pairwise.py

# -*- coding: utf-8 -*-

# Authors: Alexandre Gramfort <alexandre.gramfort@inria.fr>
#          Mathieu Blondel <mathieu@mblondel.org>
#          Robert Layton <robertlayton@gmail.com>
#          Andreas Mueller <amueller@ais.uni-bonn.de>
#          Philippe Gervais <philippe.gervais@inria.fr>
#          Lars Buitinck
#          Joel Nothman <joel.nothman@gmail.com>
# License: BSD 3 clause

import itertools
from functools import partial

import numpy as np
from scipy.spatial import distance
from scipy.sparse import csr_matrix
from scipy.sparse import issparse

from ..utils import check_array
from ..utils import gen_even_slices
from ..utils import gen_batches
from ..utils.extmath import row_norms, safe_sparse_dot
from ..preprocessing import normalize
from ..externals.joblib import Parallel
from ..externals.joblib import delayed
from ..externals.joblib import cpu_count

from .pairwise_fast import _chi2_kernel_fast, _sparse_manhattan


# Utility Functions
def _return_float_dtype(X, Y):
    """
    1. If dtype of X and Y is float32, then dtype float32 is returned.
    2. Else dtype float is returned.
    """
    if not issparse(X) and not isinstance(X, np.ndarray):
        X = np.asarray(X)

    if Y is None:
        Y_dtype = X.dtype
    elif not issparse(Y) and not isinstance(Y, np.ndarray):
        Y = np.asarray(Y)
        Y_dtype = Y.dtype
    else:
        Y_dtype = Y.dtype

    if X.dtype == Y_dtype == np.float32:
        dtype = np.float32
    else:
        dtype = np.float

    return X, Y, dtype


def check_pairwise_arrays(X, Y, precomputed=False, dtype=None):
    """ Set X and Y appropriately and checks inputs

    If Y is None, it is set as a pointer to X (i.e. not a copy).
    If Y is given, this does not happen.
    All distance metrics should use this function first to assert that the
    given parameters are correct and safe to use.

    Specifically, this function first ensures that both X and Y are arrays,
    then checks that they are at least two dimensional while ensuring that
    their elements are floats (or dtype if provided). Finally, the function
    checks that the size of the second dimension of the two arrays is equal, or
    the equivalent check for a precomputed distance matrix.

    Parameters
    ----------
    X : {array-like, sparse matrix}, shape (n_samples_a, n_features)

    Y : {array-like, sparse matrix}, shape (n_samples_b, n_features)

    precomputed : bool
        True if X is to be treated as precomputed distances to the samples in
        Y.

    dtype : string, type, list of types or None (default=None)
        Data type required for X and Y. If None, the dtype will be an
        appropriate float type selected by _return_float_dtype.

        .. versionadded:: 0.18

    Returns
    -------
    safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features)
        An array equal to X, guaranteed to be a numpy array.

    safe_Y : {array-like, sparse matrix}, shape (n_samples_b, n_features)
        An array equal to Y if Y was not None, guaranteed to be a numpy array.
        If Y was None, safe_Y will be a pointer to X.

    """
    X, Y, dtype_float = _return_float_dtype(X, Y)

    warn_on_dtype = dtype is not None
    estimator = 'check_pairwise_arrays'
    if dtype is None:
        dtype = dtype_float

    if Y is X or Y is None:
        X = Y = check_array(X, accept_sparse='csr', dtype=dtype,
                            warn_on_dtype=warn_on_dtype, estimator=estimator)
    else:
        X = check_array(X, accept_sparse='csr', dtype=dtype,
                        warn_on_dtype=warn_on_dtype, estimator=estimator)
        Y = check_array(Y, accept_sparse='csr', dtype=dtype,
                        warn_on_dtype=warn_on_dtype, estimator=estimator)

    if precomputed:
        if X.shape[1] != Y.shape[0]:
            raise ValueError("Precomputed metric requires shape "
                             "(n_queries, n_indexed). Got (%d, %d) "
                             "for %d indexed." %
                             (X.shape[0], X.shape[1], Y.shape[0]))
    elif X.shape[1] != Y.shape[1]:
        raise ValueError("Incompatible dimension for X and Y matrices: "
                         "X.shape[1] == %d while Y.shape[1] == %d" % (
                             X.shape[1], Y.shape[1]))

    return X, Y


def check_paired_arrays(X, Y):
    """ Set X and Y appropriately and checks inputs for paired distances

    All paired distance metrics should use this function first to assert that
    the given parameters are correct and safe to use.

    Specifically, this function first ensures that both X and Y are arrays,
    then checks that they are at least two dimensional while ensuring that
    their elements are floats. Finally, the function checks that the size
    of the dimensions of the two arrays are equal.

    Parameters
    ----------
    X : {array-like, sparse matrix}, shape (n_samples_a, n_features)

    Y : {array-like, sparse matrix}, shape (n_samples_b, n_features)

    Returns
    -------
    safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features)
        An array equal to X, guaranteed to be a numpy array.

    safe_Y : {array-like, sparse matrix}, shape (n_samples_b, n_features)
        An array equal to Y if Y was not None, guaranteed to be a numpy array.
        If Y was None, safe_Y will be a pointer to X.

