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author | Michael Armbrust <marmbrus@apache.org> | 2016-01-04 17:53:21 +0000 |
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committer | Michael Armbrust <marmbrus@apache.org> | 2016-01-04 17:53:21 +0000 |
commit | 0d824b0442707b6842ef19a589cb8a2af997273b (patch) | |
tree | 4ac1758b478ee6cb74382733fd3eb3157b6c3161 /site/docs/1.6.0/mllib-dimensionality-reduction.html | |
parent | ebcd167cbcb4ba33d8926b66850eff0cce12d365 (diff) | |
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</li> + + +<ul> + + <li> + <a href="mllib-dimensionality-reduction.html#singular-value-decomposition-svd"> + + singular value decomposition (SVD) + + </a> + </li> + + + <li> + <a href="mllib-dimensionality-reduction.html#principal-component-analysis-pca"> + + principal component analysis (PCA) + + </a> + </li> + + +</ul> + + + + <li> + <a href="mllib-feature-extraction.html"> + + Feature extraction and transformation + + </a> + </li> + + + <li> + <a href="mllib-frequent-pattern-mining.html"> + + Frequent pattern mining + + </a> + </li> + + + <li> + <a href="mllib-evaluation-metrics.html"> + + Evaluation metrics + + </a> + </li> + + + <li> + <a href="mllib-pmml-model-export.html"> + + PMML model export + + </a> + </li> + + + <li> + <a href="mllib-optimization.html"> + + Optimization (developer) + + </a> + </li> + + +</ul> + + </div> +</div> + <input id="nav-trigger" class="nav-trigger" checked type="checkbox"> + <label for="nav-trigger"></label> + <div class="content-with-sidebar" id="content"> + + <h1 class="title">Dimensionality Reduction - spark.mllib</h1> + + + <ul id="markdown-toc"> + <li><a href="#singular-value-decomposition-svd" id="markdown-toc-singular-value-decomposition-svd">Singular value decomposition (SVD)</a> <ul> + <li><a href="#performance" id="markdown-toc-performance">Performance</a></li> + <li><a href="#svd-example" id="markdown-toc-svd-example">SVD Example</a></li> + </ul> + </li> + <li><a href="#principal-component-analysis-pca" id="markdown-toc-principal-component-analysis-pca">Principal component analysis (PCA)</a></li> +</ul> + +<p><a href="http://en.wikipedia.org/wiki/Dimensionality_reduction">Dimensionality reduction</a> is the process +of reducing the number of variables under consideration. +It can be used to extract latent features from raw and noisy features +or compress data while maintaining the structure. +<code>spark.mllib</code> provides support for dimensionality reduction on the <a href="mllib-data-types.html#rowmatrix">RowMatrix</a> class.</p> + +<h2 id="singular-value-decomposition-svd">Singular value decomposition (SVD)</h2> + +<p><a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Singular value decomposition (SVD)</a> +factorizes a matrix into three matrices: $U$, $\Sigma$, and $V$ such that</p> + +<p><code>\[ +A = U \Sigma V^T, +\]</code></p> + +<p>where</p> + +<ul> + <li>$U$ is an orthonormal matrix, whose columns are called left singular vectors,</li> + <li>$\Sigma$ is a diagonal matrix with non-negative diagonals in descending order, +whose diagonals are called singular values,</li> + <li>$V$ is an orthonormal matrix, whose columns are called right singular vectors.</li> +</ul> + +<p>For large matrices, usually we don’t need the complete factorization but only the top singular +values and its associated singular vectors. This can save storage, de-noise +and recover the low-rank structure of the matrix.</p> + +<p>If we keep the top $k$ singular values, then the dimensions of the resulting low-rank matrix will be:</p> + +<ul> + <li><code>$U$</code>: <code>$m \times k$</code>,</li> + <li><code>$\Sigma$</code>: <code>$k \times k$</code>,</li> + <li><code>$V$</code>: <code>$n \times k$</code>.</li> +</ul> + +<h3 id="performance">Performance</h3> +<p>We assume $n$ is smaller than $m$. The singular values and the right singular vectors are derived +from the eigenvalues and the eigenvectors of the Gramian matrix $A^T A$. The matrix +storing the left singular vectors $U$, is computed via matrix multiplication as +$U = A (V S^{-1})$, if requested by the user via the computeU parameter. +The actual method to use is determined automatically based on the computational cost:</p> + +<ul> + <li>If $n$ is small ($n < 100$) or $k$ is large compared with $n$ ($k > n / 2$), we compute the Gramian matrix +first and then compute its top eigenvalues and eigenvectors locally on the driver. +This requires a single pass with $O(n^2)$ storage on each executor and on the driver, and +$O(n^2 k)$ time on the driver.</li> + <li>Otherwise, we compute $(A^T A) v$ in a distributive way and send it to +<a href="http://www.caam.rice.edu/software/ARPACK/">ARPACK</a> to +compute $(A^T A)$’s top eigenvalues and eigenvectors on the driver node. This requires $O(k)$ +passes, $O(n)$ storage on each executor, and $O(n k)$ storage on the driver.</li> +</ul> + +<h3 id="svd-example">SVD Example</h3> + +<p><code>spark.mllib</code> provides SVD functionality to row-oriented matrices, provided in the +<a href="mllib-data-types.html#rowmatrix">RowMatrix</a> class.</p> + +<div class="codetabs"> +<div data-lang="scala"> + <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.SingularValueDecomposition"><code>SingularValueDecomposition</code> Scala docs</a> for details on the API.