scikit-learn · ogrisel · Nov 2, 2021 · Sep 25, 2021 · Sep 25, 2021 · Sep 25, 2021
diff --git a/doc/modules/clustering.rst b/doc/modules/clustering.rst
@@ -492,11 +492,15 @@ computed using a function of a gradient of the image.
 
 .. |coin_kmeans| image:: ../auto_examples/cluster/images/sphx_glr_plot_coin_segmentation_001.png
     :target: ../auto_examples/cluster/plot_coin_segmentation.html
-    :scale: 65
+    :scale: 35
 
 .. |coin_discretize| image:: ../auto_examples/cluster/images/sphx_glr_plot_coin_segmentation_002.png
     :target: ../auto_examples/cluster/plot_coin_segmentation.html
-    :scale: 65
+    :scale: 35
+
+.. |coin_cluster_qr| image:: ../auto_examples/cluster/images/sphx_glr_plot_coin_segmentation_003.png
+    :target: ../auto_examples/cluster/plot_coin_segmentation.html
+    :scale: 35
 
 Different label assignment strategies
 -------------------------------------
@@ -508,12 +512,24 @@ In particular, unless you control the ``random_state``, it may not be
 reproducible from run-to-run, as it depends on random initialization.
 The alternative ``"discretize"`` strategy is 100% reproducible, but tends
 to create parcels of fairly even and geometrical shape.
+The recently added ``"cluster_qr"`` option is a deterministic alternative that
+tends to create the visually best partitioning on the example application
+below.
+
+================================  ================================  ================================
+ ``assign_labels="kmeans"``        ``assign_labels="discretize"``    ``assign_labels="cluster_qr"``
+================================  ================================  ================================
+|coin_kmeans|                          |coin_discretize|                  |coin_cluster_qr|
+================================  ================================  ================================
+
+.. topic:: References:
+
+ * `"Multiclass spectral clustering"
+   <https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf>`_
+   Stella X. Yu, Jianbo Shi, 2003
 
-=====================================  =====================================
- ``assign_labels="kmeans"``              ``assign_labels="discretize"``
-=====================================  =====================================
-|coin_kmeans|                          |coin_discretize|
-=====================================  =====================================
+ * :doi:`"Simple, direct, and efficient multi-way spectral clustering"<10.1093/imaiai/iay008>`
+    Anil Damle, Victor Minden, Lexing Ying, 2019
 
 Spectral Clustering Graphs
 --------------------------

diff --git a/doc/whats_new/v1.1.rst b/doc/whats_new/v1.1.rst
@@ -49,6 +49,13 @@ Changelog
   add this information to the plot.
   :pr:`21038` by :user:`Guillaume Lemaitre <glemaitre>`.
 
+- |Enhancement| :class:`cluster.SpectralClustering` and :func:`cluster.spectral` 
+  now include the new `'cluster_qr'` method from :func:`cluster.cluster_qr`
+  that clusters samples in the embedding space as an alternative to the existing
+  `'kmeans'` and `'discrete'` methods.
+  See :func:`cluster.spectral_clustering` for more details.
+  :pr:`21148` by :user:`Andrew Knyazev <lobpcg>`
+
 :mod:`sklearn.cross_decomposition`
 ..................................
 

diff --git a/examples/cluster/plot_coin_segmentation.py b/examples/cluster/plot_coin_segmentation.py
@@ -10,16 +10,19 @@
 This procedure (spectral clustering on an image) is an efficient
 approximate solution for finding normalized graph cuts.
 