    """
    X, Y = check_pairwise_arrays(X, Y)
    if X.shape != Y.shape:
        raise ValueError("X and Y should be of same shape. They were "
                         "respectively %r and %r long." % (X.shape, Y.shape))
    return X, Y


# Pairwise distances
def euclidean_distances(X, Y=None, Y_norm_squared=None, squared=False,
                        X_norm_squared=None):
    """
    Considering the rows of X (and Y=X) as vectors, compute the
    distance matrix between each pair of vectors.

    For efficiency reasons, the euclidean distance between a pair of row
    vector x and y is computed as::

        dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y))

    This formulation has two advantages over other ways of computing distances.
    First, it is computationally efficient when dealing with sparse data.
    Second, if one argument varies but the other remains unchanged, then
    `dot(x, x)` and/or `dot(y, y)` can be pre-computed.

    However, this is not the most precise way of doing this computation, and
    the distance matrix returned by this function may not be exactly
    symmetric as required by, e.g., ``scipy.spatial.distance`` functions.

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

    Parameters
    ----------
    X : {array-like, sparse matrix}, shape (n_samples_1, n_features)

    Y : {array-like, sparse matrix}, shape (n_samples_2, n_features)

    Y_norm_squared : array-like, shape (n_samples_2, ), optional
        Pre-computed dot-products of vectors in Y (e.g.,
        ``(Y**2).sum(axis=1)``)

    squared : boolean, optional
        Return squared Euclidean distances.

    X_norm_squared : array-like, shape = [n_samples_1], optional
        Pre-computed dot-products of vectors in X (e.g.,
        ``(X**2).sum(axis=1)``)

    Returns
    -------
    distances : {array, sparse matrix}, shape (n_samples_1, n_samples_2)

    Examples
    --------
    >>> from sklearn.metrics.pairwise import euclidean_distances
    >>> X = [[0, 1], [1, 1]]
    >>> # distance between rows of X
    >>> euclidean_distances(X, X)
    array([[ 0.,  1.],
           [ 1.,  0.]])
    >>> # get distance to origin
    >>> euclidean_distances(X, [[0, 0]])
    array([[ 1.        ],
           [ 1.41421356]])

    See also
    --------
    paired_distances : distances betweens pairs of elements of X and Y.
    """
    X, Y = check_pairwise_arrays(X, Y)

    if X_norm_squared is not None:
        XX = check_array(X_norm_squared)
        if XX.shape == (1, X.shape[0]):
            XX = XX.T
        elif XX.shape != (X.shape[0], 1):
            raise ValueError(
                "Incompatible dimensions for X and X_norm_squared")
    else:
        XX = row_norms(X, squared=True)[:, np.newaxis]

    if X is Y:  # shortcut in the common case euclidean_distances(X, X)
        YY = XX.T
    elif Y_norm_squared is not None:
        YY = np.atleast_2d(Y_norm_squared)

        if YY.shape != (1, Y.shape[0]):
            raise ValueError(
                "Incompatible dimensions for Y and Y_norm_squared")
    else:
        YY = row_norms(Y, squared=True)[np.newaxis, :]

    distances = safe_sparse_dot(X, Y.T, dense_output=True)
    distances *= -2
    distances += XX
    distances += YY
    np.maximum(distances, 0, out=distances)

    if X is Y:
        # Ensure that distances between vectors and themselves are set to 0.0.
        # This may not be the case due to floating point rounding errors.
        distances.flat[::distances.shape[0] + 1] = 0.0

    return distances if squared else np.sqrt(distances, out=distances)


def pairwise_distances_argmin_min(X, Y, axis=1, metric="euclidean",
                                  batch_size=500, metric_kwargs=None):
    """Compute minimum distances between one point and a set of points.

    This function computes for each row in X, the index of the row of Y which
    is closest (according to the specified distance). The minimal distances are
    also returned.

    This is mostly equivalent to calling:

        (pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis),
         pairwise_distances(X, Y=Y, metric=metric).min(axis=axis))

    but uses much less memory, and is faster for large arrays.

    Parameters
    ----------
    X, Y : {array-like, sparse matrix}
        Arrays containing points. Respective shapes (n_samples1, n_features)
        and (n_samples2, n_features)

    batch_size : integer
        To reduce memory consumption over the naive solution, data are
        processed in batches, comprising batch_size rows of X and
        batch_size rows of Y. The default value is quite conservative, but
        can be changed for fine-tuning. The larger the number, the larger the
        memory usage.

    metric : string or callable, default 'euclidean'
        metric to use for distance computation. Any metric from scikit-learn
        or scipy.spatial.distance can be used.

        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 as input and return one value indicating the
        distance between them. This works for Scipy's metrics, but is less
        efficient than passing the metric name as a string.

        Distance matrices are not supported.

        Valid values for metric are:

        - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2',
          'manhattan']

        - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev',
          'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski',
          'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto',
          'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath',
          'sqeuclidean', 'yule']

        See the documentation for scipy.spatial.distance for details on these
        metrics.

    metric_kwargs : dict, optional
        Keyword arguments to pass to specified metric function.

    axis : int, optional, default 1
        Axis along which the argmin and distances are to be computed.

