text: # -*- 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 warnings
  
  import numpy as np
  from scipy.spatial import distance
  from scipy.sparse import csr_matrix
  from scipy.sparse import issparse
  from joblib import Parallel, effective_n_jobs
  
  from ..utils.validation import _num_samples
  from ..utils.validation import check_non_negative
  from ..utils import check_array
  from ..utils import gen_even_slices
  from ..utils import gen_batches, get_chunk_n_rows
  from ..utils import is_scalar_nan
  from ..utils.extmath import row_norms, safe_sparse_dot
  from ..preprocessing import normalize
  from ..utils._mask import _get_mask
  from ..utils.fixes import delayed
  from ..utils.fixes import sp_version, parse_version
  
  from ._pairwise_fast import _chi2_kernel_fast, _sparse_manhattan
  from ..exceptions import DataConversionWarning
  
  
  # 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 = float
  
      return X, Y, dtype
  
  
  def check_pairwise_arrays(
      X,
      Y,
      *,
      precomputed=False,
      dtype=None,
      accept_sparse="csr",
      force_all_finite=True,
      copy=False,
  ):
      """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} of shape (n_samples_X, n_features)
  
      Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features)
  
      precomputed : bool, default=False
          True if X is to be treated as precomputed distances to the samples in
          Y.
  
      dtype : str, type, list of type, 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
  
      accept_sparse : str, bool or list/tuple of str, default='csr'
          String[s] representing allowed sparse matrix formats, such as 'csc',
          'csr', etc. If the input is sparse but not in the allowed format,
          it will be converted to the first listed format. True allows the input
          to be any format. False means that a sparse matrix input will
          raise an error.
  
      force_all_finite : bool or 'allow-nan', default=True
          Whether to raise an error on np.inf, np.nan, pd.NA in array. The
          possibilities are:
  
          - True: Force all values of array to be finite.
          - False: accepts np.inf, np.nan, pd.NA in array.
          - 'allow-nan': accepts only np.nan and pd.NA values in array. Values
            cannot be infinite.
  
          .. versionadded:: 0.22
             ``force_all_finite`` accepts the string ``'allow-nan'``.
  
          .. versionchanged:: 0.23
             Accepts `pd.NA` and converts it into `np.nan`.
  
      copy : bool, default=False
          Whether a forced copy will be triggered. If copy=False, a copy might
          be triggered by a conversion.
  
          .. versionadded:: 0.22
  
      Returns
      -------
      safe_X : {array-like, sparse matrix} of shape (n_samples_X, n_features)
          An array equal to X, guaranteed to be a numpy array.
  
      safe_Y : {array-like, sparse matrix} of shape (n_samples_Y, 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)
  
      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=accept_sparse,
              dtype=dtype,
              copy=copy,
              force_all_finite=force_all_finite,
              estimator=estimator,
          )
      else:
          X = check_array(
              X,
              accept_sparse=accept_sparse,
              dtype=dtype,
              copy=copy,
              force_all_finite=force_all_finite,
              estimator=estimator,
          )
          Y = check_array(
              Y,
              accept_sparse=accept_sparse,
              dtype=dtype,
              copy=copy,
              force_all_finite=force_all_finite,
              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} of shape (n_samples_X, n_features)
  
      Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features)
  
      Returns
      -------
      safe_X : {array-like, sparse matrix} of shape (n_samples_X, n_features)
          An array equal to X, guaranteed to be a numpy array.
  
      safe_Y : {array-like, sparse matrix} of shape (n_samples_Y, 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,
      because this equation potentially suffers from "catastrophic cancellation".
      Also, 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} of shape (n_samples_X, n_features)
  
      Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features), \
              default=None
          If `None`, uses `Y=X`.
  
      Y_norm_squared : array-like of shape (n_samples_Y,) or (n_samples_Y, 1) \
              or (1, n_samples_Y), default=None
          Pre-computed dot-products of vectors in Y (e.g.,
          ``(Y**2).sum(axis=1)``)
          May be ignored in some cases, see the note below.
  
      squared : bool, default=False
          Return squared Euclidean distances.
  
      X_norm_squared : array-like of shape (n_samples_X,) or (n_samples_X, 1) \
              or (1, n_samples_X), default=None
          Pre-computed dot-products of vectors in X (e.g.,
          ``(X**2).sum(axis=1)``)
          May be ignored in some cases, see the note below.
  
      Notes
      -----
      To achieve better accuracy, `X_norm_squared` and `Y_norm_squared` may be
      unused if they are passed as ``float32``.
  
      Returns
      -------
      distances : ndarray of shape (n_samples_X, n_samples_Y)
  
      See Also
      --------
      paired_distances : Distances betweens pairs of elements of X and Y.
  
