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</ul> - </div> - </div> -</div> + <!--</ul>--> + </div><!-- /.navbar-collapse --> </div> +</nav> - <div class="col-md-8"> - <div class="page-header"> - <h1>(Deprecated) Spectral Clustering </h1> - </div> - <h1 id="spectral-clustering-overview">Spectral Clustering Overview</h1> + <div class="container mt-5 pb-4"> + + <div class="row"> + + <div class="col-lg-8"> + <h1 id="spectral-clustering-overview">Spectral Clustering Overview</h1> <p>Spectral clustering, as its name implies, makes use of the spectrum (or eigenvalues) of the similarity matrix of the data. It examines the <em>connectedness</em> of the data, whereas other clustering algorithms such as k-means use the <em>compactness</em> to assign clusters. Consequently, in situations where k-means performs well, spectral clustering will also perform well. Additionally, there are situations in which k-means will underperform (e.g. concentric circles), but spectral clustering will be able to segment the underlying clusters. Spectral clustering is also very useful for image segmentation.</p> @@ -314,16 +171,16 @@ <ol> <li> - <p>Computing a similarity (or <em>affinity</em>) matrix <code>\(\mathbf{A}\)</code> from the data. This involves determining a pairwise distance function <code>\(f\)</code> that takes a pair of data points and returns a scalar.</p> + <p>Computing a similarity (or <em>affinity</em>) matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code> from the data. This involves determining a pairwise distance function <code class="highlighter-rouge">\(f\)</code> that takes a pair of data points and returns a scalar.</p> </li> <li> - <p>Computing a graph Laplacian <code>\(\mathbf{L}\)</code> from the affinity matrix. There are several types of graph Laplacians; which is used will often depends on the situation.</p> + <p>Computing a graph Laplacian <code class="highlighter-rouge">\(\mathbf{L}\)</code> from the affinity matrix. There are several types of graph Laplacians; which is used will often depends on the situation.</p> </li> <li> - <p>Computing the eigenvectors and eigenvalues of <code>\(\mathbf{L}\)</code>. The degree of this decomposition is often modulated by <code>\(k\)</code>, or the number of clusters. Put another way, <code>\(k\)</code> eigenvectors and eigenvalues are computed.</p> + <p>Computing the eigenvectors and eigenvalues of <code class="highlighter-rouge">\(\mathbf{L}\)</code>. The degree of this decomposition is often modulated by <code class="highlighter-rouge">\(k\)</code>, or the number of clusters. Put another way, <code class="highlighter-rouge">\(k\)</code> eigenvectors and eigenvalues are computed.</p> </li> <li> - <p>The <code>\(k\)</code> eigenvectors are used as âproxyâ data for the original dataset, and fed into k-means clustering. The resulting cluster assignments are transparently passed back to the original data.</p> + <p>The <code class="highlighter-rouge">\(k\)</code> eigenvectors are used as âproxyâ data for the original dataset, and fed into k-means clustering. The resulting cluster assignments are transparently passed back to the original data.</p> </li> </ol> @@ -337,13 +194,13 @@ <h2 id="input">Input</h2> -<p>The input format for the algorithm currently takes the form of a Hadoop-backed affinity matrix in the form of text files. Each line of the text file specifies a single element of the affinity matrix: the row index <code>\(i\)</code>, the column index <code>\(j\)</code>, and the value:</p> +<p>The input format for the algorithm currently takes the form of a Hadoop-backed affinity matrix in the form of text files. Each line of the text file specifies a single element of the affinity matrix: the row index <code class="highlighter-rouge">\(i\)</code>, the column index <code class="highlighter-rouge">\(j\)</code>, and the value:</p> -<p><code>i, j, value</code></p> +<p><code class="highlighter-rouge">i, j, value</code></p> -<p>The affinity matrix is symmetric, and any unspecified <code>\(i, j\)</code> pairs are assumed to be 0 for sparsity. The row and column indices are 0-indexed. Thus, only the non-zero entries of either the upper or lower triangular need be specified.</p> +<p>The affinity matrix is symmetric, and any unspecified <code class="highlighter-rouge">\(i, j\)</code> pairs are assumed to be 0 for sparsity. The row and column indices are 0-indexed. Thus, only the non-zero entries of either the upper or lower triangular need be specified.</p> -<p>The matrix elements specified in the text files are collected into a Mahout <code>DistributedRowMatrix</code>.</p> +<p>The matrix elements specified in the text files are collected into a Mahout <code class="highlighter-rouge">DistributedRowMatrix</code>.</p> <p><strong>(<a href="https://issues.apache.org/jira/browse/MAHOUT-1539";>MAHOUT-1539</a> will allow for the creation of the affinity matrix to occur as part of the core spectral clustering algorithm, as opposed to the current requirement that the user create this matrix themselves and provide it, rather than the original data, to the algorithm)</strong></p> @@ -353,21 +210,22 @@ <p>Spectral clustering can be invoked with the following arguments.