    Returns
    -------
    argmin : numpy.ndarray
        Y[argmin[i], :] is the row in Y that is closest to X[i, :].

    distances : numpy.ndarray
        distances[i] is the distance between the i-th row in X and the
        argmin[i]-th row in Y.

    See also
    --------
    sklearn.metrics.pairwise_distances
    sklearn.metrics.pairwise_distances_argmin
    """
    dist_func = None
    if metric in PAIRWISE_DISTANCE_FUNCTIONS:
        dist_func = PAIRWISE_DISTANCE_FUNCTIONS[metric]
    elif not callable(metric) and not isinstance(metric, str):
        raise ValueError("'metric' must be a string or a callable")

    X, Y = check_pairwise_arrays(X, Y)

    if metric_kwargs is None:
        metric_kwargs = {}

    if axis == 0:
        X, Y = Y, X

    # Allocate output arrays
    indices = np.empty(X.shape[0], dtype=np.intp)
    values = np.empty(X.shape[0])
    values.fill(np.infty)

    for chunk_x in gen_batches(X.shape[0], batch_size):
        X_chunk = X[chunk_x, :]

        for chunk_y in gen_batches(Y.shape[0], batch_size):
            Y_chunk = Y[chunk_y, :]

            if dist_func is not None:
                if metric == 'euclidean':  # special case, for speed
                    d_chunk = safe_sparse_dot(X_chunk, Y_chunk.T,
                                              dense_output=True)
                    d_chunk *= -2
                    d_chunk += row_norms(X_chunk, squared=True)[:, np.newaxis]
                    d_chunk += row_norms(Y_chunk, squared=True)[np.newaxis, :]
                    np.maximum(d_chunk, 0, d_chunk)
                else:
                    d_chunk = dist_func(X_chunk, Y_chunk, **metric_kwargs)
            else:
                d_chunk = pairwise_distances(X_chunk, Y_chunk,
                                             metric=metric, **metric_kwargs)

            # Update indices and minimum values using chunk
            min_indices = d_chunk.argmin(axis=1)
            min_values = d_chunk[np.arange(chunk_x.stop - chunk_x.start),
                                 min_indices]

            flags = values[chunk_x] > min_values
            indices[chunk_x][flags] = min_indices[flags] + chunk_y.start
            values[chunk_x][flags] = min_values[flags]

    if metric == "euclidean" and not metric_kwargs.get("squared", False):
        np.sqrt(values, values)
    return indices, values


def pairwise_distances_argmin(X, Y, axis=1, metric="euclidean",
                              batch_size=500, metric_kwargs=None):
    """Compute minimum distances between one point and a set of points.

    This function computes for each row in X, the index of the row of Y which
    is closest (according to the specified distance).

    This is mostly equivalent to calling:

        pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis)

    but uses much less memory, and is faster for large arrays.

    This function works with dense 2D arrays only.

    Parameters
    ----------
    X : array-like
        Arrays containing points. Respective shapes (n_samples1, n_features)
        and (n_samples2, n_features)

    Y : array-like
        Arrays containing points. Respective shapes (n_samples1, n_features)
        and (n_samples2, n_features)

    batch_size : integer
        To reduce memory consumption over the naive solution, data are
        processed in batches, comprising batch_size rows of X and
        batch_size rows of Y. The default value is quite conservative, but
        can be changed for fine-tuning. The larger the number, the larger the
        memory usage.

    metric : string or callable
        metric to use for distance computation. Any metric from scikit-learn
        or scipy.spatial.distance can be used.

        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 as input and return one value indicating the
        distance between them. This works for Scipy's metrics, but is less
        efficient than passing the metric name as a string.

        Distance matrices are not supported.

        Valid values for metric are:

        - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2',
          'manhattan']

        - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev',
          'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski',
          'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto',
          'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath',
          'sqeuclidean', 'yule']

        See the documentation for scipy.spatial.distance for details on these
        metrics.

    metric_kwargs : dict
        keyword arguments to pass to specified metric function.

    axis : int, optional, default 1
        Axis along which the argmin and distances are to be computed.

    Returns
    -------
    argmin : numpy.ndarray
        Y[argmin[i], :] is the row in Y that is closest to X[i, :].

    See also
    --------
    sklearn.metrics.pairwise_distances
    sklearn.metrics.pairwise_distances_argmin_min
    """
    if metric_kwargs is None:
        metric_kwargs = {}

    return pairwise_distances_argmin_min(X, Y, axis, metric, batch_size,
                                         metric_kwargs)[0]


def manhattan_distances(X, Y=None, sum_over_features=True,
                        size_threshold=5e8):
    """ Compute the L1 distances between the vectors in X and Y.

    With sum_over_features equal to False it returns the componentwise
    distances.

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

    Parameters
    ----------
    X : array_like
        An array with shape (n_samples_X, n_features).

    Y : array_like, optional
        An array with shape (n_samples_Y, n_features).

    sum_over_features : bool, default=True
        If True the function returns the pairwise distance matrix
        else it returns the componentwise L1 pairwise-distances.
        Not supported for sparse matrix inputs.

    size_threshold : int, default=5e8
        Unused parameter.

    Returns
    -------
    D : array
        If sum_over_features is False shape is
        (n_samples_X * n_samples_Y, n_features) and D contains the
        componentwise L1 pairwise-distances (ie. absolute difference),
        else shape is (n_samples_X, n_samples_Y) and D contains
        the pairwise L1 distances.

