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authorPatrick Wendell <pwendell@apache.org>2014-07-11 17:23:23 +0000
committerPatrick Wendell <pwendell@apache.org>2014-07-11 17:23:23 +0000
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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">Singular value decomposition (SVD)</a></li>
+ <li><a href="#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 is used to extract latent features from raw and noisy features,
+or compress data while maintaining the structure.
+In this release, we provide preliminary support for dimensionality reduction on tall-and-skinny matrices.</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&#8217;t need the complete factorization but only the top singular
+values and its associated singular vectors. This can save storage, and more importantly, 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 return 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>
+
+<p>In this release, we provide SVD computation to row-oriented matrices that have only a few columns,
+say, less than $1000$, but many rows, which we call <em>tall-and-skinny</em>.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <div class="highlight"><pre><code class="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>
+
+ </div>
+Same code applies to `IndexedRowMatrix`.
+The only difference that the `U` matrix becomes an `IndexedRowMatrix`.
+</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>In this release, we implement PCA for tall-and-skinny matrices stored in row-oriented format.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>The following code demonstrates how to compute principal components on a tall-and-skinny <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="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>
+
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