<filename>sklearn/cluster/_agglomerative.py
"""Hierarchical Agglomerative Clustering

These routines perform some hierarchical agglomerative clustering of some
input data.

Authors : <NAME>, <NAME>, <NAME>,
          <NAME>
License: BSD 3 clause
"""
import warnings
from heapq import heapify, heappop, heappush, heappushpop

import numpy as np
from scipy import sparse
from scipy.sparse.csgraph import connected_components

from ..base import BaseEstimator, ClusterMixin
from ..metrics.pairwise import paired_distances
from ..metrics import DistanceMetric
from ..metrics._dist_metrics import METRIC_MAPPING
from ..utils import check_array
from ..utils._fast_dict import IntFloatDict
from ..utils.fixes import _astype_copy_false
from ..utils.graph import _fix_connected_components
from ..utils.validation import check_memory

# mypy error: Module 'sklearn.cluster' has no attribute '_hierarchical_fast'
from . import _hierarchical_fast as _hierarchical  # type: ignore
from ._feature_agglomeration import AgglomerationTransform

###############################################################################
# For non fully-connected graphs


def _fix_connectivity(X, connectivity, affinity):
    """
    Fixes the connectivity matrix.

    The different steps are:

    - copies it
    - makes it symmetric
    - converts it to LIL if necessary
    - completes it if necessary.

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is `None`, i.e, the Ward algorithm is unstructured.

    affinity : {"euclidean", "precomputed"}, default="euclidean"
        Which affinity to use. At the moment `precomputed` and
        ``euclidean`` are supported. `euclidean` uses the
        negative squared Euclidean distance between points.

    Returns
    -------
    connectivity : sparse matrix
        The fixed connectivity matrix.

    n_connected_components : int
        The number of connected components in the graph.
    """
    n_samples = X.shape[0]
    if connectivity.shape[0] != n_samples or connectivity.shape[1] != n_samples:
        raise ValueError(
            "Wrong shape for connectivity matrix: %s when X is %s"
            % (connectivity.shape, X.shape)
        )

    # Make the connectivity matrix symmetric:
    connectivity = connectivity + connectivity.T

    # Convert connectivity matrix to LIL
    if not sparse.isspmatrix_lil(connectivity):
        if not sparse.isspmatrix(connectivity):
            connectivity = sparse.lil_matrix(connectivity)
        else:
            connectivity = connectivity.tolil()

    # Compute the number of nodes
    n_connected_components, labels = connected_components(connectivity)

    if n_connected_components > 1:
        warnings.warn(
            "the number of connected components of the "
            "connectivity matrix is %d > 1. Completing it to avoid "
            "stopping the tree early." % n_connected_components,
            stacklevel=2,
        )
        # XXX: Can we do without completing the matrix?
        connectivity = _fix_connected_components(
            X=X,
            graph=connectivity,
            n_connected_components=n_connected_components,
            component_labels=labels,
            metric=affinity,
            mode="connectivity",
        )

    return connectivity, n_connected_components


def _single_linkage_tree(
    connectivity,
    n_samples,
    n_nodes,
    n_clusters,
    n_connected_components,
    return_distance,
):
    """
    Perform single linkage clustering on sparse data via the minimum
    spanning tree from scipy.sparse.csgraph, then using union-find to label.
    The parent array is then generated by walking through the tree.
    """
    from scipy.sparse.csgraph import minimum_spanning_tree

    # explicitly cast connectivity to ensure safety
    connectivity = connectivity.astype("float64", **_astype_copy_false(connectivity))

    # Ensure zero distances aren't ignored by setting them to "epsilon"
    epsilon_value = np.finfo(dtype=connectivity.data.dtype).eps
    connectivity.data[connectivity.data == 0] = epsilon_value

    # Use scipy.sparse.csgraph to generate a minimum spanning tree
    mst = minimum_spanning_tree(connectivity.tocsr())

    # Convert the graph to scipy.cluster.hierarchy array format
    mst = mst.tocoo()

    # Undo the epsilon values
    mst.data[mst.data == epsilon_value] = 0

    mst_array = np.vstack([mst.row, mst.col, mst.data]).T

    # Sort edges of the min_spanning_tree by weight
    mst_array = mst_array[np.argsort(mst_array.T[2], kind="mergesort"), :]

    # Convert edge list into standard hierarchical clustering format
    single_linkage_tree = _hierarchical._single_linkage_label(mst_array)
    children_ = single_linkage_tree[:, :2].astype(int)

    # Compute parents
    parent = np.arange(n_nodes, dtype=np.intp)
    for i, (left, right) in enumerate(children_, n_samples):
        if n_clusters is not None and i >= n_nodes:
            break
        if left < n_nodes:
            parent[left] = i
        if right < n_nodes:
            parent[right] = i

    if return_distance:
        distances = single_linkage_tree[:, 2]
        return children_, n_connected_components, n_samples, parent, distances
    return children_, n_connected_components, n_samples, parent


###############################################################################
# Hierarchical tree building functions


def ward_tree(X, *, connectivity=None, n_clusters=None, return_distance=False):
    """Ward clustering based on a Feature matrix.

    Recursively merges the pair of clusters that minimally increases
    within-cluster variance.

    The inertia matrix uses a Heapq-based representation.

    This is the structured version, that takes into account some topological
    structure between samples.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is None, i.e, the Ward algorithm is unstructured.

    n_clusters : int, default=None
        `n_clusters` should be less than `n_samples`.  Stop early the
        construction of the tree at `n_clusters.` This is useful to decrease
        computation time if the number of clusters is not small compared to the
        number of samples. In this case, the complete tree is not computed, thus
        the 'children' output is of limited use, and the 'parents' output should
        rather be used. This option is valid only when specifying a connectivity
        matrix.

    return_distance : bool, default=False
        If `True`, return the distance between the clusters.

    Returns
    -------
    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_connected_components : int
        The number of connected components in the graph.

    n_leaves : int
        The number of leaves in the tree.

    parents : ndarray of shape (n_nodes,) or None
        The parent of each node. Only returned when a connectivity matrix
        is specified, elsewhere 'None' is returned.

    distances : ndarray of shape (n_nodes-1,)
        Only returned if `return_distance` is set to `True` (for compatibility).
        The distances between the centers of the nodes. `distances[i]`
        corresponds to a weighted Euclidean distance between
        the nodes `children[i, 1]` and `children[i, 2]`. If the nodes refer to
        leaves of the tree, then `distances[i]` is their unweighted Euclidean
        distance. Distances are updated in the following way
        (from scipy.hierarchy.linkage):

        The new entry :math:`d(u,v)` is computed as follows,

        .. math::

           d(u,v) = \\sqrt{\\frac{|v|+|s|}
                               {T}d(v,s)^2
                        + \\frac{|v|+|t|}
                               {T}d(v,t)^2
                        - \\frac{|v|}
                               {T}d(s,t)^2}

        where :math:`u` is the newly joined cluster consisting of
        clusters :math:`s` and :math:`t`, :math:`v` is an unused
        cluster in the forest, :math:`T=|v|+|s|+|t|`, and
        :math:`|*|` is the cardinality of its argument. This is also
        known as the incremental algorithm.
    """
    X = np.asarray(X)
    if X.ndim == 1:
        X = np.reshape(X, (-1, 1))
    n_samples, n_features = X.shape

    if connectivity is None:
        from scipy.cluster import hierarchy  # imports PIL

        if n_clusters is not None:
            warnings.warn(
                "Partial build of the tree is implemented "
                "only for structured clustering (i.e. with "
                "explicit connectivity). The algorithm "
                "will build the full tree and only "
                "retain the lower branches required "
                "for the specified number of clusters",
                stacklevel=2,
            )
        X = np.require(X, requirements="W")
        out = hierarchy.ward(X)
        children_ = out[:, :2].astype(np.intp)

        if return_distance:
            distances = out[:, 2]
            return children_, 1, n_samples, None, distances
        else:
            return children_, 1, n_samples, None

    connectivity, n_connected_components = _fix_connectivity(
        X, connectivity, affinity="euclidean"
    )
    if n_clusters is None:
        n_nodes = 2 * n_samples - 1
    else:
        if n_clusters > n_samples:
            raise ValueError(
                "Cannot provide more clusters than samples. "
                "%i n_clusters was asked, and there are %i "
                "samples." % (n_clusters, n_samples)
            )
        n_nodes = 2 * n_samples - n_clusters

    # create inertia matrix
    coord_row = []
    coord_col = []
    A = []
    for ind, row in enumerate(connectivity.rows):
        A.append(row)
        # We keep only the upper triangular for the moments
        # Generator expressions are faster than arrays on the following
        row = [i for i in row if i < ind]
        coord_row.extend(
            len(row)
            * [
                ind,
            ]
        )
        coord_col.extend(row)

    coord_row = np.array(coord_row, dtype=np.intp, order="C")
    coord_col = np.array(coord_col, dtype=np.intp, order="C")

    # build moments as a list
    moments_1 = np.zeros(n_nodes, order="C")
    moments_1[:n_samples] = 1
    moments_2 = np.zeros((n_nodes, n_features), order="C")
    moments_2[:n_samples] = X
    inertia = np.empty(len(coord_row), dtype=np.float64, order="C")
    _hierarchical.compute_ward_dist(moments_1, moments_2, coord_row, coord_col, inertia)
    inertia = list(zip(inertia, coord_row, coord_col))
    heapify(inertia)

    # prepare the main fields
    parent = np.arange(n_nodes, dtype=np.intp)
    used_node = np.ones(n_nodes, dtype=bool)
    children = []
    if return_distance:
        distances = np.empty(n_nodes - n_samples)

    not_visited = np.empty(n_nodes, dtype=np.int8, order="C")

    # recursive merge loop
    for k in range(n_samples, n_nodes):
        # identify the merge
        while True:
            inert, i, j = heappop(inertia)
            if used_node[i] and used_node[j]:
                break
        parent[i], parent[j] = k, k
        children.append((i, j))
        used_node[i] = used_node[j] = False
        if return_distance:  # store inertia value
            distances[k - n_samples] = inert

        # update the moments
        moments_1[k] = moments_1[i] + moments_1[j]
        moments_2[k] = moments_2[i] + moments_2[j]

        # update the structure matrix A and the inertia matrix
        coord_col = []
        not_visited.fill(1)
        not_visited[k] = 0
        _hierarchical._get_parents(A[i], coord_col, parent, not_visited)
        _hierarchical._get_parents(A[j], coord_col, parent, not_visited)
        # List comprehension is faster than a for loop
        [A[col].append(k) for col in coord_col]
        A.append(coord_col)
        coord_col = np.array(coord_col, dtype=np.intp, order="C")
        coord_row = np.empty(coord_col.shape, dtype=np.intp, order="C")
        coord_row.fill(k)
        n_additions = len(coord_row)
        ini = np.empty(n_additions, dtype=np.float64, order="C")

        _hierarchical.compute_ward_dist(moments_1, moments_2, coord_row, coord_col, ini)

        # List comprehension is faster than a for loop
        [heappush(inertia, (ini[idx], k, coord_col[idx])) for idx in range(n_additions)]

    # Separate leaves in children (empty lists up to now)
    n_leaves = n_samples
    # sort children to get consistent output with unstructured version
    children = [c[::-1] for c in children]
    children = np.array(children)  # return numpy array for efficient caching

    if return_distance:
        # 2 is scaling factor to compare w/ unstructured version
        distances = np.sqrt(2.0 * distances)
        return children, n_connected_components, n_leaves, parent, distances
    else:
        return children, n_connected_components, n_leaves, parent


# single average and complete linkage
def linkage_tree(
    X,
    connectivity=None,
    n_clusters=None,
    linkage="complete",
    affinity="euclidean",
    return_distance=False,
):
    """Linkage agglomerative clustering based on a Feature matrix.