    Examples
    --------
    >>> from sklearn.metrics.pairwise import manhattan_distances
    >>> manhattan_distances([[3]], [[3]])#doctest:+ELLIPSIS
    array([[ 0.]])
    >>> manhattan_distances([[3]], [[2]])#doctest:+ELLIPSIS
    array([[ 1.]])
    >>> manhattan_distances([[2]], [[3]])#doctest:+ELLIPSIS
    array([[ 1.]])
    >>> manhattan_distances([[1, 2], [3, 4]],\
         [[1, 2], [0, 3]])#doctest:+ELLIPSIS
    array([[ 0.,  2.],
           [ 4.,  4.]])
    >>> import numpy as np
    >>> X = np.ones((1, 2))
    >>> y = 2 * np.ones((2, 2))
    >>> manhattan_distances(X, y, sum_over_features=False)#doctest:+ELLIPSIS
    array([[ 1.,  1.],
           [ 1.,  1.]]...)
    """
    X, Y = check_pairwise_arrays(X, Y)

    if issparse(X) or issparse(Y):
        if not sum_over_features:
            raise TypeError("sum_over_features=%r not supported"
                            " for sparse matrices" % sum_over_features)

        X = csr_matrix(X, copy=False)
        Y = csr_matrix(Y, copy=False)
        D = np.zeros((X.shape[0], Y.shape[0]))
        _sparse_manhattan(X.data, X.indices, X.indptr,
                          Y.data, Y.indices, Y.indptr,
                          X.shape[1], D)
        return D

    if sum_over_features:
        return distance.cdist(X, Y, 'cityblock')

    D = X[:, np.newaxis, :] - Y[np.newaxis, :, :]
    D = np.abs(D, D)
    return D.reshape((-1, X.shape[1]))


def cosine_distances(X, Y=None):
    """Compute cosine distance between samples in X and Y.

    Cosine distance is defined as 1.0 minus the cosine similarity.

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

    Parameters
    ----------
    X : array_like, sparse matrix
        with shape (n_samples_X, n_features).

    Y : array_like, sparse matrix (optional)
        with shape (n_samples_Y, n_features).

    Returns
    -------
    distance matrix : array
        An array with shape (n_samples_X, n_samples_Y).

    See also
    --------
    sklearn.metrics.pairwise.cosine_similarity
    scipy.spatial.distance.cosine (dense matrices only)
    """
    # 1.0 - cosine_similarity(X, Y) without copy
    S = cosine_similarity(X, Y)
    S *= -1
    S += 1
    np.clip(S, 0, 2, out=S)
    if X is Y or Y is None:
        # Ensure that distances between vectors and themselves are set to 0.0.
        # This may not be the case due to floating point rounding errors.
        S[np.diag_indices_from(S)] = 0.0
    return S


# Paired distances
def paired_euclidean_distances(X, Y):
    """
    Computes the paired euclidean distances between X and Y

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

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)

    Y : array-like, shape (n_samples, n_features)

    Returns
    -------
    distances : ndarray (n_samples, )
    """
    X, Y = check_paired_arrays(X, Y)
    return row_norms(X - Y)


def paired_manhattan_distances(X, Y):
    """Compute the L1 distances between the vectors in X and Y.

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

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)

    Y : array-like, shape (n_samples, n_features)

    Returns
    -------
    distances : ndarray (n_samples, )
    """
    X, Y = check_paired_arrays(X, Y)
    diff = X - Y
    if issparse(diff):
        diff.data = np.abs(diff.data)
        return np.squeeze(np.array(diff.sum(axis=1)))
    else:
        return np.abs(diff).sum(axis=-1)


def paired_cosine_distances(X, Y):
    """
    Computes the paired cosine distances between X and Y

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

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)

    Y : array-like, shape (n_samples, n_features)

    Returns
    -------
    distances : ndarray, shape (n_samples, )

    Notes
    ------
    The cosine distance is equivalent to the half the squared
    euclidean distance if each sample is normalized to unit norm
    """
    X, Y = check_paired_arrays(X, Y)
    return .5 * row_norms(normalize(X) - normalize(Y), squared=True)


PAIRED_DISTANCES = {
    'cosine': paired_cosine_distances,
    'euclidean': paired_euclidean_distances,
    'l2': paired_euclidean_distances,
    'l1': paired_manhattan_distances,
    'manhattan': paired_manhattan_distances,
    'cityblock': paired_manhattan_distances}


def paired_distances(X, Y, metric="euclidean", **kwds):
    """
    Computes the paired distances between X and Y.

    Computes the distances between (X[0], Y[0]), (X[1], Y[1]), etc...

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

    Parameters
    ----------
    X : ndarray (n_samples, n_features)
        Array 1 for distance computation.

    Y : ndarray (n_samples, n_features)
        Array 2 for distance computation.

    metric : string or callable
        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
        specified in PAIRED_DISTANCES, including "euclidean",
        "manhattan", or "cosine".
        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.

    Returns
    -------
    distances : ndarray (n_samples, )

    Examples
    --------
    >>> from sklearn.metrics.pairwise import paired_distances
    >>> X = [[0, 1], [1, 1]]
    >>> Y = [[0, 1], [2, 1]]
    >>> paired_distances(X, Y)
    array([ 0.,  1.])