</p> + + <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.SingularValueDecomposition</span> + +<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="o">...</span> + +<span class="c1">// Compute the top 20 singular values and corresponding singular vectors.</span> +<span class="k">val</span> <span class="n">svd</span><span class="k">:</span> <span class="kt">SingularValueDecomposition</span><span class="o">[</span><span class="kt">RowMatrix</span>, <span class="kt">Matrix</span><span class="o">]</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">computeSVD</span><span class="o">(</span><span class="mi">20</span><span class="o">,</span> <span class="n">computeU</span> <span class="k">=</span> <span class="kc">true</span><span class="o">)</span> +<span class="k">val</span> <span class="n">U</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="n">U</span> <span class="c1">// The U factor is a RowMatrix.</span> +<span class="k">val</span> <span class="n">s</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="n">s</span> <span class="c1">// The singular values are stored in a local dense vector.</span> +<span class="k">val</span> <span class="n">V</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="n">V</span> <span class="c1">// The V factor is a local dense matrix.</span></code></pre></div> + + <p>The same code applies to <code>IndexedRowMatrix</code> if <code>U</code> is defined as an +<code>IndexedRowMatrix</code>.</p> + </div> +<div data-lang="java"> + <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/SingularValueDecomposition.html"><code>SingularValueDecomposition</code> Java docs</a> for details on the API.</p> + + <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">java.util.LinkedList</span><span class="o">;</span> + +<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.*</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.SingularValueDecomposition</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.rdd.RDD</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.SparkConf</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.SparkContext</span><span class="o">;</span> + +<span class="kd">public</span> <span class="kd">class</span> <span class="nc">SVD</span> <span class="o">{</span> + <span class="kd">public</span> <span class="kd">static</span> <span class="kt">void</span> <span class="nf">main</span><span class="o">(</span><span class="n">String</span><span class="o">[]</span> <span class="n">args</span><span class="o">)</span> <span class="o">{</span> + <span class="n">SparkConf</span> <span class="n">conf</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">SparkConf</span><span class="o">().</span><span class="na">setAppName</span><span class="o">(</span><span class="s">"SVD Example"</span><span class="o">);</span> + <span class="n">SparkContext</span> <span class="n">sc</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">SparkContext</span><span class="o">(</span><span class="n">conf</span><span class="o">);</span> + + <span class="kt">double</span><span class="o">[][]</span> <span class="n">array</span> <span class="o">=</span> <span class="o">...</span> + <span class="n">LinkedList</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">rowsList</span> <span class="o">=</span> <span class="k">new</span> <span class="n">LinkedList</span><span class="o"><</span><span class="n">Vector</span><span class="o">>();</span> + <span class="k">for</span> <span class="o">(</span><span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span><span class="o">;</span> <span class="n">i</span> <span class="o"><</span> <span class="n">array</span><span class="o">.</span><span class="na">length</span><span class="o">;</span> <span class="n">i</span><span class="o">++)</span> <span class="o">{</span> + <span class="n">Vector</span> <span class="n">currentRow</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="n">array</span><span class="o">[</span><span class="n">i</span><span class="o">]);</span> + <span class="n">rowsList</span><span class="o">.</span><span class="na">add</span><span class="o">(</span><span class="n">currentRow</span><span class="o">);</span> + <span class="o">}</span> + <span class="n">JavaRDD</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">rows</span> <span class="o">=</span> <span class="n">JavaSparkContext</span><span class="o">.</span><span class="na">fromSparkContext</span><span class="o">(</span><span class="n">sc</span><span class="o">).</span><span class="na">parallelize</span><span class="o">(</span><span class="n">rowsList</span><span class="o">);</span> + + <span class="c1">// Create a RowMatrix from JavaRDD<Vector>.</span> + <span class="n">RowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span> + + <span class="c1">// Compute the top 4 singular values and corresponding singular vectors.