-There are two options to assign labels:
+There are three options to assign labels:
 
-* with 'kmeans' spectral clustering will cluster samples in the embedding space
+* 'kmeans' spectral clustering clusters samples in the embedding space
   using a kmeans algorithm
-* whereas 'discrete' will iteratively search for the closest partition
-  space to the embedding space.
-
+* 'discrete' iteratively searches for the closest partition
+  space to the embedding space of spectral clustering.
+* 'cluster_qr' assigns labels using the QR factorization with pivoting
+  that directly determines the partition in the embedding space.
 """
 
-# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>, Brian Cheung
+# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
+#         Brian Cheung
+#         Andrew Knyazev <Andrew.Knyazev@ucdenver.edu>
 # License: BSD 3 clause
 
 import time
@@ -61,28 +64,51 @@
 eps = 1e-6
 graph.data = np.exp(-beta * graph.data / graph.data.std()) + eps
 
-# Apply spectral clustering (this step goes much faster if you have pyamg
-# installed)
-N_REGIONS = 25
+# The number of segmented regions to display needs to be chosen manually.
+# The current version of 'spectral_clustering' does not support determining
+# the number of good quality clusters automatically.
+n_regions = 26
 
 # %%
-# Visualize the resulting regions
-
-for assign_labels in ("kmeans", "discretize"):
+# Compute and visualize the resulting regions
+
+# Computing a few extra eigenvectors may speed up the eigen_solver.
+# The spectral clustering quality may also benetif from requesting
+# extra regions for segmentation.
+n_regions_plus = 3
+
+# Apply spectral clustering using the default eigen_solver='arpack'.
+# Any implemented solver can be used: eigen_solver='arpack', 'lobpcg', or 'amg'.
+# Choosing eigen_solver='amg' requires an extra package called 'pyamg'.
+# The quality of segmentation and the speed of calculations is mostly determined
+# by the choice of the solver and the value of the tolerance 'eigen_tol'.
+# TODO: varying eigen_tol seems to have no effect for 'lobpcg' and 'amg' #21243.
+for assign_labels in ("kmeans", "discretize", "cluster_qr"):
     t0 = time.time()
     labels = spectral_clustering(
-        graph, n_clusters=N_REGIONS, assign_labels=assign_labels, random_state=42
+        graph,
+        n_clusters=(n_regions + n_regions_plus),
+        eigen_tol=1e-7,
+        assign_labels=assign_labels,
+        random_state=42,
     )
+
     t1 = time.time()
     labels = labels.reshape(rescaled_coins.shape)
-
     plt.figure(figsize=(5, 5))
     plt.imshow(rescaled_coins, cmap=plt.cm.gray)
-    for l in range(N_REGIONS):
-        plt.contour(labels == l, colors=[plt.cm.nipy_spectral(l / float(N_REGIONS))])
+
     plt.xticks(())
     plt.yticks(())
     title = "Spectral clustering: %s, %.2fs" % (assign_labels, (t1 - t0))
     print(title)
     plt.title(title)
+    for l in range(n_regions):
+        colors = [plt.cm.nipy_spectral((l + 4) / float(n_regions + 4))]
+        plt.contour(labels == l, colors=colors)
+        # To view individual segments as appear comment in plt.pause(0.5)
 plt.show()
+
+# TODO: After #21194 is merged and #21243 is fixed, check which eigen_solver
+# is the best and set eigen_solver='arpack', 'lobpcg', or 'amg' and eigen_tol
+# explicitly in this example.
diff --git a/sklearn/cluster/_spectral.py b/sklearn/cluster/_spectral.py
@@ -1,14 +1,18 @@
 # -*- coding: utf-8 -*-
 """Algorithms for spectral clustering"""
 
-# Author: Gael Varo
F438
quaux gael.varoquaux@normalesup.org
+# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
 #         Brian Cheung
 #         Wei LI <kuantkid@gmail.com>
+#         Andrew Knyazev <Andrew.Knyazev@ucdenver.edu>
 # License: BSD 3 clause
 import warnings
 
 import numpy as np
 
+from scipy.linalg import LinAlgError, qr, svd
+from scipy.sparse import csc_matrix
+
 from ..base import BaseEstimator, ClusterMixin
 from ..utils import check_random_state, as_float_array
 from ..utils.deprecation import deprecated
@@ -18,6 +22,38 @@
 from ._kmeans import k_means
 