    See also
    --------
    pairwise_distances : pairwise distances.
    """

    if metric in PAIRED_DISTANCES:
        func = PAIRED_DISTANCES[metric]
        return func(X, Y)
    elif callable(metric):
        # Check the matrix first (it is usually done by the metric)
        X, Y = check_paired_arrays(X, Y)
        distances = np.zeros(len(X))
        for i in range(len(X)):
            distances[i] = metric(X[i], Y[i])
        return distances
    else:
        raise ValueError('Unknown distance %s' % metric)


# Kernels
def linear_kernel(X, Y=None):
    """
    Compute the linear kernel between X and Y.

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

    Parameters
    ----------
    X : array of shape (n_samples_1, n_features)

    Y : array of shape (n_samples_2, n_features)

    Returns
    -------
    Gram matrix : array of shape (n_samples_1, n_samples_2)
    """
    X, Y = check_pairwise_arrays(X, Y)
    return safe_sparse_dot(X, Y.T, dense_output=True)


def polynomial_kernel(X, Y=None, degree=3, gamma=None, coef0=1):
    """
    Compute the polynomial kernel between X and Y::

        K(X, Y) = (gamma <X, Y> + coef0)^degree

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

    Parameters
    ----------
    X : ndarray of shape (n_samples_1, n_features)

    Y : ndarray of shape (n_samples_2, n_features)

    degree : int, default 3

    gamma : float, default None
        if None, defaults to 1.0 / n_features

    coef0 : int, default 1

    Returns
    -------
    Gram matrix : array of shape (n_samples_1, n_samples_2)
    """
    X, Y = check_pairwise_arrays(X, Y)
    if gamma is None:
        gamma = 1.0 / X.shape[1]

    K = safe_sparse_dot(X, Y.T, dense_output=True)
    K *= gamma
    K += coef0
    K **= degree
    return K


def sigmoid_kernel(X, Y=None, gamma=None, coef0=1):
    """
    Compute the sigmoid kernel between X and Y::

        K(X, Y) = tanh(gamma <X, Y> + coef0)

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

    Parameters
    ----------
    X : ndarray of shape (n_samples_1, n_features)

    Y : ndarray of shape (n_samples_2, n_features)

    gamma : float, default None
        If None, defaults to 1.0 / n_features

    coef0 : int, default 1

    Returns
    -------
    Gram matrix : array of shape (n_samples_1, n_samples_2)
    """
    X, Y = check_pairwise_arrays(X, Y)
    if gamma is None:
        gamma = 1.0 / X.shape[1]

    K = safe_sparse_dot(X, Y.T, dense_output=True)
    K *= gamma
    K += coef0
    np.tanh(K, K)   # compute tanh in-place
    return K


def rbf_kernel(X, Y=None, gamma=None):
    """
    Compute the rbf (gaussian) kernel between X and Y::

        K(x, y) = exp(-gamma ||x-y||^2)

    for each pair of rows x in X and y in Y.

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

    Parameters
    ----------
    X : array of shape (n_samples_X, n_features)

    Y : array of shape (n_samples_Y, n_features)

    gamma : float, default None
        If None, defaults to 1.0 / n_features

    Returns
    -------
    kernel_matrix : array of shape (n_samples_X, n_samples_Y)
    """
    X, Y = check_pairwise_arrays(X, Y)
    if gamma is None:
        gamma = 1.0 / X.shape[1]

    K = euclidean_distances(X, Y, squared=True)
    K *= -gamma
    np.exp(K, K)    # exponentiate K in-place
    return K


def laplacian_kernel(X, Y=None, gamma=None):
    """Compute the laplacian kernel between X and Y.

    The laplacian kernel is defined as::

        K(x, y) = exp(-gamma ||x-y||_1)

    for each pair of rows x in X and y in Y.
    Read more in the :ref:`User Guide <laplacian_kernel>`.

    .. versionadded:: 0.17

    Parameters
    ----------
    X : array of shape (n_samples_X, n_features)

    Y : array of shape (n_samples_Y, n_features)

    gamma : float, default None
        If None, defaults to 1.0 / n_features

    Returns
    -------
    kernel_matrix : array of shape (n_samples_X, n_samples_Y)
    """
    X, Y = check_pairwise_arrays(X, Y)
    if gamma is None:
        gamma = 1.0 / X.shape[1]

    K = -gamma * manhattan_distances(X, Y)
    np.exp(K, K)    # exponentiate K in-place
    return K


def cosine_similarity(X, Y=None, dense_output=True):
    """Compute cosine similarity between samples in X and Y.

    Cosine similarity, or the cosine kernel, computes similarity as the
    normalized dot product of X and Y:

        K(X, Y) = <X, Y> / (||X||*||Y||)

    On L2-normalized data, this function is equivalent to linear_kernel.

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

    Parameters
    ----------
    X : ndarray or sparse array, shape: (n_samples_X, n_features)
        Input data.

    Y : ndarray or sparse array, shape: (n_samples_Y, n_features)
        Input data. If ``None``, the output will be the pairwise
        similarities between all samples in ``X``.

    dense_output : boolean (optional), default True
        Whether to return dense output even when the input is sparse. If
        ``False``, the output is sparse if both input arrays are sparse.