    The inertia matrix uses a Heapq-based representation.

    This is the structured version, that takes into account some topological
    structure between samples.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is `None`, i.e, the Ward algorithm is unstructured.

    n_clusters : int, default=None
        Stop early the construction of the tree at `n_clusters`. This is
        useful to decrease computation time if the number of clusters is
        not small compared to the number of samples. In this case, the
        complete tree is not computed, thus the 'children' output is of
        limited use, and the 'parents' output should rather be used.
        This option is valid only when specifying a connectivity matrix.

    linkage : {"average", "complete", "single"}, default="complete"
        Which linkage criteria to use. The linkage criterion determines which
        distance to use between sets of observation.
            - "average" uses the average of the distances of each observation of
              the two sets.
            - "complete" or maximum linkage uses the maximum distances between
              all observations of the two sets.
            - "single" uses the minimum of the distances between all
              observations of the two sets.

    affinity : str or callable, default='euclidean'
        Which metric to use. Can be 'euclidean', 'manhattan', or any
        distance known to paired distance (see metric.pairwise).

    return_distance : bool, default=False
        Whether or not to return the distances between the clusters.

    Returns
    -------
    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_connected_components : int
        The number of connected components in the graph.

    n_leaves : int
        The number of leaves in the tree.

    parents : ndarray of shape (n_nodes, ) or None
        The parent of each node. Only returned when a connectivity matrix
        is specified, elsewhere 'None' is returned.

    distances : ndarray of shape (n_nodes-1,)
        Returned when `return_distance` is set to `True`.

        distances[i] refers to the distance between children[i][0] and
        children[i][1] when they are merged.

    See Also
    --------
    ward_tree : Hierarchical clustering with ward linkage.
    """
    X = np.asarray(X)
    if X.ndim == 1:
        X = np.reshape(X, (-1, 1))
    n_samples, n_features = X.shape

    linkage_choices = {
        "complete": _hierarchical.max_merge,
        "average": _hierarchical.average_merge,
        "single": None,
    }  # Single linkage is handled differently
    try:
        join_func = linkage_choices[linkage]
    except KeyError as e:
        raise ValueError(
            "Unknown linkage option, linkage should be one of %s, but %s was given"
            % (linkage_choices.keys(), linkage)
        ) from e

    if affinity == "cosine" and np.any(~np.any(X, axis=1)):
        raise ValueError("Cosine affinity cannot be used when X contains zero vectors")

    if connectivity is None:
        from scipy.cluster import hierarchy  # imports PIL

        if n_clusters is not None:
            warnings.warn(
                "Partial build of the tree is implemented "
                "only for structured clustering (i.e. with "
                "explicit connectivity). The algorithm "
                "will build the full tree and only "
                "retain the lower branches required "
                "for the specified number of clusters",
                stacklevel=2,
            )

        if affinity == "precomputed":
            # for the linkage function of hierarchy to work on precomputed
            # data, provide as first argument an ndarray of the shape returned
            # by sklearn.metrics.pairwise_distances.
            if X.shape[0] != X.shape[1]:
                raise ValueError(
                    "Distance matrix should be square, Got matrix of shape {X.shape}"
                )
            i, j = np.triu_indices(X.shape[0], k=1)
            X = X[i, j]
        elif affinity == "l2":
            # Translate to something understood by scipy
            affinity = "euclidean"
        elif affinity in ("l1", "manhattan"):
            affinity = "cityblock"
        elif callable(affinity):
            X = affinity(X)
            i, j = np.triu_indices(X.shape[0], k=1)
            X = X[i, j]
        if (
            linkage == "single"
            and affinity != "precomputed"
            and not callable(affinity)
            and affinity in METRIC_MAPPING
        ):

            # We need the fast cythonized metric from neighbors
            dist_metric = DistanceMetric.get_metric(affinity)

            # The Cython routines used require contiguous arrays
            X = np.ascontiguousarray(X, dtype=np.double)

            mst = _hierarchical.mst_linkage_core(X, dist_metric)
            # Sort edges of the min_spanning_tree by weight
            mst = mst[np.argsort(mst.T[2], kind="mergesort"), :]

            # Convert edge list into standard hierarchical clustering format
            out = _hierarchical.single_linkage_label(mst)
        else:
            out = hierarchy.linkage(X, method=linkage, metric=affinity)
        children_ = out[:, :2].astype(int, copy=False)

        if return_distance:
            distances = out[:, 2]
            return children_, 1, n_samples, None, distances
        return children_, 1, n_samples, None

    connectivity, n_connected_components = _fix_connectivity(
        X, connectivity, affinity=affinity
    )
    connectivity = connectivity.tocoo()
    # Put the diagonal to zero
    diag_mask = connectivity.row != connectivity.col
    connectivity.row = connectivity.row[diag_mask]
    connectivity.col = connectivity.col[diag_mask]
    connectivity.data = connectivity.data[diag_mask]
    del diag_mask

    if affinity == "precomputed":
        distances = X[connectivity.row, connectivity.col].astype(
            "float64", **_astype_copy_false(X)
        )
    else:
        # FIXME We compute all the distances, while we could have only computed
        # the "interesting" distances
        distances = paired_distances(
            X[connectivity.row], X[connectivity.col], metric=affinity
        )
    connectivity.data = distances

    if n_clusters is None:
        n_nodes = 2 * n_samples - 1
    else:
        assert n_clusters <= n_samples
        n_nodes = 2 * n_samples - n_clusters

    if linkage == "single":
        return _single_linkage_tree(
            connectivity,
            n_samples,
            n_nodes,
            n_clusters,
            n_connected_components,
            return_distance,
        )

    if return_distance:
        distances = np.empty(n_nodes - n_samples)
    # create inertia heap and connection matrix
    A = np.empty(n_nodes, dtype=object)
    inertia = list()

    # LIL seems to the best format to access the rows quickly,
    # without the numpy overhead of slicing CSR indices and data.
    connectivity = connectivity.tolil()
    # We are storing the graph in a list of IntFloatDict
    for ind, (data, row) in enumerate(zip(connectivity.data, connectivity.rows)):
        A[ind] = IntFloatDict(
            np.asarray(row, dtype=np.intp), np.asarray(data, dtype=np.float64)
        )
        # We keep only the upper triangular for the heap
        # Generator expressions are faster than arrays on the following
        inertia.extend(
            _hierarchical.WeightedEdge(d, ind, r) for r, d in zip(row, data) if r < ind
        )
    del connectivity

    heapify(inertia)

    # prepare the main fields
    parent = np.arange(n_nodes, dtype=np.intp)
    used_node = np.ones(n_nodes, dtype=np.intp)
    children = []

    # recursive merge loop
    for k in range(n_samples, n_nodes):
        # identify the merge
        while True:
            edge = heappop(inertia)
            if used_node[edge.a] and used_node[edge.b]:
                break
        i = edge.a
        j = edge.b

        if return_distance:
            # store distances
            distances[k - n_samples] = edge.weight

        parent[i] = parent[j] = k
        children.append((i, j))
        # Keep track of the number of elements per cluster
        n_i = used_node[i]
        n_j = used_node[j]
        used_node[k] = n_i + n_j
        used_node[i] = used_node[j] = False

        # update the structure matrix A and the inertia matrix
        # a clever 'min', or 'max' operation between A[i] and A[j]
        coord_col = join_func(A[i], A[j], used_node, n_i, n_j)
        for col, d in coord_col:
            A[col].append(k, d)
            # Here we use the information from coord_col (containing the
            # distances) to update the heap
            heappush(inertia, _hierarchical.WeightedEdge(d, k, col))
        A[k] = coord_col
        # Clear A[i] and A[j] to save memory
        A[i] = A[j] = 0

    # Separate leaves in children (empty lists up to now)
    n_leaves = n_samples

    # # return numpy array for efficient caching
    children = np.array(children)[:, ::-1]

    if return_distance:
        return children, n_connected_components, n_leaves, parent, distances
    return children, n_connected_components, n_leaves, parent


# Matching names to tree-building strategies
def _complete_linkage(*args, **kwargs):
    kwargs["linkage"] = "complete"
    return linkage_tree(*args, **kwargs)


def _average_linkage(*args, **kwargs):
    kwargs["linkage"] = "average"
    return linkage_tree(*args, **kwargs)


def _single_linkage(*args, **kwargs):
    kwargs["linkage"] = "single"
    return linkage_tree(*args, **kwargs)


_TREE_BUILDERS = dict(
    ward=ward_tree,
    complete=_complete_linkage,
    average=_average_linkage,
    single=_single_linkage,
)

###############################################################################
# Functions for cutting hierarchical clustering tree


def _hc_cut(n_clusters, children, n_leaves):
    """Function cutting the ward tree for a given number of clusters.

    Parameters
    ----------
    n_clusters : int or ndarray
        The number of clusters to form.

    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_leaves : int
        Number of leaves of the tree.

    Returns
    -------
    labels : array [n_samples]
        Cluster labels for each point.
    """
    if n_clusters > n_leaves:
        raise ValueError(
            "Cannot extract more clusters than samples: "
            "%s clusters where given for a tree with %s leaves."
            % (n_clusters, n_leaves)
        )
    # In this function, we store nodes as a heap to avoid recomputing
    # the max of the nodes: the first element is always the smallest
    # We use negated indices as heaps work on smallest elements, and we
    # are interested in largest elements
    # children[-1] is the root of the tree
    nodes = [-(max(children[-1]) + 1)]
    for _ in range(n_clusters - 1):
        # As we have a heap, nodes[0] is the smallest element
        these_children = children[-nodes[0] - n_leaves]
        # Insert the 2 children and remove the largest node
        heappush(nodes, -these_children[0])
        heappushpop(nodes, -these_children[1])
    label = np.zeros(n_leaves, dtype=np.intp)
    for i, node in enumerate(nodes):
        label[_hierarchical._hc_get_descendent(-node, children, n_leaves)] = i
    return label


###############################################################################


class AgglomerativeClustering(ClusterMixin, BaseEstimator):
    """
    Agglomerative Clustering.

    Recursively merges pair of clusters of sample data; uses linkage distance.

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

    Parameters
    ----------
    n_clusters : int or None, default=2
        The number of clusters to find. It must be ``None`` if
        ``distance_threshold`` is not ``None``.

    affinity : str or callable, default='euclidean'
        Metric used to compute the linkage. Can be "euclidean", "l1", "l2",
        "manhattan", "cosine", or "precomputed".
        If linkage is "ward", only "euclidean" is accepted.
        If "precomputed", a distance matrix (instead of a similarity matrix)
        is needed as input for the fit method.

    memory : str or object with the joblib.Memory interface, default=None
        Used to cache the output of the computation of the tree.
        By default, no caching is done. If a string is given, it is the
        path to the caching directory.

    connectivity : array-like or callable, default=None
        Connectivity matrix. Defines for each sample the neighboring
        samples following a given structure of the data.
        This can be a connectivity matrix itself or a callable that transforms
        the data into a connectivity matrix, such as derived from
        `kneighbors_graph`. Default is ``None``, i.e, the
        hierarchical clustering algorithm is unstructured.

    compute_full_tree : 'auto' or bool, default='auto'
        Stop early the construction of the tree at ``n_clusters``. This is
        useful to decrease computation time if the number of clusters is not
        small compared to the number of samples. This option is useful only
        when specifying a connectivity matrix. Note also that when varying the
        number of clusters and using caching, it may be advantageous to compute
        the full tree. It must be ``True`` if ``distance_threshold`` is not
        ``None``. By default `compute_full_tree` is "auto", which is equivalent
        to `True` when `distance_threshold` is not `None` or that `n_clusters`
        is inferior to the maximum between 100 or `0.02 * n_samples`.
        Otherwise, "auto" is equivalent to `False`.

    linkage : {'ward', 'complete', 'average', 'single'}, default='ward'
        Which linkage criterion to use. The linkage criterion determines which
        distance to use between sets of observation. The algorithm will merge
        the pairs of cluster that minimize this criterion.

