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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.cluster</span></code>.spectral_clustering</a><ul>
<li><a class="reference internal" href="#sklearn.cluster.spectral_clustering"><code class="docutils literal notranslate"><span class="pre">spectral_clustering</span></code></a></li>
<li><a class="reference internal" href="#examples-using-sklearn-cluster-spectral-clustering">Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.cluster.spectral_clustering</span></code></a></li>
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<section id="sklearn-cluster-spectral-clustering">
<h1><a class="reference internal" href="../classes.html#module-sklearn.cluster" title="sklearn.cluster"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.cluster</span></code></a>.spectral_clustering<a class="headerlink" href="#sklearn-cluster-spectral-clustering" title="Permalink to this heading">¶</a></h1>
<dl class="py function">
<dt class="sig sig-object py" id="sklearn.cluster.spectral_clustering">
<span class="sig-prename descclassname"><span class="pre">sklearn.cluster.</span></span><span class="sig-name descname"><span class="pre">spectral_clustering</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">affinity</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_clusters</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">8</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_components</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_solver</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">random_state</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">n_init</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">eigen_tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'auto'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">assign_labels</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'kmeans'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/scikit-learn/scikit-learn/blob/3f89022fa/sklearn/cluster/_spectral.py#L192"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sklearn.cluster.spectral_clustering" title="Permalink to this definition">¶</a></dt>
<dd><p>Apply clustering to a projection of the normalized Laplacian.</p>
<p>In practice Spectral Clustering is very useful when the structure of
the individual clusters is highly non-convex or more generally when
a measure of the center and spread of the cluster is not a suitable
description of the complete cluster. For instance, when clusters are
nested circles on the 2D plane.</p>
<p>If affinity is the adjacency matrix of a graph, this method can be
used to find normalized graph cuts <a class="reference internal" href="#r89dec4780971-1" id="id1">[1]</a>, <a class="reference internal" href="#r89dec4780971-2" id="id2">[2]</a>.</p>
<p>Read more in the <a class="reference internal" href="../clustering.html#spectral-clustering"><span class="std std-ref">User Guide</span></a>.</p>
<dl class="field-list">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl>
<dt><strong>affinity</strong><span class="classifier">{array-like, sparse matrix} of shape (n_samples, n_samples)</span></dt><dd><p>The affinity matrix describing the relationship of the samples to
embed. <strong>Must be symmetric</strong>.</p>
<dl class="simple">
<dt>Possible examples:</dt><dd><ul class="simple">
<li><p>adjacency matrix of a graph,</p></li>
<li><p>heat kernel of the pairwise distance matrix of the samples,</p></li>
<li><p>symmetric k-nearest neighbours connectivity matrix of the samples.</p></li>
</ul>
</dd>
</dl>
</dd>
<dt><strong>n_clusters</strong><span class="classifier">int, default=None</span></dt><dd><p>Number of clusters to extract.</p>
</dd>
<dt><strong>n_components</strong><span class="classifier">int, default=n_clusters</span></dt><dd><p>Number of eigenvectors to use for the spectral embedding.</p>
</dd>
<dt><strong>eigen_solver</strong><span class="classifier">{None, ‘arpack’, ‘lobpcg’, or ‘amg’}</span></dt><dd><p>The eigenvalue decomposition method. If None then <code class="docutils literal notranslate"><span class="pre">'arpack'</span></code> is used.
See <a class="reference internal" href="#r89dec4780971-4" id="id3">[4]</a> for more details regarding <code class="docutils literal notranslate"><span class="pre">'lobpcg'</span></code>.
Eigensolver <code class="docutils literal notranslate"><span class="pre">'amg'</span></code> runs <code class="docutils literal notranslate"><span class="pre">'lobpcg'</span></code> with optional
Algebraic MultiGrid preconditioning and requires pyamg to be installed.
It can be faster on very large sparse problems <a class="reference internal" href="#r89dec4780971-6" id="id4">[6]</a> and <a class="reference internal" href="#r89dec4780971-7" id="id5">[7]</a>.</p>
</dd>
<dt><strong>random_state</strong><span class="classifier">int, RandomState instance, default=None</span></dt><dd><p>A pseudo random number generator used for the initialization
of the lobpcg eigenvectors decomposition when <code class="docutils literal notranslate"><span class="pre">eigen_solver</span> <span class="pre">==</span>
<span class="pre">'amg'</span></code>, and for the K-Means initialization. Use an int to make
the results deterministic across calls (See
<a class="reference internal" href="../../glossary.html#term-random_state"><span class="xref std std-term">Glossary</span></a>).</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>When using <code class="docutils literal notranslate"><span class="pre">eigen_solver</span> <span class="pre">==</span> <span class="pre">'amg'</span></code>,
it is necessary to also fix the global numpy seed with
<code class="docutils literal notranslate"><span class="pre">np.random.seed(int)</span></code> to get deterministic results. See
<a class="reference external" href="https://github.com/pyamg/pyamg/issues/139">https://github.com/pyamg/pyamg/issues/139</a> for further
information.</p>
</div>
</dd>
<dt><strong>n_init</strong><span class="classifier">int, default=10</span></dt><dd><p>Number of time the k-means algorithm will be run with different
centroid seeds. The final results will be the best output of n_init
consecutive runs in terms of inertia. Only used if
<code class="docutils literal notranslate"><span class="pre">assign_labels='kmeans'</span></code>.</p>
</dd>
<dt><strong>eigen_tol</strong><span class="classifier">float, default=”auto”</span></dt><dd><p>Stopping criterion for eigendecomposition of the Laplacian matrix.