        .. versionadded:: 0.17
           parameter ``dense_output`` for dense output.

    Returns
    -------
    kernel matrix : array
        An array with shape (n_samples_X, n_samples_Y).
    """
    # to avoid recursive import

    X, Y = check_pairwise_arrays(X, Y)

    X_normalized = normalize(X, copy=True)
    if X is Y:
        Y_normalized = X_normalized
    else:
        Y_normalized = normalize(Y, copy=True)

    K = safe_sparse_dot(X_normalized, Y_normalized.T, dense_output=dense_output)

    return K


def additive_chi2_kernel(X, Y=None):
    """Computes the additive chi-squared kernel between observations in X and Y

    The chi-squared kernel is computed between each pair of rows in X and Y.  X
    and Y have to be non-negative. This kernel is most commonly applied to
    histograms.

    The chi-squared kernel is given by::

        k(x, y) = -Sum [(x - y)^2 / (x + y)]

    It can be interpreted as a weighted difference per entry.

    Read more in the :ref:`User Guide <chi2_kernel>`.

    Notes
    -----
    As the negative of a distance, this kernel is only conditionally positive
    definite.


    Parameters
    ----------
    X : array-like of shape (n_samples_X, n_features)

    Y : array of shape (n_samples_Y, n_features)

    Returns
    -------
    kernel_matrix : array of shape (n_samples_X, n_samples_Y)

    References
    ----------
    * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C.
      Local features and kernels for classification of texture and object
      categories: A comprehensive study
      International Journal of Computer Vision 2007
      http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf


    See also
    --------
    chi2_kernel : The exponentiated version of the kernel, which is usually
        preferable.

    sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation
        to this kernel.
    """
    if issparse(X) or issparse(Y):
        raise ValueError("additive_chi2 does not support sparse matrices.")
    X, Y = check_pairwise_arrays(X, Y)
    if (X < 0).any():
        raise ValueError("X contains negative values.")
    if Y is not X and (Y < 0).any():
        raise ValueError("Y contains negative values.")

    result = np.zeros((X.shape[0], Y.shape[0]), dtype=X.dtype)
    _chi2_kernel_fast(X, Y, result)
    return result


def chi2_kernel(X, Y=None, gamma=1.):
    """Computes the exponential chi-squared kernel X and Y.

    The chi-squared kernel is computed between each pair of rows in X and Y.  X
    and Y have to be non-negative. This kernel is most commonly applied to
    histograms.

    The chi-squared kernel is given by::

        k(x, y) = exp(-gamma Sum [(x - y)^2 / (x + y)])

    It can be interpreted as a weighted difference per entry.

    Read more in the :ref:`User Guide <chi2_kernel>`.

    Parameters
    ----------
    X : array-like of shape (n_samples_X, n_features)

    Y : array of shape (n_samples_Y, n_features)

    gamma : float, default=1.
        Scaling parameter of the chi2 kernel.

    Returns
    -------
    kernel_matrix : array of shape (n_samples_X, n_samples_Y)

    References
    ----------
    * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C.
      Local features and kernels for classification of texture and object
      categories: A comprehensive study
      International Journal of Computer Vision 2007
      http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf

    See also
    --------
    additive_chi2_kernel : The additive version of this kernel

    sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation
        to the additive version of this kernel.
    """
    K = additive_chi2_kernel(X, Y)
    K *= gamma
    return np.exp(K, K)


# Helper functions - distance
PAIRWISE_DISTANCE_FUNCTIONS = {
    # If updating this dictionary, update the doc in both distance_metrics()
    # and also in pairwise_distances()!
    'cityblock': manhattan_distances,
    'cosine': cosine_distances,
    'euclidean': euclidean_distances,
    'l2': euclidean_distances,
    'l1': manhattan_distances,
    'manhattan': manhattan_distances,
    'precomputed': None,  # HACK: precomputed is always allowed, never called
}


def distance_metrics():
    """Valid metrics for pairwise_distances.

    This function simply returns the valid pairwise distance metrics.
    It exists to allow for a description of the mapping for
    each of the valid strings.

    The valid distance metrics, and the function they map to, are:

    ============     ====================================
    metric           Function
    ============     ====================================
    'cityblock'      metrics.pairwise.manhattan_distances
    'cosine'         metrics.pairwise.cosine_distances
    'euclidean'      metrics.pairwise.euclidean_distances
    'l1'             metrics.pairwise.manhattan_distances
    'l2'             metrics.pairwise.euclidean_distances
    'manhattan'      metrics.pairwise.manhattan_distances
    ============     ====================================

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

    """
    return PAIRWISE_DISTANCE_FUNCTIONS


def _parallel_pairwise(X, Y, func, n_jobs, **kwds):
    """Break the pairwise matrix in n_jobs even slices
    and compute them in parallel"""
    if n_jobs < 0:
        n_jobs = max(cpu_count() + 1 + n_jobs, 1)

    if Y is None:
        Y = X

    if n_jobs == 1:
        # Special case to avoid picklability checks in delayed
        return func(X, Y, **kwds)