        - 'ward' minimizes the variance of the clusters being merged.
        - 'average' uses the average of the distances of each observation of
          the two sets.
        - 'complete' or 'maximum' linkage uses the maximum distances between
          all observations of the two sets.
        - 'single' uses the minimum of the distances between all observations
          of the two sets.

        .. versionadded:: 0.20
            Added the 'single' option

    distance_threshold : float, default=None
        The linkage distance threshold above which, clusters will not be
        merged. If not ``None``, ``n_clusters`` must be ``None`` and
        ``compute_full_tree`` must be ``True``.

        .. versionadded:: 0.21

    compute_distances : bool, default=False
        Computes distances between clusters even if `distance_threshold` is not
        used. This can be used to make dendrogram visualization, but introduces
        a computational and memory overhead.

        .. versionadded:: 0.24

    Attributes
    ----------
    n_clusters_ : int
        The number of clusters found by the algorithm. If
        ``distance_threshold=None``, it will be equal to the given
        ``n_clusters``.

    labels_ : ndarray of shape (n_samples)
        Cluster labels for each point.

    n_leaves_ : int
        Number of leaves in the hierarchical tree.

    n_connected_components_ : int
        The estimated number of connected components in the graph.

        .. versionadded:: 0.21
            ``n_connected_components_`` was added to replace ``n_components_``.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    children_ : array-like of shape (n_samples-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    distances_ : array-like of shape (n_nodes-1,)
        Distances between nodes in the corresponding place in `children_`.
        Only computed if `distance_threshold` is used or `compute_distances`
        is set to `True`.

    See Also
    --------
    FeatureAgglomeration : Agglomerative clustering but for features instead of
        samples.
    ward_tree : Hierarchical clustering with ward linkage.

    Examples
    --------
    >>> from sklearn.cluster import AgglomerativeClustering
    >>> import numpy as np
    >>> X = np.array([[1, 2], [1, 4], [1, 0],
    ...               [4, 2], [4, 4], [4, 0]])
    >>> clustering = AgglomerativeClustering().fit(X)
    >>> clustering
    AgglomerativeClustering()
    >>> clustering.labels_
    array([1, 1, 1, 0, 0, 0])
    """

    def __init__(
        self,
        n_clusters=2,
        *,
        affinity="euclidean",
        memory=None,
        connectivity=None,
        compute_full_tree="auto",
        linkage="ward",
        distance_threshold=None,
        compute_distances=False,
    ):
        self.n_clusters = n_clusters
        self.distance_threshold = distance_threshold
        self.memory = memory
        self.connectivity = connectivity
        self.compute_full_tree = compute_full_tree
        self.linkage = linkage
        self.affinity = affinity
        self.compute_distances = compute_distances

    def fit(self, X, y=None):
        """Fit the hierarchical clustering from features, or distance matrix.

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features) or \
                (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        self : object
            Returns the fitted instance.
        """
        X = self._validate_data(X, ensure_min_samples=2)
        return self._fit(X)

    def _fit(self, X):
        """Fit without validation

        Parameters
        ----------
        X : ndarray of shape (n_samples, n_features) or (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        Returns
        -------
        self : object
            Returns the fitted instance.
        """
        memory = check_memory(self.memory)

        if self.n_clusters is not None and self.n_clusters <= 0:
            raise ValueError(
                "n_clusters should be an integer greater than 0. %s was provided."
                % str(self.n_clusters)
            )

        if not ((self.n_clusters is None) ^ (self.distance_threshold is None)):
            raise ValueError(
                "Exactly one of n_clusters and "
                "distance_threshold has to be set, and the other "
                "needs to be None."
            )

        if self.distance_threshold is not None and not self.compute_full_tree:
            raise ValueError(
                "compute_full_tree must be True if distance_threshold is set."
            )

        if self.linkage == "ward" and self.affinity != "euclidean":
            raise ValueError(
                "%s was provided as affinity. Ward can only "
                "work with euclidean distances." % (self.affinity,)
            )

        if self.linkage not in _TREE_BUILDERS:
            raise ValueError(
                "Unknown linkage type %s. Valid options are %s"
                % (self.linkage, _TREE_BUILDERS.keys())
            )
        tree_builder = _TREE_BUILDERS[self.linkage]

        connectivity = self.connectivity
        if self.connectivity is not None:
            if callable(self.connectivity):
                connectivity = self.connectivity(X)
            connectivity = check_array(
                connectivity, accept_sparse=["csr", "coo", "lil"]
            )

        n_samples = len(X)
        compute_full_tree = self.compute_full_tree
        if self.connectivity is None:
            compute_full_tree = True
        if compute_full_tree == "auto":
            if self.distance_threshold is not None:
                compute_full_tree = True
            else:
                # Early stopping is likely to give a speed up only for
                # a large number of clusters. The actual threshold
                # implemented here is heuristic
                compute_full_tree = self.n_clusters < max(100, 0.02 * n_samples)
        n_clusters = self.n_clusters
        if compute_full_tree:
            n_clusters = None

        # Construct the tree
        kwargs = {}
        if self.linkage != "ward":
            kwargs["linkage"] = self.linkage
            kwargs["affinity"] = self.affinity

        distance_threshold = self.distance_threshold

        return_distance = (distance_threshold is not None) or self.compute_distances

        out = memory.cache(tree_builder)(
            X,
            connectivity=connectivity,
            n_clusters=n_clusters,
            return_distance=return_distance,
            **kwargs,
        )
        (self.children_, self.n_connected_components_, self.n_leaves_, parents) = out[
            :4
        ]

        if return_distance:
            self.distances_ = out[-1]

        if self.distance_threshold is not None:  # distance_threshold is used
            self.n_clusters_ = (
                np.count_nonzero(self.distances_ >= distance_threshold) + 1
            )
        else:  # n_clusters is used
            self.n_clusters_ = self.n_clusters

        # Cut the tree
        if compute_full_tree:
            self.labels_ = _hc_cut(self.n_clusters_, self.children_, self.n_leaves_)
        else:
            labels = _hierarchical.hc_get_heads(parents, copy=False)
            # copy to avoid holding a reference on the original array
            labels = np.copy(labels[:n_samples])
            # Reassign cluster numbers
            self.labels_ = np.searchsorted(np.unique(labels), labels)
        return self

    def fit_predict(self, X, y=None):
        """Fit and return the result of each sample's clustering assignment.

        In addition to fitting, this method also return the result of the
        clustering assignment for each sample in the training set.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features) or \
                (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        labels : ndarray of shape (n_samples,)
            Cluster labels.
        """
        return super().fit_predict(X, y)


class FeatureAgglomeration(AgglomerativeClustering, AgglomerationTransform):
    """Agglomerate features.

    Recursively merges pair of clusters of features.

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

    Parameters
    ----------
    n_clusters : int, default=2
        The number of clusters to find. It must be ``None`` if
        ``distance_threshold`` is not ``None``.

    affinity : str or callable, default='euclidean'
        Metric used to compute the linkage. Can be "euclidean", "l1", "l2",
        "manhattan", "cosine", or 'precomputed'.
        If linkage is "ward", only "euclidean" is accepted.

    memory : str or object with the joblib.Memory interface, default=None
        Used to cache the output of the computation of the tree.
        By default, no caching is done. If a string is given, it is the
        path to the caching directory.

    connectivity : array-like or callable, default=None
        Connectivity matrix. Defines for each feature the neighboring
        features following a given structure of the data.
        This can be a connectivity matrix itself or a callable that transforms
        the data into a connectivity matrix, such as derived from
        `kneighbors_graph`. Default is `None`, i.e, the
        hierarchical clustering algorithm is unstructured.

    compute_full_tree : 'auto' or bool, default='auto'
        Stop early the construction of the tree at `n_clusters`. This is useful
        to decrease computation time if the number of clusters is not small
        compared to the number of features. This option is useful only when
        specifying a connectivity matrix. Note also that when varying the
        number of clusters and using caching, it may be advantageous to compute
        the full tree. It must be ``True`` if ``distance_threshold`` is not
        ``None``. By default `compute_full_tree` is "auto", which is equivalent
        to `True` when `distance_threshold` is not `None` or that `n_clusters`
        is inferior to the maximum between 100 or `0.02 * n_samples`.
        Otherwise, "auto" is equivalent to `False`.

    linkage : {"ward", "complete", "average", "single"}, default="ward"
        Which linkage criterion to use. The linkage criterion determines which
        distance to use between sets of features. The algorithm will merge
        the pairs of cluster that minimize this criterion.

        - "ward" minimizes the variance of the clusters being merged.
        - "complete" or maximum linkage uses the maximum distances between
          all features of the two sets.
        - "average" uses the average of the distances of each feature of
          the two sets.
        - "single" uses the minimum of the distances between all features
          of the two sets.

    pooling_func : callable, default=np.mean
        This combines the values of agglomerated features into a single
        value, and should accept an array of shape [M, N] and the keyword
        argument `axis=1`, and reduce it to an array of size [M].

    distance_threshold : float, default=None
        The linkage distance threshold above which, clusters will not be
        merged. If not ``None``, ``n_clusters`` must be ``None`` and
        ``compute_full_tree`` must be ``True``.

        .. versionadded:: 0.21

    compute_distances : bool, default=False
        Computes distances between clusters even if `distance_threshold` is not
        used. This can be used to make dendrogram visualization, but introduces
        a computational and memory overhead.

        .. versionadded:: 0.24

    Attributes
    ----------
    n_clusters_ : int
        The number of clusters found by the algorithm. If
        ``distance_threshold=None``, it will be equal to the given
        ``n_clusters``.

    labels_ : array-like of (n_features,)
        Cluster labels for each feature.

    n_leaves_ : int
        Number of leaves in the hierarchical tree.

    n_connected_components_ : int
        The estimated number of connected components in the graph.

        .. versionadded:: 0.21
            ``n_connected_components_`` was added to replace ``n_components_``.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    children_ : array-like of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_features`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_features` is a non-leaf
        node and has children `children_[i - n_features]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_features + i`.

    distances_ : array-like of shape (n_nodes-1,)
        Distances between nodes in the corresponding place in `children_`.
        Only computed if `distance_threshold` is used or `compute_distances`
        is set to `True`.