      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]])
      """
      X, Y = check_pairwise_arrays(X, Y)
  
      if X_norm_squared is not None:
          X_norm_squared = check_array(X_norm_squared, ensure_2d=False)
          original_shape = X_norm_squared.shape
          if X_norm_squared.shape == (X.shape[0],):
              X_norm_squared = X_norm_squared.reshape(-1, 1)
          if X_norm_squared.shape == (1, X.shape[0]):
              X_norm_squared = X_norm_squared.T
          if X_norm_squared.shape != (X.shape[0], 1):
              raise ValueError(
                  f"Incompatible dimensions for X of shape {X.shape} and "
                  f"X_norm_squared of shape {original_shape}."
              )
  
      if Y_norm_squared is not None:
          Y_norm_squared = check_array(Y_norm_squared, ensure_2d=False)
          original_shape = Y_norm_squared.shape
          if Y_norm_squared.shape == (Y.shape[0],):
              Y_norm_squared = Y_norm_squared.reshape(1, -1)
          if Y_norm_squared.shape == (Y.shape[0], 1):
              Y_norm_squared = Y_norm_squared.T
          if Y_norm_squared.shape != (1, Y.shape[0]):
              raise ValueError(
                  f"Incompatible dimensions for Y of shape {Y.shape} and "
                  f"Y_norm_squared of shape {original_shape}."
              )
  
      return _euclidean_distances(X, Y, X_norm_squared, Y_norm_squared, squared)
  
  
  def _euclidean_distances(X, Y, X_norm_squared=None, Y_norm_squared=None, squared=False):
      """Computational part of euclidean_distances
  
      Assumes inputs are already checked.
  
      If norms are passed as float32, they are unused. If arrays are passed as
      float32, norms needs to be recomputed on upcast chunks.
      TODO: use a float64 accumulator in row_norms to avoid the latter.
      """
      if X_norm_squared is not None:
          if X_norm_squared.dtype == np.float32:
              XX = None
          else:
              XX = X_norm_squared.reshape(-1, 1)
      elif X.dtype == np.float32:
          XX = None
      else:
          XX = row_norms(X, squared=True)[:, np.newaxis]
  
      if Y is X:
          YY = None if XX is None else XX.T
      else:
          if Y_norm_squared is not None:
              if Y_norm_squared.dtype == np.float32:
                  YY = None
              else:
                  YY = Y_norm_squared.reshape(1, -1)
          elif Y.dtype == np.float32:
              YY = None
          else:
              YY = row_norms(Y, squared=True)[np.newaxis, :]
  
      if X.dtype == np.float32:
          # To minimize precision issues with float32, we compute the distance
          # matrix on chunks of X and Y upcast to float64
          distances = _euclidean_distances_upcast(X, XX, Y, YY)
      else:
          # if dtype is already float64, no need to chunk and upcast
          distances = -2 * safe_sparse_dot(X, Y.T, dense_output=True)
          distances += XX
          distances += YY
      np.maximum(distances, 0, out=distances)
  
      # Ensure that distances between vectors and themselves are set to 0.0.
      # This may not be the case due to floating point rounding errors.
      if X is Y:
          np.fill_diagonal(distances, 0)
  
      return distances if squared else np.sqrt(distances, out=distances)
  
  
  def nan_euclidean_distances(
      X, Y=None, *, squared=False, missing_values=np.nan, copy=True
  ):
      """Calculate the euclidean distances in the presence of missing values.
  
      Compute the euclidean distance between each pair of samples in X and Y,
      where Y=X is assumed if Y=None. When calculating the distance between a
      pair of samples, this formulation ignores feature coordinates with a
      missing value in either sample and scales up the weight of the remaining
      coordinates:
  
          dist(x,y) = sqrt(weight * sq. distance from present coordinates)
          where,
          weight = Total # of coordinates / # of present coordinates
  
      For example, the distance between ``[3, na, na, 6]`` and ``[1, na, 4, 5]``
      is:
  
          .. math::
              \\sqrt{\\frac{4}{2}((3-1)^2 + (6-5)^2)}
  
      If all the coordinates are missing or if there are no common present
      coordinates then NaN is returned for that pair.
  
      Read more in the :ref:`User Guide <metrics>`.
  
      .. versionadded:: 0.22
  
      Parameters
      ----------
      X : array-like of shape=(n_samples_X, n_features)
  
      Y : array-like of shape=(n_samples_Y, n_features), default=None
  
      squared : bool, default=False
          Return squared Euclidean distances.
  
      missing_values : np.nan or int, default=np.nan
          Representation of missing value.
  
      copy : bool, default=True
          Make and use a deep copy of X and Y (if Y exists).
  
      Returns
      -------
      distances : ndarray of shape (n_samples_X, n_samples_Y)
  
      See Also
      --------
      paired_distances : Distances between pairs of elements of X and Y.
  
