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| |
| <h1 class="title"><a href="mllib-guide.html">MLlib</a> - Dimensionality Reduction</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. |
| MLlib 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>MLlib 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"> |
| |
| <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"> |
| |
| <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>MLlib 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> |
| |
| <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> |
| |
| <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> |
| |
| <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> |
| |
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