    See Also
    --------
    AgglomerativeClustering : Agglomerative clustering samples instead of
        features.
    ward_tree : Hierarchical clustering with ward linkage.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn import datasets, cluster
    >>> digits = datasets.load_digits()
    >>> images = digits.images
    >>> X = np.reshape(images, (len(images), -1))
    >>> agglo = cluster.FeatureAgglomeration(n_clusters=32)
    >>> agglo.fit(X)
    FeatureAgglomeration(n_clusters=32)
    >>> X_reduced = agglo.transform(X)
    >>> X_reduced.shape
    (1797, 32)
    """

    def __init__(
        self,
        n_clusters=2,
        *,
        affinity="euclidean",
        memory=None,
        connectivity=None,
        compute_full_tree="auto",
        linkage="ward",
        pooling_func=np.mean,
        distance_threshold=None,
        compute_distances=False,
    ):
        super().__init__(
            n_clusters=n_clusters,
            memory=memory,
            connectivity=connectivity,
            compute_full_tree=compute_full_tree,
            linkage=linkage,
            affinity=affinity,
            distance_threshold=distance_threshold,
            compute_distances=compute_distances,
        )
        self.pooling_func = pooling_func

    def fit(self, X, y=None):
        """Fit the hierarchical clustering on the data.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            The data.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        self : object
            Returns the transformer.
        """
        X = self._validate_data(X, ensure_min_features=2)
        super()._fit(X.T)
        return self

    @property
    def fit_predict(self):
        """Fit and return the result of each sample's clustering assignment."""
        raise AttributeError

<filename>sklearn/cluster/_agglomerative.py
"""Hierarchical Agglomerative Clustering

These routines perform some hierarchical agglomerative clustering of some
input data.

Authors : <NAME>, <NAME>, <NAME>,
          <NAME>
License: BSD 3 clause
"""
import warnings
from heapq import heapify, heappop, heappush, heappushpop

import numpy as np
from scipy import sparse
from scipy.sparse.csgraph import connected_components

from ..base import BaseEstimator, ClusterMixin
from ..metrics.pairwise import paired_distances
from ..metrics import DistanceMetric
from ..metrics._dist_metrics import METRIC_MAPPING
from ..utils import check_array
from ..utils._fast_dict import IntFloatDict
from ..utils.fixes import _astype_copy_false
from ..utils.graph import _fix_connected_components
from ..utils.validation import check_memory

# mypy error: Module 'sklearn.cluster' has no attribute '_hierarchical_fast'
from . import _hierarchical_fast as _hierarchical  # type: ignore
from ._feature_agglomeration import AgglomerationTransform

###############################################################################
# For non fully-connected graphs


def _fix_connectivity(X, connectivity, affinity):
    """
    Fixes the connectivity matrix.

    The different steps are:

    - copies it
    - makes it symmetric
    - converts it to LIL if necessary
    - completes it if necessary.

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is `None`, i.e, the Ward algorithm is unstructured.

    affinity : {"euclidean", "precomputed"}, default="euclidean"
        Which affinity to use. At the moment `precomputed` and
        ``euclidean`` are supported. `euclidean` uses the
        negative squared Euclidean distance between points.

    Returns
    -------
    connectivity : sparse matrix
        The fixed connectivity matrix.

    n_connected_components : int
        The number of connected components in the graph.
    """
    n_samples = X.shape[0]
    if connectivity.shape[0] != n_samples or connectivity.shape[1] != n_samples:
        raise ValueError(
            "Wrong shape for connectivity matrix: %s when X is %s"
            % (connectivity.shape, X.shape)
        )

    # Make the connectivity matrix symmetric:
    connectivity = connectivity + connectivity.T

    # Convert connectivity matrix to LIL
    if not sparse.isspmatrix_lil(connectivity):
        if not sparse.isspmatrix(connectivity):
            connectivity = sparse.lil_matrix(connectivity)
        else:
            connectivity = connectivity.tolil()

    # Compute the number of nodes
    n_connected_components, labels = connected_components(connectivity)

    if n_connected_components > 1:
        warnings.warn(
            "the number of connected components of the "
            "connectivity matrix is %d > 1. Completing it to avoid "
            "stopping the tree early." % n_connected_components,
            stacklevel=2,
        )
        # XXX: Can we do without completing the matrix?
        connectivity = _fix_connected_components(
            X=X,
            graph=connectivity,
            n_connected_components=n_connected_components,
            component_labels=labels,
            metric=affinity,
            mode="connectivity",
        )

    return connectivity, n_connected_components


def _single_linkage_tree(
    connectivity,
    n_samples,
    n_nodes,
    n_clusters,
    n_connected_components,
    return_distance,
):
    """
    Perform single linkage clustering on sparse data via the minimum
    spanning tree from scipy.sparse.csgraph, then using union-find to label.
    The parent array is then generated by walking through the tree.
    """
    from scipy.sparse.csgraph import minimum_spanning_tree

    # explicitly cast connectivity to ensure safety
    connectivity = connectivity.astype("float64", **_astype_copy_false(connectivity))

    # Ensure zero distances aren't ignored by setting them to "epsilon"
    epsilon_value = np.finfo(dtype=connectivity.data.dtype).eps
    connectivity.data[connectivity.data == 0] = epsilon_value

    # Use scipy.sparse.csgraph to generate a minimum spanning tree
    mst = minimum_spanning_tree(connectivity.tocsr())

    # Convert the graph to scipy.cluster.hierarchy array format
    mst = mst.tocoo()

    # Undo the epsilon values
    mst.data[mst.data == epsilon_value] = 0

    mst_array = np.vstack([mst.row, mst.col, mst.data]).T

    # Sort edges of the min_spanning_tree by weight
    mst_array = mst_array[np.argsort(mst_array.T[2], kind="mergesort"), :]

    # Convert edge list into standard hierarchical clustering format
    single_linkage_tree = _hierarchical._single_linkage_label(mst_array)
    children_ = single_linkage_tree[:, :2].astype(int)

    # Compute parents
    parent = np.arange(n_nodes, dtype=np.intp)
    for i, (left, right) in enumerate(children_, n_samples):
        if n_clusters is not None and i >= n_nodes:
            break
        if left < n_nodes:
            parent[left] = i
        if right < n_nodes:
            parent[right] = i

    if return_distance:
        distances = single_linkage_tree[:, 2]
        return children_, n_connected_components, n_samples, parent, distances
    return children_, n_connected_components, n_samples, parent


###############################################################################
# Hierarchical tree building functions


def ward_tree(X, *, connectivity=None, n_clusters=None, return_distance=False):
    """Ward clustering based on a Feature matrix.

    Recursively merges the pair of clusters that minimally increases
    within-cluster variance.

    The inertia matrix uses a Heapq-based representation.

    This is the structured version, that takes into account some topological
    structure between samples.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is None, i.e, the Ward algorithm is unstructured.

    n_clusters : int, default=None
        `n_clusters` should be less than `n_samples`.  Stop early the
        construction of the tree at `n_clusters.` This is useful to decrease
        computation time if the number of clusters is not small compared to the
        number of samples. In this case, the complete tree is not computed, thus
        the 'children' output is of limited use, and the 'parents' output should
        rather be used. This option is valid only when specifying a connectivity
        matrix.

    return_distance : bool, default=False
        If `True`, return the distance between the clusters.

    Returns
    -------
    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_connected_components : int
        The number of connected components in the graph.

    n_leaves : int
        The number of leaves in the tree.

    parents : ndarray of shape (n_nodes,) or None
        The parent of each node. Only returned when a connectivity matrix
        is specified, elsewhere 'None' is returned.

    distances : ndarray of shape (n_nodes-1,)
        Only returned if `return_distance` is set to `True` (for compatibility).
        The distances between the centers of the nodes. `distances[i]`
        corresponds to a weighted Euclidean distance between
        the nodes `children[i, 1]` and `children[i, 2]`. If the nodes refer to
        leaves of the tree, then `distances[i]` is their unweighted Euclidean
        distance. Distances are updated in the following way
        (from scipy.hierarchy.linkage):

        The new entry :math:`d(u,v)` is computed as follows,

        .. math::

           d(u,v) = \\sqrt{\\frac{|v|+|s|}
                               {T}d(v,s)^2
                        + \\frac{|v|+|t|}
                               {T}d(v,t)^2
                        - \\frac{|v|}
                               {T}d(s,t)^2}

        where :math:`u` is the newly joined cluster consisting of
        clusters :math:`s` and :math:`t`, :math:`v` is an unused
        cluster in the forest, :math:`T=|v|+|s|+|t|`, and
        :math:`|*|` is the cardinality of its argument. This is also
        known as the incremental algorithm.
    """
    X = np.asarray(X)
    if X.ndim == 1:
        X = np.reshape(X, (-1, 1))
    n_samples, n_features = X.shape

    if connectivity is None:
        from scipy.cluster import hierarchy  # imports PIL

        if n_clusters is not None:
            warnings.warn(
                "Partial build of the tree is implemented "
                "only for structured clustering (i.e. with "
                "explicit connectivity). The algorithm "
                "will build the full tree and only "
                "retain the lower branches required "
                "for the specified number of clusters",
                stacklevel=2,
            )
        X = np.require(X, requirements="W")
        out = hierarchy.ward(X)
        children_ = out[:, :2].astype(np.intp)

        if return_distance:
            distances = out[:, 2]
            return children_, 1, n_samples, None, distances
        else:
            return children_, 1, n_samples, None

    connectivity, n_connected_components = _fix_connectivity(
        X, connectivity, affinity="euclidean"
    )
    if n_clusters is None:
        n_nodes = 2 * n_samples - 1
    else:
        if n_clusters > n_samples:
            raise ValueError(
                "Cannot provide more clusters than samples. "
                "%i n_clusters was asked, and there are %i "
                "samples." % (n_clusters, n_samples)
            )
        n_nodes = 2 * n_samples - n_clusters

    # create inertia matrix
    coord_row = []
    coord_col = []
    A = []
    for ind, row in enumerate(connectivity.rows):
        A.append(row)
        # We keep only the upper triangular for the moments
        # Generator expressions are faster than arrays on the following
        row = [i for i in row if i < ind]
        coord_row.extend(
            len(row)
            * [
                ind,
            ]
        )
        coord_col.extend(row)

    coord_row = np.array(coord_row, dtype=np.intp, order="C")
    coord_col = np.array(coord_col, dtype=np.intp, order="C")

    # build moments as a list
    moments_1 = np.zeros(n_nodes, order="C")
    moments_1[:n_samples] = 1
    moments_2 = np.zeros((n_nodes, n_features), order="C")
    moments_2[:n_samples] = X
    inertia = np.empty(len(coord_row), dtype=np.float64, order="C")
    _hierarchical.compute_ward_dist(moments_1, moments_2, coord_row, coord_col, inertia)
    inertia = list(zip(inertia, coord_row, coord_col))
    heapify(inertia)

    # prepare the main fields
    parent = np.arange(n_nodes, dtype=np.intp)
    used_node = np.ones(n_nodes, dtype=bool)
    children = []
    if return_distance:
        distances = np.empty(n_nodes - n_samples)

    not_visited = np.empty(n_nodes, dtype=np.int8, order="C")

    # recursive merge loop
    for k in range(n_samples, n_nodes):
        # identify the merge
        while True:
            inert, i, j = heappop(inertia)
            if used_node[i] and used_node[j]:
                break
        parent[i], parent[j] = k, k
        children.append((i, j))
        used_node[i] = used_node[j] = False
        if return_distance:  # store inertia value
            distances[k - n_samples] = inert

        # update the moments
        moments_1[k] = moments_1[i] + moments_1[j]
        moments_2[k] = moments_2[i] + moments_2[j]

        # update the structure matrix A and the inertia matrix
        coord_col = []
        not_visited.fill(1)
        not_visited[k] = 0
        _hierarchical._get_parents(A[i], coord_col, parent, not_visited)
        _hierarchical._get_parents(A[j], coord_col, parent, not_visited)
        # List comprehension is faster than a for loop
        [A[col].append(k) for col in coord_col]
        A.append(coord_col)
        coord_col = np.array(coord_col, dtype=np.intp, order="C")
        coord_row = np.empty(coord_col.shape, dtype=np.intp, order="C")
        coord_row.fill(k)
        n_additions = len(coord_row)
        ini = np.empty(n_additions, dtype=np.float64, order="C")

        _hierarchical.compute_ward_dist(moments_1, moments_2, coord_row, coord_col, ini)

        # List comprehension is faster than a for loop
        [heappush(inertia, (ini[idx], k, coord_col[idx])) for idx in range(n_additions)]

    # Separate leaves in children (empty lists up to now)
    n_leaves = n_samples
    # sort children to get consistent output with unstructured version
    children = [c[::-1] for c in children]
    children = np.array(children)  # return numpy array for efficient caching

    if return_distance:
        # 2 is scaling factor to compare w/ unstructured version
        distances = np.sqrt(2.0 * distances)
        return children, n_connected_components, n_leaves, parent, distances
    else:
        return children, n_connected_components, n_leaves, parent


# single average and complete linkage
def linkage_tree(
    X,
    connectivity=None,
    n_clusters=None,
    linkage="complete",
    affinity="euclidean",
    return_distance=False,
):
    """Linkage agglomerative clustering based on a Feature matrix.