      Examples
      --------
      >>> from sklearn.metrics.pairwise import nan_euclidean_distances
      >>> nan = float("NaN")
      >>> X = [[0, 1], [1, nan]]
      >>> nan_euclidean_distances(X, X) # distance between rows of X
      array([[0.        , 1.41421356],
             [1.41421356, 0.        ]])
  
      >>> # get distance to origin
      >>> nan_euclidean_distances(X, [[0, 0]])
      array([[1.        ],
             [1.41421356]])
  
      References
      ----------
      * John K. Dixon, "Pattern Recognition with Partly Missing Data",
        IEEE Transactions on Systems, Man, and Cybernetics, Volume: 9, Issue:
        10, pp. 617 - 621, Oct. 1979.
        http://ieeexplore.ieee.org/abstract/document/4310090/
      """
  
      force_all_finite = "allow-nan" if is_scalar_nan(missing_values) else True
      X, Y = check_pairwise_arrays(
          X, Y, accept_sparse=False, force_all_finite=force_all_finite, copy=copy
      )
      # Get missing mask for X
      missing_X = _get_mask(X, missing_values)
  
      # Get missing mask for Y
      missing_Y = missing_X if Y is X else _get_mask(Y, missing_values)
  
      # set missing values to zero
      X[missing_X] = 0
      Y[missing_Y] = 0
  
      distances = euclidean_distances(X, Y, squared=True)
  
      # Adjust distances for missing values
      XX = X * X
      YY = Y * Y
      distances -= np.dot(XX, missing_Y.T)
      distances -= np.dot(missing_X, YY.T)
  
      np.clip(distances, 0, None, 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.
          np.fill_diagonal(distances, 0.0)
  
      present_X = 1 - missing_X
      present_Y = present_X if Y is X else ~missing_Y
      present_count = np.dot(present_X, present_Y.T)
      distances[present_count == 0] = np.nan
      # avoid divide by zero
      np.maximum(1, present_count, out=present_count)
      distances /= present_count
      distances *= X.shape[1]
  
      if not squared:
          np.sqrt(distances, out=distances)
  
      return distances
  
  
  def _euclidean_distances_upcast(X, XX=None, Y=None, YY=None, batch_size=None):
      """Euclidean distances between X and Y.
  
      Assumes X and Y have float32 dtype.
      Assumes XX and YY have float64 dtype or are None.
  
      X and Y are upcast to float64 by chunks, which size is chosen to limit
      memory increase by approximately 10% (at least 10MiB).
      """
      n_samples_X = X.shape[0]
      n_samples_Y = Y.shape[0]
      n_features = X.shape[1]
  
      distances = np.empty((n_samples_X, n_samples_Y), dtype=np.float32)
  
      if batch_size is None:
          x_density = X.nnz / np.prod(X.shape) if issparse(X) else 1
          y_density = Y.nnz / np.prod(Y.shape) if issparse(Y) else 1
  
          # Allow 10% more memory than X, Y and the distance matrix take (at
          # least 10MiB)
          maxmem = max(
              (
                  (x_density * n_samples_X + y_density * n_samples_Y) * n_features
                  + (x_density * n_samples_X * y_density * n_samples_Y)
              )
              / 10,
              10 * 2 ** 17,
          )
  
          # The increase amount of memory in 8-byte blocks is:
          # - x_density * batch_size * n_features (copy of chunk of X)
          # - y_density * batch_size * n_features (copy of chunk of Y)
          # - batch_size * batch_size (chunk of distance matrix)
          # Hence x² + (xd+yd)kx = M, where x=batch_size, k=n_features, M=maxmem
          #                                 xd=x_density and yd=y_density
          tmp = (x_density + y_density) * n_features
          batch_size = (-tmp + np.sqrt(tmp ** 2 + 4 * maxmem)) / 2
          batch_size = max(int(batch_size), 1)
  
      x_batches = gen_batches(n_samples_X, batch_size)
  
      for i, x_slice in enumerate(x_batches):
          X_chunk = X[x_slice].astype(np.float64)
          if XX is None:
              XX_chunk = row_norms(X_chunk, squared=True)[:, np.newaxis]
          else:
              XX_chunk = XX[x_slice]
  
          y_batches = gen_batches(n_samples_Y, batch_size)
  
          for j, y_slice in enumerate(y_batches):
              if X is Y and j < i:
                  # when X is Y the distance matrix is symmetric so we only need
                  # to compute half of it.
                  d = distances[y_slice, x_slice].T
  
              else:
                  Y_chunk = Y[y_slice].astype(np.float64)
                  if YY is None:
                      YY_chunk = row_norms(Y_chunk, squared=True)[np.newaxis, :]
                  else:
                      YY_chunk = YY[:, y_slice]
  
                  d = -2 * safe_sparse_dot(X_chunk, Y_chunk.T, dense_output=True)
                  d += XX_chunk
                  d += YY_chunk
  
              distances[x_slice, y_slice] = d.astype(np.float32, copy=False)
  
      return distances
  
  
  def _argmin_min_reduce(dist, start):
      indices = dist.argmin(axis=1)
      values = dist[np.arange(dist.shape[0]), indices]
      return indices, values
  
  
  def pairwise_distances_argmin_min(
      X, Y, *, axis=1, metric="euclidean", 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 : {array-like, sparse matrix} of shape (n_samples_X, n_features)
          Array containing points.
  
      Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features)
          Array containing points.
  
      axis : int, default=1
          Axis along which the argmin and distances are to be computed.
  
      metric : str 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', 'minkowski', 'rogerstanimoto', 'russellrao',
            'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean',
            'yule']
  
          See the documentation for scipy.spatial.distance for details on these
          metrics.
  
      metric_kwargs : dict, default=None
          Keyword arguments to pass to specified metric function.
  
      Returns
      -------
      argmin : ndarray
          Y[argmin[i], :] is the row in Y that is closest to X[i, :].
  
      distances : 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
      """
      X, Y = check_pairwise_arrays(X, Y)
  
      if metric_kwargs is None:
          metric_kwargs = {}
  
      if axis == 0:
          X, Y = Y, X
  
      indices, values = zip(
          *pairwise_distances_chunked(
              X, Y, reduce_func=_argmin_min_reduce, metric=metric, **metric_kwargs
          )
      )
      indices = np.concatenate(indices)
      values = np.concatenate(values)
  
      return indices, values
  
  
  def pairwise_distances_argmin(X, Y, *, axis=1, metric="euclidean", 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 of shape (n_samples_X, n_features)
          Array containing points.
  
      Y : array-like of shape (n_samples_Y, n_features)
          Arrays containing points.
  
      axis : int, default=1
          Axis along which the argmin and distances are to be computed.
  
      metric : str 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', 'minkowski', 'rogerstanimoto', 'russellrao',
            'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean',
            'yule']
  
          See the documentation for scipy.spatial.distance for details on these
          metrics.
  
      metric_kwargs : dict, default=None
          Keyword arguments to pass to specified metric function.
  
      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=axis, metric=metric, metric_kwargs=metric_kwargs
      )[0]
  
  
  def haversine_distances(X, Y=None):
      """Compute the Haversine distance between samples in X and Y.
  
      The Haversine (or great circle) distance is the angular distance between
      two points on the surface of a sphere. The first coordinate of each point
      is assumed to be the latitude, the second is the longitude, given
      in radians. The dimension of the data must be 2.
  
      .. math::
         D(x, y) = 2\\arcsin[\\sqrt{\\sin^2((x1 - y1) / 2)
                                  + \\cos(x1)\\cos(y1)\\sin^2((x2 - y2) / 2)}]
  
      Parameters
      ----------
      X : array-like of shape (n_samples_X, 2)
  
      Y : array-like of shape (n_samples_Y, 2), default=None
  
      Returns
      -------
      distance : ndarray of shape (n_samples_X, n_samples_Y)
  
      Notes
      -----
      As the Earth is nearly spherical, the haversine formula provides a good
      approximation of the distance between two points of the Earth surface, with
      a less than 1% error on average.
  
      Examples
      --------
      We want to calculate the distance between the Ezeiza Airport
      (Buenos Aires, Argentina) and the Charles de Gaulle Airport (Paris,
      France).
  
      >>> from sklearn.metrics.pairwise import haversine_distances
      >>> from math import radians
      >>> bsas = [-34.83333, -58.5166646]
      >>> paris = [49.0083899664, 2.53844117956]
      >>> bsas_in_radians = [radians(_) for _ in bsas]
      >>> paris_in_radians = [radians(_) for _ in paris]
      >>> result = haversine_distances([bsas_in_radians, paris_in_radians])
      >>> result * 6371000/1000  # multiply by Earth radius to get kilometers
      array([[    0.        , 11099.54035582],
             [11099.54035582,     0.        ]])
      """
      from ..neighbors import DistanceMetric
  
      return DistanceMetric.get_metric("haversine").pairwise(X, Y)
  
  
  def manhattan_distances(X, Y=None, *, sum_over_features=True):
      """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 of shape (n_samples_X, n_features)
  
      Y : array-like of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      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.
  
      Returns
      -------
      D : ndarray of shape (n_samples_X * n_samples_Y, n_features) or \
              (n_samples_X, n_samples_Y)
          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.
  
      Notes
      --------
      When X and/or Y are CSR sparse matrices and they are not already
      in canonical format, this function modifies them in-place to
      make them canonical.
  