 
+def cluster_qr(vectors):
+    """Find the discrete partition closest to the eigenvector embedding.
+
+        This implementation was proposed in [1]_.
+
+    .. versionadded:: 1.1
+
+        Parameters
+        ----------
+        vectors : array-like, shape: (n_samples, n_clusters)
+            The embedding space of the samples.
+
+        Returns
+        -------
+        labels : array of integers, shape: n_samples
+            The cluster labels of vectors.
+
+        References
+        ----------
+        .. [1] `Simple, direct, and efficient multi-way spectral clustering, 2019
+            Anil Damle, Victor Minden, Lexing Ying
+            <:doi:`10.1093/imaiai/iay008`>`_
+
+    """
+
+    k = vectors.shape[1]
+    _, _, piv = qr(vectors.T, pivoting=True)
+    ut, _, v = svd(vectors[piv[:k], :].T)
+    vectors = abs(np.dot(vectors, np.dot(ut, v.conj())))
+    return vectors.argmax(axis=1)
+
+
 def discretize(
     vectors, *, copy=True, max_svd_restarts=30, n_iter_max=20, random_state=None
 ):
@@ -73,9 +109,6 @@ def discretize(
 
     """
 
-    from scipy.sparse import csc_matrix
-    from scipy.linalg import LinAlgError
-
     random_state = check_random_state(random_state)
 
     vectors = as_float_array(vectors, copy=copy)
@@ -200,10 +233,11 @@ def spectral_clustering(
         Number of eigenvectors to use for the spectral embedding
 
     eigen_solver : {None, 'arpack', 'lobpcg', or 'amg'}
-        The eigenvalue decomposition strategy to use. AMG requires pyamg
-        to be installed. It can be faster on very large, sparse problems,
-        but may also lead to instabilities. If None, then ``'arpack'`` is
-        used. See [4]_ for more details regarding `'lobpcg'`.
+        The eigenvalue decomposition method. If None then ``'arpack'`` is used.
+        See [4]_ for more details regarding ``'lobpcg'``.
+        Eigensolver ``'amg'`` runs ``'lobpcg'`` with optional
+        Algebraic MultiGrid preconditioning and requires pyamg to be installed.
+        It can be faster on very large sparse problems [6]_ and [7]_.
 
     random_state : int, RandomState instance, default=None
         A pseudo random number generator used for the initialization
@@ -229,12 +263,19 @@ def spectral_clustering(
         Stopping criterion for eigendecomposition of the Laplacian matrix
         when using arpack eigen_solver.
 
-    assign_labels : {'kmeans', 'discretize'}, default='kmeans'
+    assign_labels : {'kmeans', 'discretize', 'cluster_qr'}, default='kmeans'
         The strategy to use to assign labels in the embedding
-        space.  There are two ways to assign labels after the Laplacian
+        space.  There are three ways to assign labels after the Laplacian
         embedding.  k-means can be applied and is a popular choice. But it can
         also be sensitive to initialization. Discretization is another
         approach which is less sensitive to random initialization [3]_.
+        The cluster_qr method [5]_ directly extracts clusters from eigenvectors
+        in spectral clustering. In contrast to k-means and discretization, cluster_qr
+        has no tuning parameters and is not an iterative method, yet may outperform
+        k-means and discretization in terms of both quality and speed.
+
+        .. versionchanged:: 1.1
+           Added new labeling method 'cluster_qr'.
 
     verbose : bool, default=False
         Verbosity mode.
@@ -262,23 +303,38 @@ def spectral_clustering(
            <https://www1.icsi.berkeley.edu/~stellayu/publication/doc/2003kwayICCV.pdf>`_
 