    The inertia matrix uses a Heapq-based representation.

    This is the structured version, that takes into account some topological
    structure between samples.

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

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Feature matrix representing `n_samples` samples to be clustered.

    connectivity : sparse matrix, default=None
        Connectivity matrix. Defines for each sample the neighboring samples
        following a given structure of the data. The matrix is assumed to
        be symmetric and only the upper triangular half is used.
        Default is `None`, i.e, the Ward algorithm is unstructured.

    n_clusters : int, default=None
        Stop early the construction of the tree at `n_clusters`. This is
        useful to decrease computation time if the number of clusters is
        not small compared to the number of samples. In this case, the
        complete tree is not computed, thus the 'children' output is of
        limited use, and the 'parents' output should rather be used.
        This option is valid only when specifying a connectivity matrix.

    linkage : {"average", "complete", "single"}, default="complete"
        Which linkage criteria to use. The linkage criterion determines which
        distance to use between sets of observation.
            - "average" uses the average of the distances of each observation of
              the two sets.
            - "complete" or maximum linkage uses the maximum distances between
              all observations of the two sets.
            - "single" uses the minimum of the distances between all
              observations of the two sets.

    affinity : str or callable, default='euclidean'
        Which metric to use. Can be 'euclidean', 'manhattan', or any
        distance known to paired distance (see metric.pairwise).

    return_distance : bool, default=False
        Whether or not to return the distances between the clusters.

    Returns
    -------
    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_connected_components : int
        The number of connected components in the graph.

    n_leaves : int
        The number of leaves in the tree.

    parents : ndarray of shape (n_nodes, ) or None
        The parent of each node. Only returned when a connectivity matrix
        is specified, elsewhere 'None' is returned.

    distances : ndarray of shape (n_nodes-1,)
        Returned when `return_distance` is set to `True`.

        distances[i] refers to the distance between children[i][0] and
        children[i][1] when they are merged.

    See Also
    --------
    ward_tree : Hierarchical clustering with ward linkage.
    """
    X = np.asarray(X)
    if X.ndim == 1:
        X = np.reshape(X, (-1, 1))
    n_samples, n_features = X.shape

    linkage_choices = {
        "complete": _hierarchical.max_merge,
        "average": _hierarchical.average_merge,
        "single": None,
    }  # Single linkage is handled differently
    try:
        join_func = linkage_choices[linkage]
    except KeyError as e:
        raise ValueError(
            "Unknown linkage option, linkage should be one of %s, but %s was given"
            % (linkage_choices.keys(), linkage)
        ) from e

    if affinity == "cosine" and np.any(~np.any(X, axis=1)):
        raise ValueError("Cosine affinity cannot be used when X contains zero vectors")

    if connectivity is None:
        from scipy.cluster import hierarchy  # imports PIL

        if n_clusters is not None:
            warnings.warn(
                "Partial build of the tree is implemented "
                "only for structured clustering (i.e. with "
                "explicit connectivity). The algorithm "
                "will build the full tree and only "
                "retain the lower branches required "
                "for the specified number of clusters",
                stacklevel=2,
            )

        if affinity == "precomputed":
            # for the linkage function of hierarchy to work on precomputed
            # data, provide as first argument an ndarray of the shape returned
            # by sklearn.metrics.pairwise_distances.
            if X.shape[0] != X.shape[1]:
                raise ValueError(
                    "Distance matrix should be square, Got matrix of shape {X.shape}"
                )
            i, j = np.triu_indices(X.shape[0], k=1)
            X = X[i, j]
        elif affinity == "l2":
            # Translate to something understood by scipy
            affinity = "euclidean"
        elif affinity in ("l1", "manhattan"):
            affinity = "cityblock"
        elif callable(affinity):
            X = affinity(X)
            i, j = np.triu_indices(X.shape[0], k=1)
            X = X[i, j]
        if (
            linkage == "single"
            and affinity != "precomputed"
            and not callable(affinity)
            and affinity in METRIC_MAPPING
        ):

            # We need the fast cythonized metric from neighbors
            dist_metric = DistanceMetric.get_metric(affinity)

            # The Cython routines used require contiguous arrays
            X = np.ascontiguousarray(X, dtype=np.double)

            mst = _hierarchical.mst_linkage_core(X, dist_metric)
            # Sort edges of the min_spanning_tree by weight
            mst = mst[np.argsort(mst.T[2], kind="mergesort"), :]

            # Convert edge list into standard hierarchical clustering format
            out = _hierarchical.single_linkage_label(mst)
        else:
            out = hierarchy.linkage(X, method=linkage, metric=affinity)
        children_ = out[:, :2].astype(int, copy=False)

        if return_distance:
            distances = out[:, 2]
            return children_, 1, n_samples, None, distances
        return children_, 1, n_samples, None

    connectivity, n_connected_components = _fix_connectivity(
        X, connectivity, affinity=affinity
    )
    connectivity = connectivity.tocoo()
    # Put the diagonal to zero
    diag_mask = connectivity.row != connectivity.col
    connectivity.row = connectivity.row[diag_mask]
    connectivity.col = connectivity.col[diag_mask]
    connectivity.data = connectivity.data[diag_mask]
    del diag_mask

    if affinity == "precomputed":
        distances = X[connectivity.row, connectivity.col].astype(
            "float64", **_astype_copy_false(X)
        )
    else:
        # FIXME We compute all the distances, while we could have only computed
        # the "interesting" distances
        distances = paired_distances(
            X[connectivity.row], X[connectivity.col], metric=affinity
        )
    connectivity.data = distances

    if n_clusters is None:
        n_nodes = 2 * n_samples - 1
    else:
        assert n_clusters <= n_samples
        n_nodes = 2 * n_samples - n_clusters

    if linkage == "single":
        return _single_linkage_tree(
            connectivity,
            n_samples,
            n_nodes,
            n_clusters,
            n_connected_components,
            return_distance,
        )

    if return_distance:
        distances = np.empty(n_nodes - n_samples)
    # create inertia heap and connection matrix
    A = np.empty(n_nodes, dtype=object)
    inertia = list()

    # LIL seems to the best format to access the rows quickly,
    # without the numpy overhead of slicing CSR indices and data.
    connectivity = connectivity.tolil()
    # We are storing the graph in a list of IntFloatDict
    for ind, (data, row) in enumerate(zip(connectivity.data, connectivity.rows)):
        A[ind] = IntFloatDict(
            np.asarray(row, dtype=np.intp), np.asarray(data, dtype=np.float64)
        )
        # We keep only the upper triangular for the heap
        # Generator expressions are faster than arrays on the following
        inertia.extend(
            _hierarchical.WeightedEdge(d, ind, r) for r, d in zip(row, data) if r < ind
        )
    del connectivity

    heapify(inertia)

    # prepare the main fields
    parent = np.arange(n_nodes, dtype=np.intp)
    used_node = np.ones(n_nodes, dtype=np.intp)
    children = []

    # recursive merge loop
    for k in range(n_samples, n_nodes):
        # identify the merge
        while True:
            edge = heappop(inertia)
            if used_node[edge.a] and used_node[edge.b]:
                break
        i = edge.a
        j = edge.b

        if return_distance:
            # store distances
            distances[k - n_samples] = edge.weight

        parent[i] = parent[j] = k
        children.append((i, j))
        # Keep track of the number of elements per cluster
        n_i = used_node[i]
        n_j = used_node[j]
        used_node[k] = n_i + n_j
        used_node[i] = used_node[j] = False

        # update the structure matrix A and the inertia matrix
        # a clever 'min', or 'max' operation between A[i] and A[j]
        coord_col = join_func(A[i], A[j], used_node, n_i, n_j)
        for col, d in coord_col:
            A[col].append(k, d)
            # Here we use the information from coord_col (containing the
            # distances) to update the heap
            heappush(inertia, _hierarchical.WeightedEdge(d, k, col))
        A[k] = coord_col
        # Clear A[i] and A[j] to save memory
        A[i] = A[j] = 0

    # Separate leaves in children (empty lists up to now)
    n_leaves = n_samples

    # # return numpy array for efficient caching
    children = np.array(children)[:, ::-1]

    if return_distance:
        return children, n_connected_components, n_leaves, parent, distances
    return children, n_connected_components, n_leaves, parent


# Matching names to tree-building strategies
def _complete_linkage(*args, **kwargs):
    kwargs["linkage"] = "complete"
    return linkage_tree(*args, **kwargs)


def _average_linkage(*args, **kwargs):
    kwargs["linkage"] = "average"
    return linkage_tree(*args, **kwargs)


def _single_linkage(*args, **kwargs):
    kwargs["linkage"] = "single"
    return linkage_tree(*args, **kwargs)


_TREE_BUILDERS = dict(
    ward=ward_tree,
    complete=_complete_linkage,
    average=_average_linkage,
    single=_single_linkage,
)

###############################################################################
# Functions for cutting hierarchical clustering tree


def _hc_cut(n_clusters, children, n_leaves):
    """Function cutting the ward tree for a given number of clusters.

    Parameters
    ----------
    n_clusters : int or ndarray
        The number of clusters to form.

    children : ndarray of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    n_leaves : int
        Number of leaves of the tree.

    Returns
    -------
    labels : array [n_samples]
        Cluster labels for each point.
    """
    if n_clusters > n_leaves:
        raise ValueError(
            "Cannot extract more clusters than samples: "
            "%s clusters where given for a tree with %s leaves."
            % (n_clusters, n_leaves)
        )
    # In this function, we store nodes as a heap to avoid recomputing
    # the max of the nodes: the first element is always the smallest
    # We use negated indices as heaps work on smallest elements, and we
    # are interested in largest elements
    # children[-1] is the root of the tree
    nodes = [-(max(children[-1]) + 1)]
    for _ in range(n_clusters - 1):
        # As we have a heap, nodes[0] is the smallest element
        these_children = children[-nodes[0] - n_leaves]
        # Insert the 2 children and remove the largest node
        heappush(nodes, -these_children[0])
        heappushpop(nodes, -these_children[1])
    label = np.zeros(n_leaves, dtype=np.intp)
    for i, node in enumerate(nodes):
        label[_hierarchical._hc_get_descendent(-node, children, n_leaves)] = i
    return label


###############################################################################


class AgglomerativeClustering(ClusterMixin, BaseEstimator):
    """
    Agglomerative Clustering.

    Recursively merges pair of clusters of sample data; uses linkage distance.