      Examples
      --------
      >>> from sklearn.metrics.pairwise import manhattan_distances
      >>> manhattan_distances([[3]], [[3]])
      array([[0.]])
      >>> manhattan_distances([[3]], [[2]])
      array([[1.]])
      >>> manhattan_distances([[2]], [[3]])
      array([[1.]])
      >>> manhattan_distances([[1, 2], [3, 4]],\
           [[1, 2], [0, 3]])
      array([[0., 2.],
             [4., 4.]])
      >>> import numpy as np
      >>> X = np.ones((1, 2))
      >>> y = np.full((2, 2), 2.)
      >>> manhattan_distances(X, y, sum_over_features=False)
      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)
          X.sum_duplicates()  # this also sorts indices in-place
          Y.sum_duplicates()
          D = np.zeros((X.shape[0], Y.shape[0]))
          _sparse_manhattan(X.data, X.indices, X.indptr, Y.data, Y.indices, Y.indptr, 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} of shape (n_samples_X, n_features)
          Matrix `X`.
  
      Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features), \
              default=None
          Matrix `Y`.
  
      Returns
      -------
      distance matrix : ndarray of shape (n_samples_X, n_samples_Y)
  
      See Also
      --------
      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 of shape (n_samples, n_features)
  
      Y : array-like of shape (n_samples, n_features)
  
      Returns
      -------
      distances : ndarray of shape (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 of shape (n_samples, n_features)
  
      Y : array-like of shape (n_samples, n_features)
  
      Returns
      -------
      distances : ndarray of shape (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 of shape (n_samples, n_features)
  
      Y : array-like of shape (n_samples, n_features)
  
      Returns
      -------
      distances : ndarray of 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 0.5 * 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 of shape (n_samples, n_features)
          Array 1 for distance computation.
  
      Y : ndarray of shape (n_samples, n_features)
          Array 2 for distance computation.
  
      metric : str or callable, default="euclidean"
          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 of shape (n_samples,)
  
      See Also
      --------
      pairwise_distances : Computes the distance between every pair of 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.])
      """
  
      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, dense_output=True):
      """
      Compute the linear kernel between X and Y.
  
      Read more in the :ref:`User Guide <linear_kernel>`.
  
      Parameters
      ----------
      X : ndarray of shape (n_samples_X, n_features)
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      dense_output : bool, 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:: 0.20
  
      Returns
      -------
      Gram matrix : ndarray of shape (n_samples_X, n_samples_Y)
      """
      X, Y = check_pairwise_arrays(X, Y)
      return safe_sparse_dot(X, Y.T, dense_output=dense_output)
  
  
  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_X, n_features)
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
  
      degree : int, default=3
  
      gamma : float, default=None
          If None, defaults to 1.0 / n_features.
  
      coef0 : float, default=1
  
      Returns
      -------
      Gram matrix : ndarray 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 = 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_X, n_features)
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      gamma : float, default=None
          If None, defaults to 1.0 / n_features.
  
      coef0 : float, default=1
  
      Returns
      -------
      Gram matrix : ndarray 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 = 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 : ndarray of shape (n_samples_X, n_features)
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      gamma : float, default=None
          If None, defaults to 1.0 / n_features.
  
      Returns
      -------
      kernel_matrix : ndarray 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 : ndarray of shape (n_samples_X, n_features)
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      gamma : float, default=None
          If None, defaults to 1.0 / n_features.
  
      Returns
      -------
      kernel_matrix : ndarray 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, sparse matrix} of shape (n_samples_X, n_features)
          Input data.
  
      Y : {ndarray, sparse matrix} of shape (n_samples_Y, n_features), \
              default=None
          Input data. If ``None``, the output will be the pairwise
          similarities between all samples in ``X``.
  
      dense_output : bool, 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 : ndarray of 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 incremental_cosine_similarity(X, Y=None, dense_output=True, new_indices=None):
      """Compute cosine similarity between samples in X and Y only for entries in
      the rows with the given new_indices.
  
      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, sparse matrix} of shape (n_samples_X, n_features)
          Input data.
  
      Y : {ndarray, sparse matrix} of shape (n_samples_Y, n_features), \
              default=None
          Input data. If ``None``, the output will be the pairwise
          similarities between all samples in ``X``.
  
      dense_output : bool, 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.
  
      new_indices : list of indices to represent the newly updated or added
          input data
  
  
      Returns
      -------
      kernel matrix : ndarray of shape (length_of_new_indices, 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)
  
      new_data_normalized = X_normalized
      if new_indices is not None:
          row_idx = np.array(new_indices)
          new_data = X_normalized[row_idx]
          new_data_normalized = normalize(new_data, copy=True)
  
      K = safe_sparse_dot(new_data_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 : ndarray of shape (n_samples_Y, n_features), default=None
          If `None`, uses `Y=X`.
  
      Returns
      -------
      kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y)
  
      See Also
      --------
      chi2_kernel : The exponentiated version of the kernel, which is usually
          preferable.
      sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation
          to this kernel.
  
      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
        https://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf
      """
      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=1.0):
      """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 : ndarray of shape (n_samples_Y, n_features), default=None
  
      gamma : float, default=1.
          Scaling parameter of the chi2 kernel.
  