    # TODO: in some cases, backend='threading' may be appropriate
    fd = delayed(func)
    ret = Parallel(n_jobs=n_jobs, verbose=0)(
        fd(X, Y[s], **kwds)
        for s in gen_even_slices(Y.shape[0], n_jobs))

    return np.hstack(ret)


def _pairwise_callable(X, Y, metric, **kwds):
    """Handle the callable case for pairwise_{distances,kernels}
    """
    X, Y = check_pairwise_arrays(X, Y)

    if X is Y:
        # Only calculate metric for upper triangle
        out = np.zeros((X.shape[0], Y.shape[0]), dtype='float')
        iterator = itertools.combinations(range(X.shape[0]), 2)
        for i, j in iterator:
            out[i, j] = metric(X[i], Y[j], **kwds)

        # Make symmetric
        # NB: out += out.T will produce incorrect results
        out = out + out.T

        # Calculate diagonal
        # NB: nonzero diagonals are allowed for both metrics and kernels
        for i in range(X.shape[0]):
            x = X[i]
            out[i, i] = metric(x, x, **kwds)

    else:
        # Calculate all cells
        out = np.empty((X.shape[0], Y.shape[0]), dtype='float')
        iterator = itertools.product(range(X.shape[0]), range(Y.shape[0]))
        for i, j in iterator:
            out[i, j] = metric(X[i], Y[j], **kwds)

    return out


_VALID_METRICS = ['euclidean', 'l2', 'l1', 'manhattan', 'cityblock',
                  'braycurtis', 'canberra', 'chebyshev', 'correlation',
                  'cosine', 'dice', 'hamming', 'jaccard', 'kulsinski',
                  'mahalanobis', 'matching', 'minkowski', 'rogerstanimoto',
                  'russellrao', 'seuclidean', 'sokalmichener',
                  'sokalsneath', 'sqeuclidean', 'yule', "wminkowski"]


def pairwise_distances(X, Y=None, metric="euclidean", n_jobs=1, **kwds):
    """ Compute the distance matrix from a vector array X and optional Y.

    This method takes either a vector array or a distance matrix, and returns
    a distance matrix. If the input is a vector array, the distances are
    computed. If the input is a distances matrix, it is returned instead.

    This method provides a safe way to take a distance matrix as input, while
    preserving compatibility with many other algorithms that take a vector
    array.

    If Y is given (default is None), then the returned matrix is the pairwise
    distance between the arrays from both X and Y.

    Valid values for metric are:

    - From scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2',
      'manhattan']. These metrics support sparse matrix inputs.

    - From scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev',
      'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis',
      'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean',
      'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule']
      See the documentation for scipy.spatial.distance for details on these
      metrics. These metrics do not support sparse matrix inputs.

    Note that in the case of 'cityblock', 'cosine' and 'euclidean' (which are
    valid scipy.spatial.distance metrics), the scikit-learn implementation
    will be used, which is faster and has support for sparse matrices (except
    for 'cityblock'). For a verbose description of the metrics from
    scikit-learn, see the __doc__ of the sklearn.pairwise.distance_metrics
    function.

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

    Parameters
    ----------
    X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \
             [n_samples_a, n_features] otherwise
        Array of pairwise distances between samples, or a feature array.

    Y : array [n_samples_b, n_features], optional
        An optional second feature array. Only allowed if metric != "precomputed".

    metric : string, or callable
        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.

    n_jobs : int
        The number of jobs to use for the computation. This works by breaking
        down the pairwise matrix into n_jobs even slices and computing them in
        parallel.

        If -1 all CPUs are used. If 1 is given, no parallel computing code is
        used at all, which is useful for debugging. For n_jobs below -1,
        (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
        are used.

    **kwds : optional keyword parameters
        Any further parameters are passed directly to the distance function.
        If using a scipy.spatial.distance metric, the parameters are still
        metric dependent. See the scipy docs for usage examples.

    Returns
    -------
    D : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b]
        A distance matrix D such that D_{i, j} is the distance between the
        ith and jth vectors of the given matrix X, if Y is None.
        If Y is not None, then D_{i, j} is the distance between the ith array
        from X and the jth array from Y.

    """
    if (metric not in _VALID_METRICS and
            not callable(metric) and metric != "precomputed"):
        raise ValueError("Unknown metric %s. "
                         "Valid metrics are %s, or 'precomputed', or a "
                         "callable" % (metric, _VALID_METRICS))

    if metric == "precomputed":
        X, _ = check_pairwise_arrays(X, Y, precomputed=True)
        return X
    elif metric in PAIRWISE_DISTANCE_FUNCTIONS:
        func = PAIRWISE_DISTANCE_FUNCTIONS[metric]
    elif callable(metric):
        func = partial(_pairwise_callable, metric=metric, **kwds)
    else:
        if issparse(X) or issparse(Y):
            raise TypeError("scipy distance metrics do not"
                            " support sparse matrices.")

        dtype = bool if metric in PAIRWISE_BOOLEAN_FUNCTIONS else None

        X, Y = check_pairwise_arrays(X, Y, dtype=dtype)

        if n_jobs == 1 and X is Y:
            return distance.squareform(distance.pdist(X, metric=metric,
                                                      **kwds))
        func = partial(distance.cdist, metric=metric, **kwds)

    return _parallel_pairwise(X, Y, func, n_jobs, **kwds)