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

    Parameters
    ----------
    n_clusters : int or None, default=2
        The number of clusters to find. It must be ``None`` if
        ``distance_threshold`` is not ``None``.

    affinity : str or callable, default='euclidean'
        Metric used to compute the linkage. Can be "euclidean", "l1", "l2",
        "manhattan", "cosine", or "precomputed".
        If linkage is "ward", only "euclidean" is accepted.
        If "precomputed", a distance matrix (instead of a similarity matrix)
        is needed as input for the fit method.

    memory : str or object with the joblib.Memory interface, default=None
        Used to cache the output of the computation of the tree.
        By default, no caching is done. If a string is given, it is the
        path to the caching directory.

    connectivity : array-like or callable, default=None
        Connectivity matrix. Defines for each sample the neighboring
        samples following a given structure of the data.
        This can be a connectivity matrix itself or a callable that transforms
        the data into a connectivity matrix, such as derived from
        `kneighbors_graph`. Default is ``None``, i.e, the
        hierarchical clustering algorithm is unstructured.

    compute_full_tree : 'auto' or bool, default='auto'
        Stop early the construction of the tree at ``n_clusters``. This is
        useful to decrease computation time if the number of clusters is not
        small compared to the number of samples. This option is useful only
        when specifying a connectivity matrix. Note also that when varying the
        number of clusters and using caching, it may be advantageous to compute
        the full tree. It must be ``True`` if ``distance_threshold`` is not
        ``None``. By default `compute_full_tree` is "auto", which is equivalent
        to `True` when `distance_threshold` is not `None` or that `n_clusters`
        is inferior to the maximum between 100 or `0.02 * n_samples`.
        Otherwise, "auto" is equivalent to `False`.

    linkage : {'ward', 'complete', 'average', 'single'}, default='ward'
        Which linkage criterion to use. The linkage criterion determines which
        distance to use between sets of observation. The algorithm will merge
        the pairs of cluster that minimize this criterion.

        - 'ward' minimizes the variance of the clusters being merged.
        - 'average' uses the average of the distances of each observation of
          the two sets.
        - 'complete' or 'maximum' linkage uses the maximum distances between
          all observations of the two sets.
        - 'single' uses the minimum of the distances between all observations
          of the two sets.

        .. versionadded:: 0.20
            Added the 'single' option

    distance_threshold : float, default=None
        The linkage distance threshold above which, clusters will not be
        merged. If not ``None``, ``n_clusters`` must be ``None`` and
        ``compute_full_tree`` must be ``True``.

        .. versionadded:: 0.21

    compute_distances : bool, default=False
        Computes distances between clusters even if `distance_threshold` is not
        used. This can be used to make dendrogram visualization, but introduces
        a computational and memory overhead.

        .. versionadded:: 0.24

    Attributes
    ----------
    n_clusters_ : int
        The number of clusters found by the algorithm. If
        ``distance_threshold=None``, it will be equal to the given
        ``n_clusters``.

    labels_ : ndarray of shape (n_samples)
        Cluster labels for each point.

    n_leaves_ : int
        Number of leaves in the hierarchical tree.

    n_connected_components_ : int
        The estimated number of connected components in the graph.

        .. versionadded:: 0.21
            ``n_connected_components_`` was added to replace ``n_components_``.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    children_ : array-like of shape (n_samples-1, 2)
        The children of each non-leaf node. Values less than `n_samples`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_samples` is a non-leaf
        node and has children `children_[i - n_samples]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_samples + i`.

    distances_ : array-like of shape (n_nodes-1,)
        Distances between nodes in the corresponding place in `children_`.
        Only computed if `distance_threshold` is used or `compute_distances`
        is set to `True`.

    See Also
    --------
    FeatureAgglomeration : Agglomerative clustering but for features instead of
        samples.
    ward_tree : Hierarchical clustering with ward linkage.

    Examples
    --------
    >>> from sklearn.cluster import AgglomerativeClustering
    >>> import numpy as np
    >>> X = np.array([[1, 2], [1, 4], [1, 0],
    ...               [4, 2], [4, 4], [4, 0]])
    >>> clustering = AgglomerativeClustering().fit(X)
    >>> clustering
    AgglomerativeClustering()
    >>> clustering.labels_
    array([1, 1, 1, 0, 0, 0])
    """

    def __init__(
        self,
        n_clusters=2,
        *,
        affinity="euclidean",
        memory=None,
        connectivity=None,
        compute_full_tree="auto",
        linkage="ward",
        distance_threshold=None,
        compute_distances=False,
    ):
        self.n_clusters = n_clusters
        self.distance_threshold = distance_threshold
        self.memory = memory
        self.connectivity = connectivity
        self.compute_full_tree = compute_full_tree
        self.linkage = linkage
        self.affinity = affinity
        self.compute_distances = compute_distances

    def fit(self, X, y=None):
        """Fit the hierarchical clustering from features, or distance matrix.

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features) or \
                (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        self : object
            Returns the fitted instance.
        """
        X = self._validate_data(X, ensure_min_samples=2)
        return self._fit(X)

    def _fit(self, X):
        """Fit without validation

        Parameters
        ----------
        X : ndarray of shape (n_samples, n_features) or (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        Returns
        -------
        self : object
            Returns the fitted instance.
        """
        memory = check_memory(self.memory)

        if self.n_clusters is not None and self.n_clusters <= 0:
            raise ValueError(
                "n_clusters should be an integer greater than 0. %s was provided."
                % str(self.n_clusters)
            )

        if not ((self.n_clusters is None) ^ (self.distance_threshold is None)):
            raise ValueError(
                "Exactly one of n_clusters and "
                "distance_threshold has to be set, and the other "
                "needs to be None."
            )

        if self.distance_threshold is not None and not self.compute_full_tree:
            raise ValueError(
                "compute_full_tree must be True if distance_threshold is set."
            )

        if self.linkage == "ward" and self.affinity != "euclidean":
            raise ValueError(
                "%s was provided as affinity. Ward can only "
                "work with euclidean distances." % (self.affinity,)
            )

        if self.linkage not in _TREE_BUILDERS:
            raise ValueError(
                "Unknown linkage type %s. Valid options are %s"
                % (self.linkage, _TREE_BUILDERS.keys())
            )
        tree_builder = _TREE_BUILDERS[self.linkage]

        connectivity = self.connectivity
        if self.connectivity is not None:
            if callable(self.connectivity):
                connectivity = self.connectivity(X)
            connectivity = check_array(
                connectivity, accept_sparse=["csr", "coo", "lil"]
            )

        n_samples = len(X)
        compute_full_tree = self.compute_full_tree
        if self.connectivity is None:
            compute_full_tree = True
        if compute_full_tree == "auto":
            if self.distance_threshold is not None:
                compute_full_tree = True
            else:
                # Early stopping is likely to give a speed up only for
                # a large number of clusters. The actual threshold
                # implemented here is heuristic
                compute_full_tree = self.n_clusters < max(100, 0.02 * n_samples)
        n_clusters = self.n_clusters
        if compute_full_tree:
            n_clusters = None

        # Construct the tree
        kwargs = {}
        if self.linkage != "ward":
            kwargs["linkage"] = self.linkage
            kwargs["affinity"] = self.affinity

        distance_threshold = self.distance_threshold

        return_distance = (distance_threshold is not None) or self.compute_distances

        out = memory.cache(tree_builder)(
            X,
            connectivity=connectivity,
            n_clusters=n_clusters,
            return_distance=return_distance,
            **kwargs,
        )
        (self.children_, self.n_connected_components_, self.n_leaves_, parents) = out[
            :4
        ]

        if return_distance:
            self.distances_ = out[-1]

        if self.distance_threshold is not None:  # distance_threshold is used
            self.n_clusters_ = (
                np.count_nonzero(self.distances_ >= distance_threshold) + 1
            )
        else:  # n_clusters is used
            self.n_clusters_ = self.n_clusters

        # Cut the tree
        if compute_full_tree:
            self.labels_ = _hc_cut(self.n_clusters_, self.children_, self.n_leaves_)
        else:
            labels = _hierarchical.hc_get_heads(parents, copy=False)
            # copy to avoid holding a reference on the original array
            labels = np.copy(labels[:n_samples])
            # Reassign cluster numbers
            self.labels_ = np.searchsorted(np.unique(labels), labels)
        return self

    def fit_predict(self, X, y=None):
        """Fit and return the result of each sample's clustering assignment.

        In addition to fitting, this method also return the result of the
        clustering assignment for each sample in the training set.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features) or \
                (n_samples, n_samples)
            Training instances to cluster, or distances between instances if
            ``affinity='precomputed'``.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        labels : ndarray of shape (n_samples,)
            Cluster labels.
        """
        return super().fit_predict(X, y)


class FeatureAgglomeration(AgglomerativeClustering, AgglomerationTransform):
    """Agglomerate features.

    Recursively merges pair of clusters of features.

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

    Parameters
    ----------
    n_clusters : int, default=2
        The number of clusters to find. It must be ``None`` if
        ``distance_threshold`` is not ``None``.

    affinity : str or callable, default='euclidean'
        Metric used to compute the linkage. Can be "euclidean", "l1", "l2",
        "manhattan", "cosine", or 'precomputed'.
        If linkage is "ward", only "euclidean" is accepted.

    memory : str or object with the joblib.Memory interface, default=None
        Used to cache the output of the computation of the tree.
        By default, no caching is done. If a string is given, it is the
        path to the caching directory.

    connectivity : array-like or callable, default=None
        Connectivity matrix. Defines for each feature the neighboring
        features following a given structure of the data.
        This can be a connectivity matrix itself or a callable that transforms
        the data into a connectivity matrix, such as derived from
        `kneighbors_graph`. Default is `None`, i.e, the
        hierarchical clustering algorithm is unstructured.

    compute_full_tree : 'auto' or bool, default='auto'
        Stop early the construction of the tree at `n_clusters`. This is useful
        to decrease computation time if the number of clusters is not small
        compared to the number of features. This option is useful only when
        specifying a connectivity matrix. Note also that when varying the
        number of clusters and using caching, it may be advantageous to compute
        the full tree. It must be ``True`` if ``distance_threshold`` is not
        ``None``. By default `compute_full_tree` is "auto", which is equivalent
        to `True` when `distance_threshold` is not `None` or that `n_clusters`
        is inferior to the maximum between 100 or `0.02 * n_samples`.
        Otherwise, "auto" is equivalent to `False`.

    linkage : {"ward", "complete", "average", "single"}, default="ward"
        Which linkage criterion to use. The linkage criterion determines which
        distance to use between sets of features. The algorithm will merge
        the pairs of cluster that minimize this criterion.

        - "ward" minimizes the variance of the clusters being merged.
        - "complete" or maximum linkage uses the maximum distances between
          all features of the two sets.
        - "average" uses the average of the distances of each feature of
          the two sets.
        - "single" uses the minimum of the distances between all features
          of the two sets.

    pooling_func : callable, default=np.mean
        This combines the values of agglomerated features into a single
        value, and should accept an array of shape [M, N] and the keyword
        argument `axis=1`, and reduce it to an array of size [M].

    distance_threshold : float, default=None
        The linkage distance threshold above which, clusters will not be
        merged. If not ``None``, ``n_clusters`` must be ``None`` and
        ``compute_full_tree`` must be ``True``.

        .. versionadded:: 0.21

    compute_distances : bool, default=False
        Computes distances between clusters even if `distance_threshold` is not
        used. This can be used to make dendrogram visualization, but introduces
        a computational and memory overhead.

        .. versionadded:: 0.24

    Attributes
    ----------
    n_clusters_ : int
        The number of clusters found by the algorithm. If
        ``distance_threshold=None``, it will be equal to the given
        ``n_clusters``.

    labels_ : array-like of (n_features,)
        Cluster labels for each feature.

    n_leaves_ : int
        Number of leaves in the hierarchical tree.

    n_connected_components_ : int
        The estimated number of connected components in the graph.

