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author | Patrick Wendell <pwendell@apache.org> | 2014-07-11 17:23:23 +0000 |
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committer | Patrick Wendell <pwendell@apache.org> | 2014-07-11 17:23:23 +0000 |
commit | 0beac4e243f85e71554fe04093b09eb1745fea82 (patch) | |
tree | bc20d10426c5d57e2f189305865dc2bbec447923 /site/docs/1.0.1/mllib-dimensionality-reduction.html | |
parent | ddec2123ba6ab95543d1b250d4f20fb811c48f09 (diff) | |
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Updating docs for 1.0.1 release
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diff --git a/site/docs/1.0.1/mllib-dimensionality-reduction.html b/site/docs/1.0.1/mllib-dimensionality-reduction.html new file mode 100644 index 000000000..153b8fb7a --- /dev/null +++ b/site/docs/1.0.1/mllib-dimensionality-reduction.html @@ -0,0 +1,256 @@ +<!DOCTYPE html> +<!--[if lt IE 7]> <html class="no-js lt-ie9 lt-ie8 lt-ie7"> <![endif]--> +<!--[if IE 7]> <html class="no-js lt-ie9 lt-ie8"> <![endif]--> +<!--[if IE 8]> <html class="no-js lt-ie9"> <![endif]--> +<!--[if gt IE 8]><!--> <html class="no-js"> <!--<![endif]--> + <head> + <meta charset="utf-8"> + <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> + <title>Dimensionality Reduction - MLlib - Spark 1.0.1 Documentation</title> + <meta name="description" content=""> + + + + <link rel="stylesheet" href="css/bootstrap.min.css"> + <style> + body { + padding-top: 60px; + padding-bottom: 40px; + } + </style> + <meta name="viewport" content="width=device-width"> + <link rel="stylesheet" href="css/bootstrap-responsive.min.css"> + <link rel="stylesheet" href="css/main.css"> + + <script src="js/vendor/modernizr-2.6.1-respond-1.1.0.min.js"></script> + + <link rel="stylesheet" href="css/pygments-default.css"> + + + <!-- Google analytics script --> + <script type="text/javascript"> + var _gaq = _gaq || []; + _gaq.push(['_setAccount', 'UA-32518208-1']); + _gaq.push(['_trackPageview']); + + (function() { + var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; + ga.src = ('https:' == document.location.protocol ? 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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’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> + + </div> +</div> + + + </div> <!-- /container --> + + <script src="js/vendor/jquery-1.8.0.min.js"></script> + <script src="js/vendor/bootstrap.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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