If <code class="docutils literal notranslate"><span class="pre">eigen_tol="auto"</span></code> then the passed tolerance will depend on the
<code class="docutils literal notranslate"><span class="pre">eigen_solver</span></code>:</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">eigen_solver="arpack"</span></code>, then <code class="docutils literal notranslate"><span class="pre">eigen_tol=0.0</span></code>;</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">eigen_solver="lobpcg"</span></code> or <code class="docutils literal notranslate"><span class="pre">eigen_solver="amg"</span></code>, then
<code class="docutils literal notranslate"><span class="pre">eigen_tol=None</span></code> which configures the underlying <code class="docutils literal notranslate"><span class="pre">lobpcg</span></code> solver to
automatically resolve the value according to their heuristics. See,
<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lobpcg.html#scipy.sparse.linalg.lobpcg" title="(in SciPy v1.11.3)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.sparse.linalg.lobpcg</span></code></a> for details.</p></li>
</ul>
<p>Note that when using <code class="docutils literal notranslate"><span class="pre">eigen_solver="lobpcg"</span></code> or <code class="docutils literal notranslate"><span class="pre">eigen_solver="amg"</span></code>
values of <code class="docutils literal notranslate"><span class="pre">tol<1e-5</span></code> may lead to convergence issues and should be
avoided.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 1.2: </span>Added ‘auto’ option.</p>
</div>
</dd>
<dt><strong>assign_labels</strong><span class="classifier">{‘kmeans’, ‘discretize’, ‘cluster_qr’}, default=’kmeans’</span></dt><dd><p>The strategy to use to assign labels in the embedding
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 <a class="reference internal" href="#r89dec4780971-3" id="id6">[3]</a>.
The cluster_qr method <a class="reference internal" href="#r89dec4780971-5" id="id7">[5]</a> 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.</p>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 1.1: </span>Added new labeling method ‘cluster_qr’.</p>
</div>
</dd>
<dt><strong>verbose</strong><span class="classifier">bool, default=False</span></dt><dd><p>Verbosity mode.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 0.24.</span></p>
</div>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>labels</strong><span class="classifier">array of integers, shape: n_samples</span></dt><dd><p>The labels of the clusters.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The graph should contain only one connected component, elsewhere
the results make little sense.</p>
<p>This algorithm solves the normalized cut for <code class="docutils literal notranslate"><span class="pre">k=2</span></code>: it is a
normalized spectral clustering.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r89dec4780971-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1109/34.868688">Normalized cuts and image segmentation, 2000
Jianbo Shi, Jitendra Malik</a></p>
</div>
<div class="citation" id="r89dec4780971-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">2</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1007/s11222-007-9033-z">A Tutorial on Spectral Clustering, 2007
Ulrike von Luxburg</a></p>
</div>
<div class="citation" id="r89dec4780971-3" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">3</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://people.eecs.berkeley.edu/~jordan/courses/281B-spring04/readings/yu-shi.pdf">Multiclass spectral clustering, 2003
Stella X. Yu, Jianbo Shi</a></p>
</div>
<div class="citation" id="r89dec4780971-4" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">4</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1137/S1064827500366124">Toward the Optimal Preconditioned Eigensolver:
Locally Optimal Block Preconditioned Conjugate Gradient Method, 2001
A. V. Knyazev
SIAM Journal on Scientific Computing 23, no. 2, pp. 517-541.</a></p>
</div>
<div class="citation" id="r89dec4780971-5" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">5</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1093/imaiai/iay008">Simple, direct, and efficient multi-way spectral clustering, 2019
Anil Damle, Victor Minden, Lexing Ying</a></p>
</div>
<div class="citation" id="r89dec4780971-6" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">6</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.13140/RG.2.2.35280.02565">Multiscale Spectral Image Segmentation Multiscale preconditioning
for computing eigenvalues of graph Laplacians in image segmentation, 2006
Andrew Knyazev</a></p>
</div>
<div class="citation" id="r89dec4780971-7" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">7</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://doi.org/10.1109/HPEC.2017.8091045">Preconditioned spectral clustering for stochastic block partition
streaming graph challenge (Preliminary version at arXiv.)
David Zhuzhunashvili, Andrew Knyazev</a></p>
</div>
</div>
</dd></dl>
<section id="examples-using-sklearn-cluster-spectral-clustering">
<h2>Examples using <code class="docutils literal notranslate"><span class="pre">sklearn.cluster.spectral_clustering</span></code><a class="headerlink" href="#examples-using-sklearn-cluster-spectral-clustering" title="Permalink to this heading">¶</a></h2>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example uses spectral_clustering on a graph created from voxel-to-voxel difference on an i..."><img alt="" src="../../_images/sphx_glr_plot_coin_segmentation_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/cluster/plot_coin_segmentation.html#sphx-glr-auto-examples-cluster-plot-coin-segmentation-py"><span class="std std-ref">Segmenting the picture of greek coins in regions</span></a></p>
<div class="sphx-glr-thumbnail-title">Segmenting the picture of greek coins in regions</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="In this example, an image with connected circles is generated and spectral clustering is used t..."><img alt="" src="../../_images/sphx_glr_plot_segmentation_toy_thumb.png" />
<p><a class="reference internal" href="../../auto_examples/cluster/plot_segmentation_toy.html#sphx-glr-auto-examples-cluster-plot-segmentation-toy-py"><span class="std std-ref">Spectral clustering for image segmentation</span></a></p>
<div class="sphx-glr-thumbnail-title">Spectral clustering for image segmentation</div>
</div></div><div class="clearer"></div></section>
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