        .. versionadded:: 0.21
            ``n_connected_components_`` was added to replace ``n_components_``.

    n_features_in_ : int
        Number of features seen during :term:`fit`.

        .. versionadded:: 0.24

    feature_names_in_ : ndarray of shape (`n_features_in_`,)
        Names of features seen during :term:`fit`. Defined only when `X`
        has feature names that are all strings.

        .. versionadded:: 1.0

    children_ : array-like of shape (n_nodes-1, 2)
        The children of each non-leaf node. Values less than `n_features`
        correspond to leaves of the tree which are the original samples.
        A node `i` greater than or equal to `n_features` is a non-leaf
        node and has children `children_[i - n_features]`. Alternatively
        at the i-th iteration, children[i][0] and children[i][1]
        are merged to form node `n_features + i`.

    distances_ : array-like of shape (n_nodes-1,)
        Distances between nodes in the corresponding place in `children_`.
        Only computed if `distance_threshold` is used or `compute_distances`
        is set to `True`.

    See Also
    --------
    AgglomerativeClustering : Agglomerative clustering samples instead of
        features.
    ward_tree : Hierarchical clustering with ward linkage.

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn import datasets, cluster
    >>> digits = datasets.load_digits()
    >>> images = digits.images
    >>> X = np.reshape(images, (len(images), -1))
    >>> agglo = cluster.FeatureAgglomeration(n_clusters=32)
    >>> agglo.fit(X)
    FeatureAgglomeration(n_clusters=32)
    >>> X_reduced = agglo.transform(X)
    >>> X_reduced.shape
    (1797, 32)
    """

    def __init__(
        self,
        n_clusters=2,
        *,
        affinity="euclidean",
        memory=None,
        connectivity=None,
        compute_full_tree="auto",
        linkage="ward",
        pooling_func=np.mean,
        distance_threshold=None,
        compute_distances=False,
    ):
        super().__init__(
            n_clusters=n_clusters,
            memory=memory,
            connectivity=connectivity,
            compute_full_tree=compute_full_tree,
            linkage=linkage,
            affinity=affinity,
            distance_threshold=distance_threshold,
            compute_distances=compute_distances,
        )
        self.pooling_func = pooling_func

    def fit(self, X, y=None):
        """Fit the hierarchical clustering on the data.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            The data.

        y : Ignored
            Not used, present here for API consistency by convention.

        Returns
        -------
        self : object
            Returns the transformer.
        """
        X = self._validate_data(X, ensure_min_features=2)
        super()._fit(X.T)
        return self

    @property
    def fit_predict(self):
        """Fit and return the result of each sample's clustering assignment."""
        raise AttributeError