</span> + <span class="n">SingularValueDecomposition</span><span class="o"><</span><span class="n">RowMatrix</span><span class="o">,</span> <span class="n">Matrix</span><span class="o">></span> <span class="n">svd</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">computeSVD</span><span class="o">(</span><span class="mi">4</span><span class="o">,</span> <span class="kc">true</span><span class="o">,</span> <span class="mf">1.0</span><span class="n">E</span><span class="o">-</span><span class="mi">9</span><span class="n">d</span><span class="o">);</span> + <span class="n">RowMatrix</span> <span class="n">U</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="na">U</span><span class="o">();</span> + <span class="n">Vector</span> <span class="n">s</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="na">s</span><span class="o">();</span> + <span class="n">Matrix</span> <span class="n">V</span> <span class="o">=</span> <span class="n">svd</span><span class="o">.</span><span class="na">V</span><span class="o">();</span> + <span class="o">}</span> +<span class="o">}</span></code></pre></div> + + <p>The same code applies to <code>IndexedRowMatrix</code> if <code>U</code> is defined as an +<code>IndexedRowMatrix</code>.</p> + + <p>In order to run the above application, follow the instructions +provided in the <a href="quick-start.html#self-contained-applications">Self-Contained +Applications</a> section of the Spark +quick-start guide. Be sure to also include <em>spark-mllib</em> to your build file as +a dependency.</p> + + </div> +</div> + +<h2 id="principal-component-analysis-pca">Principal component analysis (PCA)</h2> + +<p><a href="http://en.wikipedia.org/wiki/Principal_component_analysis">Principal component analysis (PCA)</a> is a +statistical method to find a rotation such that the first coordinate has the largest variance +possible, and each succeeding coordinate in turn has the largest variance possible. The columns of +the rotation matrix are called principal components. PCA is used widely in dimensionality reduction.</p> + +<p><code>spark.mllib</code> supports PCA for tall-and-skinny matrices stored in row-oriented format and any Vectors.</p> + +<div class="codetabs"> +<div data-lang="scala"> + + <p>The following code demonstrates how to compute principal components on a <code>RowMatrix</code> +and use them to project the vectors into a low-dimensional space.</p> + + <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code> Scala docs</a> for details on the API.</p> + + <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span> + +<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="o">...</span> + +<span class="c1">// Compute the top 10 principal components.</span> +<span class="k">val</span> <span class="n">pc</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">computePrincipalComponents</span><span class="o">(</span><span class="mi">10</span><span class="o">)</span> <span class="c1">// Principal components are stored in a local dense matrix.</span> + +<span class="c1">// Project the rows to the linear space spanned by the top 10 principal components.</span> +<span class="k">val</span> <span class="n">projected</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">multiply</span><span class="o">(</span><span class="n">pc</span><span class="o">)</span></code></pre></div> + + <p>The following code demonstrates how to compute principal components on source vectors +and use them to project the vectors into a low-dimensional space while keeping associated labels:</p> + + <p>Refer to the <a href="api/scala/index.html#org.apache.spark.mllib.feature.PCA"><code>PCA</code> Scala docs</a> for details on the API.</p> + + <div class="highlight"><pre><code class="language-scala" data-lang="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span> +<span class="k">import</span> <span class="nn">org.apache.spark.mllib.feature.PCA</span> + +<span class="k">val</span> <span class="n">data</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">LabeledPoint</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> + +<span class="c1">// Compute the top 10 principal components.</span> +<span class="k">val</span> <span class="n">pca</span> <span class="k">=</span> <span class="k">new</span> <span class="nc">PCA</span><span class="o">(</span><span class="mi">10</span><span class="o">).</span><span class="n">fit</span><span class="o">(</span><span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="k">_</span><span class="o">.</span><span class="n">features</span><span class="o">))</span> + +<span class="c1">// Project vectors to the linear space spanned by the top 10 principal components, keeping the label</span> +<span class="k">val</span> <span class="n">projected</span> <span class="k">=</span> <span class="n">data</span><span class="o">.</span><span class="n">map</span><span class="o">(</span><span class="n">p</span> <span class="k">=></span> <span class="n">p</span><span class="o">.</span><span class="n">copy</span><span class="o">(</span><span class="n">features</span> <span class="k">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">transform</span><span class="o">(</span><span class="n">p</span><span class="o">.</span><span class="n">features</span><span class="o">)))</span></code></pre></div> + + </div> + +<div data-lang="java"> + + <p>The following code demonstrates how to compute principal components on a <code>RowMatrix</code> +and use them to project the vectors into a low-dimensional space. +The number of columns should be small, e.g, less than 1000.</p> + + <p>Refer to the <a href="api/java/org/apache/spark/mllib/linalg/distributed/RowMatrix.html"><code>RowMatrix</code> Java docs</a> for details on the API.