</p> -<pre><code>bin/mahout spectralkmeans \ +<div class="highlighter-rouge"><pre class="highlight"><code>bin/mahout spectralkmeans \ -i <affinity matrix directory> \ -o <output working directory> \ -d <number of data points> \ -k <number of clusters AND number of top eigenvectors to use> \ -x <maximum number of k-means iterations> </code></pre> +</div> -<p>The affinity matrix can be contained in a single text file (using the aforementioned one-line-per-entry format) or span many text files <a href="https://issues.apache.org/jira/browse/MAHOUT-978";>per (MAHOUT-978</a>, do not prefix text files with a leading underscore â_â or period â.â). The <code>-d</code> flag is required for the algorithm to know the dimensions of the affinity matrix. <code>-k</code> is the number of top eigenvectors from the normalized graph Laplacian in the SSVD step, and also the number of clusters given to k-means after the SSVD step.</p> +<p>The affinity matrix can be contained in a single text file (using the aforementioned one-line-per-entry format) or span many text files <a href="https://issues.apache.org/jira/browse/MAHOUT-978";>per (MAHOUT-978</a>, do not prefix text files with a leading underscore â_â or period â.â). The <code class="highlighter-rouge">-d</code> flag is required for the algorithm to know the dimensions of the affinity matrix. <code class="highlighter-rouge">-k</code> is the number of top eigenvectors from the normalized graph Laplacian in the SSVD step, and also the number of clusters given to k-means after the SSVD step.</p> <h2 id="example">Example</h2> -<p>To provide a simple example, take the following affinity matrix, contained in a text file called <code>affinity.txt</code>:</p> +<p>To provide a simple example, take the following affinity matrix, contained in a text file called <code class="highlighter-rouge">affinity.txt</code>:</p> -<pre><code>0, 0, 0 +<div class="highlighter-rouge"><pre class="highlight"><code>0, 0, 0 0, 1, 0.8 0, 2, 0.5 1, 0, 0.8 @@ -377,50 +235,45 @@ 2, 1, 0.9 2, 2, 0 </code></pre> +</div> -<p>With this 3-by-3 matrix, <code>-d</code> would be <code>3</code>. Furthermore, since all affinity matrices are assumed to be symmetric, the entries specifying both <code>1, 2, 0.9</code> and <code>2, 1, 0.9</code> are redundant; only one of these is needed. Additionally, any entries that are 0, such as those along the diagonal, also need not be specified at all. They are provided here for completeness.</p> +<p>With this 3-by-3 matrix, <code class="highlighter-rouge">-d</code> would be <code class="highlighter-rouge">3</code>. Furthermore, since all affinity matrices are assumed to be symmetric, the entries specifying both <code class="highlighter-rouge">1, 2, 0.9</code> and <code class="highlighter-rouge">2, 1, 0.9</code> are redundant; only one of these is needed. Additionally, any entries that are 0, such as those along the diagonal, also need not be specified at all. They are provided here for completeness.</p> <p>In general, larger values indicate a stronger âconnectednessâ, whereas smaller values indicate a weaker connectedness. This will vary somewhat depending on the distance function used, though a common one is the <a href="http://en.wikipedia.org/wiki/RBF_kernel";>RBF kernel</a> (used in the above example) which returns values in the range [0, 1], where 0 indicates completely disconnected (or completely dissimilar) and 1 is fully connected (or identical).</p> <p>The call signature with this matrix could be as follows:</p> -<pre><code>bin/mahout spectralkmeans \ +<div class="highlighter-rouge"><pre class="highlight"><code>bin/mahout spectralkmeans \ -i s3://mahout-example/input/ \ -o s3://mahout-example/output/ \ -d 3 \ -k 2 \ -x 10 </code></pre> +</div> <p>There are many other optional arguments, in particular for tweaking the SSVD process (block size, number of power iterations, etc) and the k-means clustering step (distance measure, convergence delta, etc).</p> </div> -</div> - </div> - - -</div> -<div id="footer"> - <div class="container"> - <p>© 2017 Apache Mahout - with help from <a href="http://jekyllbootstrap.com"; target="_blank" title="The Definitive Jekyll Blogging Framework">Jekyll Bootstrap</a> - and <a href="http://getbootstrap.com"; target="_blank">Bootstrap</a> - </p> </div> -</div> - - +</div> + <footer class="footer bg-light"> + <div class="container text-center small"> + Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + </div> +</footer> + <script src="/assets/vendor/jquery/jquery-slim.min.js"></script> + <script src="/assets/vendor/popper/popper.min.js"></script> + <script src="/assets/vendor/bootstrap/js/bootstrap.min.js"></script> + <script src="/assets/header.js"></script> + <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; type="text/javascript"></script> -<!-- Latest compiled and minified JavaScript, requires jQuery 1.x (2.x not supported in IE8) --> -<!-- Placed at the end of the document so the pages load faster --> -<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js";></script> -<script src="/assets/themes/mahout3/js/bootstrap.min.js"></script> </body> -</html> +</html>