Hierarchical Agglomerative Clustering  These routines perform some hierarchical agglomerative clustering of some input data.  Authors : <NAME>, <NAME>, <NAME>,           <NAME> License: BSD 3 clause # mypy error: Module 'sklearn.cluster' has no attribute '_hierarchical_fast' # type: ignore ############################################################################### # For non fully-connected graphs Fixes the connectivity matrix.      The different steps are:      - copies it     - makes it symmetric     - converts it to LIL if necessary     - completes it if necessary.      Parameters     ----------     X : array-like of shape (n_samples, n_features)         Feature matrix representing `n_samples` samples to be clustered.      connectivity : sparse matrix, default=None         Connectivity matrix. Defines for each sample the neighboring samples         following a given structure of the data. The matrix is assumed to         be symmetric and only the upper triangular half is used.         Default is `None`, i.e, the Ward algorithm is unstructured.      affinity : {"euclidean", "precomputed"}, default="euclidean"         Which affinity to use. At the moment `precomputed` and         ``euclidean`` are supported. `euclidean` uses the         negative squared Euclidean distance between points.      Returns     -------     connectivity : sparse matrix         The fixed connectivity matrix.      n_connected_components : int         The number of connected components in the graph. # Make the connectivity matrix symmetric: # Convert connectivity matrix to LIL # Compute the number of nodes # XXX: Can we do without completing the matrix? Perform single linkage clustering on sparse data via the minimum     spanning tree from scipy.sparse.csgraph, then using union-find to label.     The parent array is then generated by walking through the tree. # explicitly cast connectivity to ensure safety # Ensure zero distances aren't ignored by setting them to "epsilon" # Use scipy.sparse.csgraph to generate a minimum spanning tree # Convert the graph to scipy.cluster.hierarchy array format # Undo the epsilon values # Sort edges of the min_spanning_tree by weight # Convert edge list into standard hierarchical clustering format # Compute parents ############################################################################### # Hierarchical tree building functions Ward clustering based on a Feature matrix.      Recursively merges the pair of clusters that minimally increases     within-cluster variance.      The inertia matrix uses a Heapq-based representation.      This is the structured version, that takes into account some topological     structure between samples.      Read more in the :ref:`User Guide <hierarchical_clustering>`.      Parameters     ----------     X : array-like of shape (n_samples, n_features)         Feature matrix representing `n_samples` samples to be clustered.      connectivity : sparse matrix, default=None         Connectivity matrix. Defines for each sample the neighboring samples         following a given structure of the data. The matrix is assumed to         be symmetric and only the upper triangular half is used.         Default is None, i.e, the Ward algorithm is unstructured.      n_clusters : int, default=None         `n_clusters` should be less than `n_samples`.  Stop early the         construction of the tree at `n_clusters.` This is useful to decrease         computation time if the number of clusters is not small compared to the         number of samples. In this case, the complete tree is not computed, thus         the 'children' output is of limited use, and the 'parents' output should         rather be used. This option is valid only when specifying a connectivity         matrix.      return_distance : bool, default=False         If `True`, return the distance between the clusters.      Returns     -------     children : ndarray of shape (n_nodes-1, 2)         The children of each non-leaf node. Values less than `n_samples`         correspond to leaves of the tree which are the original samples.         A node `i` greater than or equal to `n_samples` is a non-leaf         node and has children `children_[i - n_samples]`. Alternatively         at the i-th iteration, children[i][0] and children[i][1]         are merged to form node `n_samples + i`.      n_connected_components : int         The number of connected components in the graph.      n_leaves : int         The number of leaves in the tree.      parents : ndarray of shape (n_nodes,) or None         The parent of each node. Only returned when a connectivity matrix         is specified, elsewhere 'None' is returned.      distances : ndarray of shape (n_nodes-1,)         Only returned if `return_distance` is set to `True` (for compatibility).         The distances between the centers of the nodes. `distances[i]`         corresponds to a weighted Euclidean distance between         the nodes `children[i, 1]` and `children[i, 2]`. If the nodes refer to         leaves of the tree, then `distances[i]` is their unweighted Euclidean         distance. Distances are updated in the following way         (from scipy.hierarchy.linkage):          The new entry :math:`d(u,v)` is computed as follows,          .. math::             d(u,v) = \\sqrt{\\frac{|v|+|s|}                                {T}d(v,s)^2                         + \\frac{|v|+|t|}                                {T}d(v,t)^2                         - \\frac{|v|}                                {T}d(s,t)^2}          where :math:`u` is the newly joined cluster consisting of         clusters :math:`s` and :math:`t`, :math:`v` is an unused         cluster in the forest, :math:`T=|v|+|s|+|t|`, and         :math:`|*|` is the cardinality of its argument. This is also         known as the incremental algorithm. # imports PIL # create inertia matrix # We keep only the upper triangular for the moments # Generator expressions are faster than arrays on the following # build moments as a list # prepare the main fields # recursive merge loop # identify the merge # store inertia value # update the moments # update the structure matrix A and the inertia matrix # List comprehension is faster than a for loop # List comprehension is faster than a for loop # Separate leaves in children (empty lists up to now) # sort children to get consistent output with unstructured version # return numpy array for efficient caching # 2 is scaling factor to compare w/ unstructured version # single average and complete linkage Linkage agglomerative clustering based on a Feature matrix.      The inertia matrix uses a Heapq-based representation.      This is the structured version, that takes into account some topological     structure between samples.      Read more in the :ref:`User Guide <hierarchical_clustering>`.      Parameters     ----------     X : array-like of shape (n_samples, n_features)         Feature matrix representing `n_samples` samples to be clustered.      connectivity : sparse matrix, default=None         Connectivity matrix. Defines for each sample the neighboring samples         following a given structure of the data. The matrix is assumed to         be symmetric and only the upper triangular half is used.         Default is `None`, i.e, the Ward algorithm is unstructured.      n_clusters : int, default=None         Stop early the construction of the tree at `n_clusters`. This is         useful to decrease computation time if the number of clusters is         not small compared to the number of samples. In this case, the         complete tree is not computed, thus the 'children' output is of         limited use, and the 'parents' output should rather be used.         This option is valid only when specifying a connectivity matrix.      linkage : {"average", "complete", "single"}, default="complete"         Which linkage criteria to use. The linkage criterion determines which         distance to use between sets of observation.             - "average" uses the average of the distances of each observation of               the two sets.             - "complete" or maximum linkage uses the maximum distances between               all observations of the two sets.             - "single" uses the minimum of the distances between all               observations of the two sets.      affinity : str or callable, default='euclidean'         Which metric to use. Can be 'euclidean', 'manhattan', or any         distance known to paired distance (see metric.pairwise).      return_distance : bool, default=False         Whether or not to return the distances between the clusters.      Returns     -------     children : ndarray of shape (n_nodes-1, 2)         The children of each non-leaf node. Values less than `n_samples`         correspond to leaves of the tree which are the original samples.         A node `i` greater than or equal to `n_samples` is a non-leaf         node and has children `children_[i - n_samples]`. Alternatively         at the i-th iteration, children[i][0] and children[i][1]         are merged to form node `n_samples + i`.      n_connected_components : int         The number of connected components in the graph.      n_leaves : int         The number of leaves in the tree.      parents : ndarray of shape (n_nodes, ) or None         The parent of each node. Only returned when a connectivity matrix         is specified, elsewhere 'None' is returned.      distances : ndarray of shape (n_nodes-1,)         Returned when `return_distance` is set to `True`.          distances[i] refers to the distance between children[i][0] and         children[i][1] when they are merged.      See Also     --------     ward_tree : Hierarchical clustering with ward linkage. # Single linkage is handled differently # imports PIL # for the linkage function of hierarchy to work on precomputed # data, provide as first argument an ndarray of the shape returned # by sklearn.metrics.pairwise_distances. # Translate to something understood by scipy # We need the fast cythonized metric from neighbors # The Cython routines used require contiguous arrays # Sort edges of the min_spanning_tree by weight # Convert edge list into standard hierarchical clustering format # Put the diagonal to zero # FIXME We compute all the distances, while we could have only computed # the "interesting" distances # create inertia heap and connection matrix # LIL seems to the best format to access the rows quickly, # without the numpy overhead of slicing CSR indices and data. # We are storing the graph in a list of IntFloatDict # We keep only the upper triangular for the heap # Generator expressions are faster than arrays on the following # prepare the main fields # recursive merge loop # identify the merge # store distances # Keep track of the number of elements per cluster # update the structure matrix A and the inertia matrix # a clever 'min', or 'max' operation between A[i] and A[j] # Here we use the information from coord_col (containing the # distances) to update the heap # Clear A[i] and A[j] to save memory # Separate leaves in children (empty lists up to now) # # return numpy array for efficient caching # Matching names to tree-building strategies ############################################################################### # Functions for cutting hierarchical clustering tree Function cutting the ward tree for a given number of clusters.      Parameters     ----------     n_clusters : int or ndarray         The number of clusters to form.      children : ndarray of shape (n_nodes-1, 2)         The children of each non-leaf node. Values less than `n_samples`         correspond to leaves of the tree which are the original samples.         A node `i` greater than or equal to `n_samples` is a non-leaf         node and has children `children_[i - n_samples]`. Alternatively         at the i-th iteration, children[i][0] and children[i][1]         are merged to form node `n_samples + i`.      n_leaves : int         Number of leaves of the tree.      Returns     -------     labels : array [n_samples]         Cluster labels for each point. # In this function, we store nodes as a heap to avoid recomputing # the max of the nodes: the first element is always the smallest # We use negated indices as heaps work on smallest elements, and we # are interested in largest elements # children[-1] is the root of the tree # As we have a heap, nodes[0] is the smallest element # Insert the 2 children and remove the largest node ############################################################################### Agglomerative Clustering.      Recursively merges pair of clusters of sample data; uses linkage distance.      Read more in the :ref:`User Guide <hierarchical_clustering>`.      Parameters     ----------     n_clusters : int or None, default=2         The number of clusters to find. It must be ``None`` if         ``distance_threshold`` is not ``None``.      affinity : str or callable, default='euclidean'         Metric used to compute the linkage. Can be "euclidean", "l1", "l2",         "manhattan", "cosine", or "precomputed".         If linkage is "ward", only "euclidean" is accepted.         If "precomputed", a distance matrix (instead of a similarity matrix)         is needed as input for the fit method.      memory : str or object with the joblib.Memory interface, default=None         Used to cache the output of the computation of the tree.         By default, no caching is done. If a string is given, it is the         path to the caching directory.      connectivity : array-like or callable, default=None         Connectivity matrix. Defines for each sample the neighboring         samples following a given structure of the data.         This can be a connectivity matrix itself or a callable that transforms         the data into a connectivity matrix, such as derived from         `kneighbors_graph`. Default is ``None``, i.e, the         hierarchical clustering algorithm is unstructured.      compute_full_tree : 'auto' or bool, default='auto'         Stop early the construction of the tree at ``n_clusters``. This is         useful to decrease computation time if the number of clusters is not         small compared to the number of samples. This option is useful only         when specifying a connectivity matrix. Note also that when varying the         number of clusters and using caching, it may be advantageous to compute         the full tree. It must be ``True`` if ``distance_threshold`` is not         ``None``. By default `compute_full_tree` is "auto", which is equivalent         to `True` when `distance_threshold` is not `None` or that `n_clusters`         is inferior to the maximum between 100 or `0.02 * n_samples`.         Otherwise, "auto" is equivalent to `False`.      linkage : {'ward', 'complete', 'average', 'single'}, default='ward'         Which linkage criterion to use. The linkage criterion determines which         distance to use between sets of observation. The algorithm will merge         the pairs of cluster that minimize this criterion.          - 'ward' minimizes the variance of the clusters being merged.         - 'average' uses the average of the distances of each observation of           the two sets.         - 'complete' or 'maximum' linkage uses the maximum distances between           all observations of the two sets.         - 'single' uses the minimum of the distances between all observations           of the two sets.          .. versionadded:: 0.20             Added the 'single' option      distance_threshold : float, default=None         The linkage distance threshold above which, clusters will not be         merged. If not ``None``, ``n_clusters`` must be ``None`` and         ``compute_full_tree`` must be ``True``.          .. versionadded:: 0.21      compute_distances : bool, default=False         Computes distances between clusters even if `distance_threshold` is not         used. This can be used to make dendrogram visualization, but introduces         a computational and memory overhead.          .. versionadded:: 0.24      Attributes     ----------     n_clusters_ : int         The number of clusters found by the algorithm. If         ``distance_threshold=None``, it will be equal to the given         ``n_clusters``.      labels_ : ndarray of shape (n_samples)         Cluster labels for each point.      n_leaves_ : int         Number of leaves in the hierarchical tree.      n_connected_components_ : int         The estimated number of connected components in the graph.          .. versionadded:: 0.21             ``n_connected_components_`` was added to replace ``n_components_``.      n_features_in_ : int         Number of features seen during :term:`fit`.          .. versionadded:: 0.24      feature_names_in_ : ndarray of shape (`n_features_in_`,)         Names of features seen during :term:`fit`. Defined only when `X`         has feature names that are all strings.          .. versionadded:: 1.0      children_ : array-like of shape (n_samples-1, 2)         The children of each non-leaf node. Values less than `n_samples`         correspond to leaves of the tree which are the original samples.         A node `i` greater than or equal to `n_samples` is a non-leaf         node and has children `children_[i - n_samples]`. Alternatively         at the i-th iteration, children[i][0] and children[i][1]         are merged to form node `n_samples + i`.      distances_ : array-like of shape (n_nodes-1,)         Distances between nodes in the corresponding place in `children_`.         Only computed if `distance_threshold` is used or `compute_distances`         is set to `True`.      See Also     --------     FeatureAgglomeration : Agglomerative clustering but for features instead of         samples.     ward_tree : Hierarchical clustering with ward linkage.      Examples     --------     >>> from sklearn.cluster import AgglomerativeClustering     >>> import numpy as np     >>> X = np.array([[1, 2], [1, 4], [1, 0],     ...               [4, 2], [4, 4], [4, 0]])     >>> clustering = AgglomerativeClustering().fit(X)     >>> clustering     AgglomerativeClustering()     >>> clustering.labels_     array([1, 1, 1, 0, 0, 0]) Fit the hierarchical clustering from features, or distance matrix.          Parameters         ----------         X : array-like, shape (n_samples, n_features) or \                 (n_samples, n_samples)             Training instances to cluster, or distances between instances if             ``affinity='precomputed'``.          y : Ignored             Not used, present here for API consistency by convention.          Returns         -------         self : object             Returns the fitted instance. Fit without validation          Parameters         ----------         X : ndarray of shape (n_samples, n_features) or (n_samples, n_samples)             Training instances to cluster, or distances between instances if             ``affinity='precomputed'``.          Returns         -------         self : object             Returns the fitted instance. # Early stopping is likely to give a speed up only for # a large number of clusters. The actual threshold # implemented here is heuristic # Construct the tree # distance_threshold is used # n_clusters is used # Cut the tree # copy to avoid holding a reference on the original array # Reassign cluster numbers Fit and return the result of each sample's clustering assignment.          In addition to fitting, this method also return the result of the         clustering assignment for each sample in the training set.          Parameters         ----------         X : array-like of shape (n_samples, n_features) or \                 (n_samples, n_samples)             Training instances to cluster, or distances between instances if             ``affinity='precomputed'``.          y : Ignored             Not used, present here for API consistency by convention.          Returns         -------         labels : ndarray of shape (n_samples,)             Cluster labels. Agglomerate features.      Recursively merges pair of clusters of features.      Read more in the :ref:`User Guide <hierarchical_clustering>`.      Parameters     ----------     n_clusters : int, default=2         The number of clusters to find. It must be ``None`` if         ``distance_threshold`` is not ``None``.      affinity : str or callable, default='euclidean'         Metric used to compute the linkage. Can be "euclidean", "l1", "l2",         "manhattan", "cosine", or 'precomputed'.         If linkage is "ward", only "euclidean" is accepted.      memory : str or object with the joblib.Memory interface, default=None         Used to cache the output of the computation of the tree.         By default, no caching is done. If a string is given, it is the         path to the caching directory.      connectivity : array-like or callable, default=None         Connectivity matrix. Defines for each feature the neighboring         features following a given structure of the data.         This can be a connectivity matrix itself or a callable that transforms         the data into a connectivity matrix, such as derived from         `kneighbors_graph`. Default is `None`, i.e, the         hierarchical clustering algorithm is unstructured.      compute_full_tree : 'auto' or bool, default='auto'         Stop early the construction of the tree at `n_clusters`. This is useful         to decrease computation time if the number of clusters is not small         compared to the number of features. This option is useful only when         specifying a connectivity matrix. Note also that when varying the         number of clusters and using caching, it may be advantageous to compute         the full tree. It must be ``True`` if ``distance_threshold`` is not         ``None``. By default `compute_full_tree` is "auto", which is equivalent         to `True` when `distance_threshold` is not `None` or that `n_clusters`         is inferior to the maximum between 100 or `0.02 * n_samples`.         Otherwise, "auto" is equivalent to `False`.      linkage : {"ward", "complete", "average", "single"}, default="ward"         Which linkage criterion to use. The linkage criterion determines which         distance to use between sets of features. The algorithm will merge         the pairs of cluster that minimize this criterion.          - "ward" minimizes the variance of the clusters being merged.         - "complete" or maximum linkage uses the maximum distances between           all features of the two sets.         - "average" uses the average of the distances of each feature of           the two sets.         - "single" uses the minimum of the distances between all features           of the two sets.      pooling_func : callable, default=np.mean         This combines the values of agglomerated features into a single         value, and should accept an array of shape [M, N] and the keyword         argument `axis=1`, and reduce it to an array of size [M].      distance_threshold : float, default=None         The linkage distance threshold above which, clusters will not be         merged. If not ``None``, ``n_clusters`` must be ``None`` and         ``compute_full_tree`` must be ``True``.          .. versionadded:: 0.21      compute_distances : bool, default=False         Computes distances between clusters even if `distance_threshold` is not         used. This can be used to make dendrogram visualization, but introduces         a computational and memory overhead.          .. versionadded:: 0.24      Attributes     ----------     n_clusters_ : int         The number of clusters found by the algorithm. If         ``distance_threshold=None``, it will be equal to the given         ``n_clusters``.      labels_ : array-like of (n_features,)         Cluster labels for each feature.      n_leaves_ : int         Number of leaves in the hierarchical tree.      n_connected_components_ : int         The estimated number of connected components in the graph.          .. versionadded:: 0.21             ``n_connected_components_`` was added to replace ``n_components_``.      n_features_in_ : int         Number of features seen during :term:`fit`.          .. versionadded:: 0.24      feature_names_in_ : ndarray of shape (`n_features_in_`,)         Names of features seen during :term:`fit`. Defined only when `X`         has feature names that are all strings.          .. versionadded:: 1.0      children_ : array-like of shape (n_nodes-1, 2)         The children of each non-leaf node. Values less than `n_features`         correspond to leaves of the tree which are the original samples.         A node `i` greater than or equal to `n_features` is a non-leaf         node and has children `children_[i - n_features]`. Alternatively         at the i-th iteration, children[i][0] and children[i][1]         are merged to form node `n_features + i`.      distances_ : array-like of shape (n_nodes-1,)         Distances between nodes in the corresponding place in `children_`.         Only computed if `distance_threshold` is used or `compute_distances`         is set to `True`.      See Also     --------     AgglomerativeClustering : Agglomerative clustering samples instead of         features.     ward_tree : Hierarchical clustering with ward linkage.      Examples     --------     >>> import numpy as np     >>> from sklearn import datasets, cluster     >>> digits = datasets.load_digits()     >>> images = digits.images     >>> X = np.reshape(images, (len(images), -1))     >>> agglo = cluster.FeatureAgglomeration(n_clusters=32)     >>> agglo.fit(X)     FeatureAgglomeration(n_clusters=32)     >>> X_reduced = agglo.transform(X)     >>> X_reduced.shape     (1797, 32) Fit the hierarchical clustering on the data.          Parameters         ----------         X : array-like of shape (n_samples, n_features)             The data.          y : Ignored             Not used, present here for API consistency by convention.          Returns         -------         self : object             Returns the transformer. Fit and return the result of each sample's clustering assignment.