.5,
        }
        if self.method == 'barnes_hut':
            obj_func = _kl_divergence_bh
            opt_args['kwargs']['angle'] = self.angle
            # Repeat verbose argument for _kl_divergence_bh
            opt_args['kwargs']['verbose'] = self.verbose
        else:
            obj_func = _kl_divergence

        # Learning schedule (part 1): do 250 iteration with lower momentum but
        # higher learning rate controlled via the early exageration parameter
        P *= self.early_exaggeration
        params, kl_divergence, it = _gradient_descent(obj_func, params,
                                                      **opt_args)
        if self.verbose:
            print("[t-SNE] KL divergence after %d iterations with early "
                  "exaggeration: %f" % (it + 1, kl_divergence))

        # Learning schedule (part 2): disable early exaggeration and finish
        # optimization with a higher momentum at 0.8
        P /= self.early_exaggeration
        remaining = self.n_iter - self._EXPLORATION_N_ITER
        if it < self._EXPLORATION_N_ITER or remaining > 0:
            opt_args['n_iter'] = self.n_iter
            opt_args['it'] = it + 1
            opt_args['momentum'] = 0.8
            opt_args['n_iter_without_progress'] = self.n_iter_without_progress
            params, kl_divergence, it = _gradient_descent(obj_func, params,
                                                          **opt_args)

        # Save the final number of iterations
        self.n_iter_ = it

        if self.verbose:
            print("[t-SNE] Error after %d iterations: %f"
                  % (it + 1, kl_divergence))

        X_embedded = params.reshape(n_samples, self.n_components)
        self.kl_divergence_ = kl_divergence

        return X_embedded

    def fit_transform(self, X, y=None):
        """Fit X into an embedded space and return that transformed
        output.

        Parameters
        ----------
        X : array, shape (n_samples, n_features) or (n_samples, n_samples)
            If the metric is 'precomputed' X must be a square distance
            matrix. Otherwise it contains a sample per row.

        y : Ignored.

        Returns
        -------
        X_new : array, shape (n_samples, n_components)
            Embedding of the training data in low-dimensional space.
        """
        embedding = self._fit(X)
        self.embedding_ = embedding
        return self.embedding_

    def fit(self, X, y=None):
        """Fit X into an embedded space.

        Parameters
        ----------
        X : array, shape (n_samples, n_features) or (n_samples, n_samples)
            If the metric is 'precomputed' X must be a square distance
            matrix. Otherwise it contains a sample per row. If the method
            is 'exact', X may be a sparse matrix of type 'csr', 'csc'
            or 'coo'.

        y : Ignored.
        """
        self.fit_transform(X)
        return self