</p> + + <div class="highlight"><pre><code class="language-java" data-lang="java"><span class="kn">import</span> <span class="nn">java.util.LinkedList</span><span class="o">;</span> + +<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.*</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.rdd.RDD</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.SparkConf</span><span class="o">;</span> +<span class="kn">import</span> <span class="nn">org.apache.spark.SparkContext</span><span class="o">;</span> + +<span class="kd">public</span> <span class="kd">class</span> <span class="nc">PCA</span> <span class="o">{</span> + <span class="kd">public</span> <span class="kd">static</span> <span class="kt">void</span> <span class="nf">main</span><span class="o">(</span><span class="n">String</span><span class="o">[]</span> <span class="n">args</span><span class="o">)</span> <span class="o">{</span> + <span class="n">SparkConf</span> <span class="n">conf</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">SparkConf</span><span class="o">().</span><span class="na">setAppName</span><span class="o">(</span><span class="s">"PCA Example"</span><span class="o">);</span> + <span class="n">SparkContext</span> <span class="n">sc</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">SparkContext</span><span class="o">(</span><span class="n">conf</span><span class="o">);</span> + + <span class="kt">double</span><span class="o">[][]</span> <span class="n">array</span> <span class="o">=</span> <span class="o">...</span> + <span class="n">LinkedList</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">rowsList</span> <span class="o">=</span> <span class="k">new</span> <span class="n">LinkedList</span><span class="o"><</span><span class="n">Vector</span><span class="o">>();</span> + <span class="k">for</span> <span class="o">(</span><span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">0</span><span class="o">;</span> <span class="n">i</span> <span class="o"><</span> <span class="n">array</span><span class="o">.</span><span class="na">length</span><span class="o">;</span> <span class="n">i</span><span class="o">++)</span> <span class="o">{</span> + <span class="n">Vector</span> <span class="n">currentRow</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="n">array</span><span class="o">[</span><span class="n">i</span><span class="o">]);</span> + <span class="n">rowsList</span><span class="o">.</span><span class="na">add</span><span class="o">(</span><span class="n">currentRow</span><span class="o">);</span> + <span class="o">}</span> + <span class="n">JavaRDD</span><span class="o"><</span><span class="n">Vector</span><span class="o">></span> <span class="n">rows</span> <span class="o">=</span> <span class="n">JavaSparkContext</span><span class="o">.</span><span class="na">fromSparkContext</span><span class="o">(</span><span class="n">sc</span><span class="o">).</span><span class="na">parallelize</span><span class="o">(</span><span class="n">rowsList</span><span class="o">);</span> + + <span class="c1">// Create a RowMatrix from JavaRDD<Vector>.</span> + <span class="n">RowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="nf">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span> + + <span class="c1">// Compute the top 3 principal components.</span> + <span class="n">Matrix</span> <span class="n">pc</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">computePrincipalComponents</span><span class="o">(</span><span class="mi">3</span><span class="o">);</span> + <span class="n">RowMatrix</span> <span class="n">projected</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">multiply</span><span class="o">(</span><span class="n">pc</span><span class="o">);</span> + <span class="o">}</span> +<span class="o">}</span></code></pre></div> + + </div> +</div> + +<p>In order to run the above application, follow the instructions +provided in the <a href="quick-start.html#self-contained-applications">Self-Contained Applications</a> +section of the Spark +quick-start guide. Be sure to also include <em>spark-mllib</em> to your build file as +a dependency.</p> + + + </div> + + <!-- /container --> + </div> + + <script src="js/vendor/jquery-1.8.0.min.js"></script> + <script src="js/vendor/bootstrap.min.js"></script> + <script src="js/vendor/anchor.min.js"></script> + <script src="js/main.js"></script> + + <!-- MathJax Section --> + <script type="text/x-mathjax-config"> + MathJax.Hub.Config({ + TeX: { equationNumbers: { autoNumber: "AMS" } } + }); + </script> + <script> + // Note that we load MathJax this way to work with local file (file://), HTTP and HTTPS. + // We could use "//cdn.mathjax...", but that won't support "file://". + (function(d, script) { + script = d.createElement('script'); + script.type = 'text/javascript'; + script.async = true; + script.onload = function(){ + MathJax.Hub.Config({ + tex2jax: { + inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], + displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], + processEscapes: true, + skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'] + } + }); + }; + script.src = ('https:' == document.location.protocol ? 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