      Returns
      -------
      kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y)
  
      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.
  
      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
        https://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf
      """
      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,
      "haversine": haversine_distances,
      "l2": euclidean_distances,
      "l1": manhattan_distances,
      "manhattan": manhattan_distances,
      "precomputed": None,  # HACK: precomputed is always allowed, never called
      "nan_euclidean": nan_euclidean_distances,
  }
  
  
  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
      'haversine'     metrics.pairwise.haversine_distances
      'l1'            metrics.pairwise.manhattan_distances
      'l2'            metrics.pairwise.euclidean_distances
      'manhattan'     metrics.pairwise.manhattan_distances
      'nan_euclidean' metrics.pairwise.nan_euclidean_distances
      =============== ========================================
  
      Read more in the :ref:`User Guide <metrics>`.
  
      """
      return PAIRWISE_DISTANCE_FUNCTIONS
  
  
  def _dist_wrapper(dist_func, dist_matrix, slice_, *args, **kwargs):
      """Write in-place to a slice of a distance matrix."""
      dist_matrix[:, slice_] = dist_func(*args, **kwargs)
  
  
  def _parallel_pairwise(X, Y, func, n_jobs, **kwds):
      """Break the pairwise matrix in n_jobs even slices
      and compute them in parallel."""
  
      if Y is None:
          Y = X
      X, Y, dtype = _return_float_dtype(X, Y)
  
      if effective_n_jobs(n_jobs) == 1:
          return func(X, Y, **kwds)
  
      # enforce a threading backend to prevent data communication overhead
      fd = delayed(_dist_wrapper)
      ret = np.empty((X.shape[0], Y.shape[0]), dtype=dtype, order="F")
      Parallel(backend="threading", n_jobs=n_jobs)(
          fd(func, ret, s, X, Y[s], **kwds)
          for s in gen_even_slices(_num_samples(Y), effective_n_jobs(n_jobs))
      )
  
      if (X is Y or Y is None) and func is euclidean_distances:
          # zeroing diagonal for euclidean norm.
          # TODO: do it also for other norms.
          np.fill_diagonal(ret, 0)
  
      return ret
  
  
  def _pairwise_callable(X, Y, metric, force_all_finite=True, **kwds):
      """Handle the callable case for pairwise_{distances,kernels}."""
      X, Y = check_pairwise_arrays(X, Y, force_all_finite=force_all_finite)
  
      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",
      "nan_euclidean",
      "haversine",
  ]
  
  _NAN_METRICS = ["nan_euclidean"]
  
  
  def _check_chunk_size(reduced, chunk_size):
      """Checks chunk is a sequence of expected size or a tuple of same."""
      if reduced is None:
          return
      is_tuple = isinstance(reduced, tuple)
      if not is_tuple:
          reduced = (reduced,)
      if any(isinstance(r, tuple) or not hasattr(r, "__iter__") for r in reduced):
          raise TypeError(
              "reduce_func returned %r. Expected sequence(s) of length %d."
              % (reduced if is_tuple else reduced[0], chunk_size)
          )
      if any(_num_samples(r) != chunk_size for r in reduced):
          actual_size = tuple(_num_samples(r) for r in reduced)
          raise ValueError(
              "reduce_func returned object of length %s. "
              "Expected same length as input: %d."
              % (actual_size if is_tuple else actual_size[0], chunk_size)
          )
  
  
  def _precompute_metric_params(X, Y, metric=None, **kwds):
      """Precompute data-derived metric parameters if not provided."""
      if metric == "seuclidean" and "V" not in kwds:
          # There is a bug in scipy < 1.5 that will cause a crash if
          # X.dtype != np.double (float64). See PR #15730
          dtype = np.float64 if sp_version < parse_version("1.5") else None
          if X is Y:
              V = np.var(X, axis=0, ddof=1, dtype=dtype)
          else:
              raise ValueError(
                  "The 'V' parameter is required for the seuclidean metric "
                  "when Y is passed."
              )
          return {"V": V}
      if metric == "mahalanobis" and "VI" not in kwds:
          if X is Y:
              VI = np.linalg.inv(np.cov(X.T)).T
          else:
              raise ValueError(
                  "The 'VI' parameter is required for the mahalanobis metric "
                  "when Y is passed."
              )
          return {"VI": VI}
      return {}
  
  
  def pairwise_distances_chunked(
      X,
      Y=None,
      *,
      reduce_func=None,
      metric="euclidean",
      n_jobs=None,
      working_memory=None,
      **kwds,
  ):
      """Generate a distance matrix chunk by chunk with optional reduction.
  
      In cases where not all of a pairwise distance matrix needs to be stored at
      once, this is used to calculate pairwise distances in
      ``working_memory``-sized chunks.  If ``reduce_func`` is given, it is run
      on each chunk and its return values are concatenated into lists, arrays
      or sparse matrices.
  