# These distances recquire boolean arrays, when using scipy.spatial.distance
PAIRWISE_BOOLEAN_FUNCTIONS = [
    'dice',
    'jaccard',
    'kulsinski',
    'matching',
    'rogerstanimoto',
    'russellrao',
    'sokalmichener',
    'sokalsneath',
    'yule',
]


# Helper functions - distance
PAIRWISE_KERNEL_FUNCTIONS = {
    # If updating this dictionary, update the doc in both distance_metrics()
    # and also in pairwise_distances()!
    'additive_chi2': additive_chi2_kernel,
    'chi2': chi2_kernel,
    'linear': linear_kernel,
    'polynomial': polynomial_kernel,
    'poly': polynomial_kernel,
    'rbf': rbf_kernel,
    'laplacian': laplacian_kernel,
    'sigmoid': sigmoid_kernel,
    'cosine': cosine_similarity, }


def kernel_metrics():
    """ Valid metrics for pairwise_kernels

    This function simply returns the valid pairwise distance metrics.
    It exists, however, to allow for a verbose description of the mapping for
    each of the valid strings.

    The valid distance metrics, and the function they map to, are:
      ===============   ========================================
      metric            Function
      ===============   ========================================
      'additive_chi2'   sklearn.pairwise.additive_chi2_kernel
      'chi2'            sklearn.pairwise.chi2_kernel
      'linear'          sklearn.pairwise.linear_kernel
      'poly'            sklearn.pairwise.polynomial_kernel
      'polynomial'      sklearn.pairwise.polynomial_kernel
      'rbf'             sklearn.pairwise.rbf_kernel
      'laplacian'       sklearn.pairwise.laplacian_kernel
      'sigmoid'         sklearn.pairwise.sigmoid_kernel
      'cosine'          sklearn.pairwise.cosine_similarity
      ===============   ========================================

    Read more in the :ref:`User Guide <metrics>`.
    """
    return PAIRWISE_KERNEL_FUNCTIONS


KERNEL_PARAMS = {
    "additive_chi2": (),
    "chi2": (),
    "cosine": (),
    "exp_chi2": frozenset(["gamma"]),
    "linear": (),
    "poly": frozenset(["gamma", "degree", "coef0"]),
    "polynomial": frozenset(["gamma", "degree", "coef0"]),
    "rbf": frozenset(["gamma"]),
    "laplacian": frozenset(["gamma"]),
    "sigmoid": frozenset(["gamma", "coef0"]),
}


def pairwise_kernels(X, Y=None, metric="linear", filter_params=False,
                     n_jobs=1, **kwds):
    """Compute the kernel between arrays X and optional array Y.

    This method takes either a vector array or a kernel matrix, and returns
    a kernel matrix. If the input is a vector array, the kernels are
    computed. If the input is a kernel matrix, it is returned instead.

    This method provides a safe way to take a kernel matrix as input, while
    preserving compatibility with many other algorithms that take a vector
    array.

    If Y is given (default is None), then the returned matrix is the pairwise
    kernel between the arrays from both X and Y.

    Valid values for metric are::
        ['rbf', 'sigmoid', 'polynomial', 'poly', 'linear', 'cosine']

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

    Parameters
    ----------
    X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \
             [n_samples_a, n_features] otherwise
        Array of pairwise kernels between samples, or a feature array.

    Y : array [n_samples_b, n_features]
        A second feature array only if X has shape [n_samples_a, n_features].

    metric : string, or callable
        The metric to use when calculating kernel between instances in a
        feature array. If metric is a string, it must be one of the metrics
        in pairwise.PAIRWISE_KERNEL_FUNCTIONS.
        If metric is "precomputed", X is assumed to be a kernel 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.

    n_jobs : int
        The number of jobs to use for the computation. This works by breaking
        down the pairwise matrix into n_jobs even slices and computing them in
        parallel.

        If -1 all CPUs are used. If 1 is given, no parallel computing code is
        used at all, which is useful for debugging. For n_jobs below -1,
        (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one
        are used.

    filter_params : boolean
        Whether to filter invalid parameters or not.

    **kwds : optional keyword parameters
        Any further parameters are passed directly to the kernel function.

    Returns
    -------
    K : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b]
        A kernel matrix K such that K_{i, j} is the kernel between the
        ith and jth vectors of the given matrix X, if Y is None.
        If Y is not None, then K_{i, j} is the kernel between the ith array
        from X and the jth array from Y.

    Notes
    -----
    If metric is 'precomputed', Y is ignored and X is returned.

    """
    # import GPKernel locally to prevent circular imports
    from ..gaussian_process.kernels import Kernel as GPKernel

    if metric == "precomputed":
        X, _ = check_pairwise_arrays(X, Y, precomputed=True)
        return X
    elif isinstance(metric, GPKernel):
        func = metric.__call__
    elif metric in PAIRWISE_KERNEL_FUNCTIONS:
        if filter_params:
            kwds = dict((k, kwds[k]) for k in kwds
                        if k in KERNEL_PARAMS[metric])
        func = PAIRWISE_KERNEL_FUNCTIONS[metric]
    elif callable(metric):
        func = partial(_pairwise_callable, metric=metric, **kwds)
    else:
        raise ValueError("Unknown kernel %r" % metric)

    return _parallel_pairwise(X, Y, func, n_jobs, **kwds)