     .. [4] `Toward the Optimal Preconditioned Eigensolver:
-           Locally Optimal Block Preconditioned Conjugate Gradient Method, 2001.
+           Locally Optimal Block Preconditioned Conjugate Gradient Method, 2001
            A. V. Knyazev
            SIAM Journal on Scientific Computing 23, no. 2, pp. 517-541.
-           <https://epubs.siam.org/doi/pdf/10.1137/S1064827500366124>`_
+           <:doi:`10.1137/S1064827500366124`>`_
+
+    .. [5] `Simple, direct, and efficient multi-way spectral clustering, 2019
+           Anil Damle, Victor Minden, Lexing Ying
+           <:doi:`10.1093/imaiai/iay008`>`_
+
+    .. [6] `Multiscale Spectral Image Segmentation Multiscale preconditioning
+           for computing eigenvalues of graph Laplacians in image segmentation, 2006
+           Andrew Knyazev
+           <:doi:`10.13140/RG.2.2.35280.02565`>`_
+
+    .. [7] `Preconditioned spectral clustering for stochastic block partition
+           streaming graph challenge (Preliminary version at arXiv.)
+           David Zhuzhunashvili, Andrew Knyazev
+           <:doi:`10.1109/HPEC.2017.8091045`>`_
 
     Notes
     -----
-    The graph should contain only one connect component, elsewhere
+    The graph should contain only one connected component, elsewhere
     the results make little sense.
 
     This algorithm solves the normalized cut for k=2: it is a
     normalized spectral clustering.
     """
-    if assign_labels not in ("kmeans", "discretize"):
+    if assign_labels not in ("kmeans", "discretize", "cluster_qr"):
         raise ValueError(
             "The 'assign_labels' parameter should be "
-            "'kmeans' or 'discretize', but '%s' was given" % assign_labels
+            "'kmeans' or 'discretize', or 'cluster_qr', "
+            f"but {assign_labels!r} was given"
         )
     if isinstance(affinity, np.matrix):
         raise TypeError(
@@ -312,6 +368,8 @@ def spectral_clustering(
         _, labels, _ = k_means(
             maps, n_clusters, random_state=random_state, n_init=n_init, verbose=verbose
         )
+    elif assign_labels == "cluster_qr":
+        labels = cluster_qr(maps)
     else:
         labels = discretize(maps, random_state=random_state)
 
@@ -407,12 +465,19 @@ class SpectralClustering(ClusterMixin, BaseEstimator):
         Stopping criterion for eigendecomposition of the Laplacian matrix
         when ``eigen_solver='arpack'``.
 
-    assign_labels : {'kmeans', 'discretize'}, default='kmeans'
+    assign_labels : {'kmeans', 'discretize', 'cluster_qr'}, default='kmeans'
         The strategy for assigning labels in the embedding space. There are two
         ways to assign labels after the Laplacian embedding. k-means is a
         popular choice, but it can be sensitive to initialization.
         Discretization is another approach which is less sensitive to random
         initialization [3]_.
+        The cluster_qr method [5]_ directly extract clusters from eigenvectors
+        in spectral clustering. In contrast to k-means and discretization, cluster_qr
+        has no tuning parameters and runs no iterations, yet may outperform
+        k-means and discretization in terms of both quality and speed.
+
+        .. versionchanged:: 1.1
+           Added new labeling method 'cluster_qr'.
 
     degree : float, default=3
         Degree of the polynomial kernel. Ignored by other kernels.
@@ -502,6 +567,10 @@ class SpectralClustering(ClusterMixin, BaseEstimator):
            SIAM Journal on Scientific Computing 23, no. 2, pp. 517-541.
            <https://epubs.siam.org/doi/pdf/10.1137/S1064827500366124>`_
 
+    .. [5] `Simple, direct, and efficient multi-way spectral clustering, 2019
+           Anil Damle, Victor Minden, Lexing Ying
+           <:doi:`10.1093/imaiai/iay008`>`_
+
     Examples
     --------
     >>> from sklearn.cluster import SpectralClustering