      Parameters
      ----------
      X : ndarray of shape (n_samples_X, n_samples_X) or \
              (n_samples_X, n_features)
          Array of pairwise distances between samples, or a feature array.
          The shape the array should be (n_samples_X, n_samples_X) if
          metric='precomputed' and (n_samples_X, n_features) otherwise.
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          An optional second feature array. Only allowed if
          metric != "precomputed".
  
      reduce_func : callable, default=None
          The function which is applied on each chunk of the distance matrix,
          reducing it to needed values.  ``reduce_func(D_chunk, start)``
          is called repeatedly, where ``D_chunk`` is a contiguous vertical
          slice of the pairwise distance matrix, starting at row ``start``.
          It should return one of: None; an array, a list, or a sparse matrix
          of length ``D_chunk.shape[0]``; or a tuple of such objects. Returning
          None is useful for in-place operations, rather than reductions.
  
          If None, pairwise_distances_chunked returns a generator of vertical
          chunks of the distance matrix.
  
      metric : str or callable, default='euclidean'
          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, default=None
          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.
  
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      working_memory : int, default=None
          The sought maximum memory for temporary distance matrix chunks.
          When None (default), the value of
          ``sklearn.get_config()['working_memory']`` is 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.
  
      Yields
      ------
      D_chunk : {ndarray, sparse matrix}
          A contiguous slice of distance matrix, optionally processed by
          ``reduce_func``.
  
      Examples
      --------
      Without reduce_func:
  
      >>> import numpy as np
      >>> from sklearn.metrics import pairwise_distances_chunked
      >>> X = np.random.RandomState(0).rand(5, 3)
      >>> D_chunk = next(pairwise_distances_chunked(X))
      >>> D_chunk
      array([[0.  ..., 0.29..., 0.41..., 0.19..., 0.57...],
             [0.29..., 0.  ..., 0.57..., 0.41..., 0.76...],
             [0.41..., 0.57..., 0.  ..., 0.44..., 0.90...],
             [0.19..., 0.41..., 0.44..., 0.  ..., 0.51...],
             [0.57..., 0.76..., 0.90..., 0.51..., 0.  ...]])
  
      Retrieve all neighbors and average distance within radius r:
  
      >>> r = .2
      >>> def reduce_func(D_chunk, start):
      ...     neigh = [np.flatnonzero(d < r) for d in D_chunk]
      ...     avg_dist = (D_chunk * (D_chunk < r)).mean(axis=1)
      ...     return neigh, avg_dist
      >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func)
      >>> neigh, avg_dist = next(gen)
      >>> neigh
      [array([0, 3]), array([1]), array([2]), array([0, 3]), array([4])]
      >>> avg_dist
      array([0.039..., 0.        , 0.        , 0.039..., 0.        ])
  
      Where r is defined per sample, we need to make use of ``start``:
  
      >>> r = [.2, .4, .4, .3, .1]
      >>> def reduce_func(D_chunk, start):
      ...     neigh = [np.flatnonzero(d < r[i])
      ...              for i, d in enumerate(D_chunk, start)]
      ...     return neigh
      >>> neigh = next(pairwise_distances_chunked(X, reduce_func=reduce_func))
      >>> neigh
      [array([0, 3]), array([0, 1]), array([2]), array([0, 3]), array([4])]
  
      Force row-by-row generation by reducing ``working_memory``:
  
      >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func,
      ...                                  working_memory=0)
      >>> next(gen)
      [array([0, 3])]
      >>> next(gen)
      [array([0, 1])]
      """
      n_samples_X = _num_samples(X)
      if metric == "precomputed":
          slices = (slice(0, n_samples_X),)
      else:
          if Y is None:
              Y = X
          # We get as many rows as possible within our working_memory budget to
          # store len(Y) distances in each row of output.
          #
          # Note:
          #  - this will get at least 1 row, even if 1 row of distances will
          #    exceed working_memory.
          #  - this does not account for any temporary memory usage while
          #    calculating distances (e.g. difference of vectors in manhattan
          #    distance.
          chunk_n_rows = get_chunk_n_rows(
              row_bytes=8 * _num_samples(Y),
              max_n_rows=n_samples_X,
              working_memory=working_memory,
          )
          slices = gen_batches(n_samples_X, chunk_n_rows)
  
      # precompute data-derived metric params
      params = _precompute_metric_params(X, Y, metric=metric, **kwds)
      kwds.update(**params)
  
      for sl in slices:
          if sl.start == 0 and sl.stop == n_samples_X:
              X_chunk = X  # enable optimised paths for X is Y
          else:
              X_chunk = X[sl]
          D_chunk = pairwise_distances(X_chunk, Y, metric=metric, n_jobs=n_jobs, **kwds)
          if (X is Y or Y is None) and PAIRWISE_DISTANCE_FUNCTIONS.get(
              metric, None
          ) is euclidean_distances:
              # zeroing diagonal, taking care of aliases of "euclidean",
              # i.e. "l2"
              D_chunk.flat[sl.start :: _num_samples(X) + 1] = 0
          if reduce_func is not None:
              chunk_size = D_chunk.shape[0]
              D_chunk = reduce_func(D_chunk, sl.start)
              _check_chunk_size(D_chunk, chunk_size)
          yield D_chunk
  
  
  def pairwise_distances(
      X, Y=None, metric="euclidean", *, n_jobs=None, force_all_finite=True, **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.
        ['nan_euclidean'] but it does not yet support sparse matrices.
  
      - From scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev',
        'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis',
        '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 : ndarray of shape (n_samples_X, n_samples_X) or \
              (n_samples_X, n_features)
          Array of pairwise distances between samples, or a feature array.
          The shape of the array should be (n_samples_X, n_samples_X) if
          metric == "precomputed" and (n_samples_X, n_features) otherwise.
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          An optional second feature array. Only allowed if
          metric != "precomputed".
  
      metric : str or callable, default='euclidean'
          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, default=None
          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.
  
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      force_all_finite : bool or 'allow-nan', default=True
          Whether to raise an error on np.inf, np.nan, pd.NA in array. Ignored
          for a metric listed in ``pairwise.PAIRWISE_DISTANCE_FUNCTIONS``. The
          possibilities are:
  
          - True: Force all values of array to be finite.
          - False: accepts np.inf, np.nan, pd.NA in array.
          - 'allow-nan': accepts only np.nan and pd.NA values in array. Values
            cannot be infinite.
  
          .. versionadded:: 0.22
             ``force_all_finite`` accepts the string ``'allow-nan'``.
  
          .. versionchanged:: 0.23
             Accepts `pd.NA` and converts it into `np.nan`.
  
      **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 : ndarray of shape (n_samples_X, n_samples_X) or \
              (n_samples_X, n_samples_Y)
          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.
  
      See Also
      --------
      pairwise_distances_chunked : Performs the same calculation as this
          function, but returns a generator of chunks of the distance matrix, in
          order to limit memory usage.
      paired_distances : Computes the distances between corresponding elements
          of two arrays.
      """
      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, force_all_finite=force_all_finite
          )
  
          whom = (
              "`pairwise_distances`. Precomputed distance "
              " need to have non-negative values."
          )
          check_non_negative(X, whom=whom)
          return X
      elif metric in PAIRWISE_DISTANCE_FUNCTIONS:
          func = PAIRWISE_DISTANCE_FUNCTIONS[metric]
      elif callable(metric):
          func = partial(
              _pairwise_callable, metric=metric, force_all_finite=force_all_finite, **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
  
          if dtype == bool and (X.dtype != bool or (Y is not None and Y.dtype != bool)):
              msg = "Data was converted to boolean for metric %s" % metric
              warnings.warn(msg, DataConversionWarning)
  
          X, Y = check_pairwise_arrays(
              X, Y, dtype=dtype, force_all_finite=force_all_finite
          )
  
          # precompute data-derived metric params
          params = _precompute_metric_params(X, Y, metric=metric, **kwds)
          kwds.update(**params)
  
          if effective_n_jobs(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 require 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": frozenset(["gamma"]),
      "cosine": (),
      "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=None, **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:
          ['additive_chi2', 'chi2', 'linear', 'poly', 'polynomial', 'rbf',
          'laplacian', 'sigmoid', 'cosine']
  
      Read more in the :ref:`User Guide <metrics>`.
  
      Parameters
      ----------
      X : ndarray of shape (n_samples_X, n_samples_X) or \
              (n_samples_X, n_features)
          Array of pairwise kernels between samples, or a feature array.
          The shape of the array should be (n_samples_X, n_samples_X) if
          metric == "precomputed" and (n_samples_X, n_features) otherwise.
  
      Y : ndarray of shape (n_samples_Y, n_features), default=None
          A second feature array only if X has shape (n_samples_X, n_features).
  
      metric : str or callable, default="linear"
          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 rows from X as input and return the corresponding
          kernel value as a single number. This means that callables from
          :mod:`sklearn.metrics.pairwise` are not allowed, as they operate on
          matrices, not single samples. Use the string identifying the kernel
          instead.
  
      filter_params : bool, default=False
          Whether to filter invalid parameters or not.
  
      n_jobs : int, default=None
          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.
  
          ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
          ``-1`` means using all processors. See :term:`Glossary <n_jobs>`
          for more details.
  
      **kwds : optional keyword parameters
          Any further parameters are passed directly to the kernel function.
  
      Returns
      -------
      K : ndarray of shape (n_samples_X, n_samples_X) or \
              (n_samples_X, n_samples_Y)
          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 = {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)
  