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class="highlighter-rouge">\(\mathbf{A}^{\top}\mathbf{A}\)</code> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, only <code class="highlighter-rouge">\(\mathbf{A}\)</code> and <code class="highlighter-rouge">\(\mathbf{Q}\)</code> are distributed matricies and <code class="highlighter-rouge">\(\mathbf{A^{\top}A}\)</code> and <code class="highlighter-rouge">\(\mathbf{R}\)</code> are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of <code class="highlighter-rouge">\(\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}\)</code>. After that we take <code class="highlighter-rouge">\(\mathbf{R}= \mathbf{L}^{\top}\)</co de> and <code class="highlighter-rouge">\(\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. The latter is easily achieved by multiplying each verticle block of <code class="highlighter-rouge">\(\mathbf{A}\)</code> by <code class="highlighter-rouge">\(\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. (There is no actual matrix inversion happening).</p> + +<h2 id="implementation">Implementation</h2> + +<p>Mahout <code class="highlighter-rouge">dqrThin(...)</code> is implemented in the mahout <code class="highlighter-rouge">math-scala</code> algebraic optimizer which translates Mahoutâs R-like linear algebra operators into a physical plan for both Spark and H2O distributed engines.</p> + +<div class="highlighter-rouge"><pre class="highlight"><code>def dqrThin[K: ClassTag](A: DrmLike[K], checkRankDeficiency: Boolean = true): (DrmLike[K], Matrix) = { + if (drmA.ncol > 5000) + log.warn("A is too fat. A'A must fit in memory and easily broadcasted.") + implicit val ctx = drmA.context + val AtA = (drmA.t %*% drmA).checkpoint() + val inCoreAtA = AtA.collect + val ch = chol(inCoreAtA) + val inCoreR = (ch.getL cloned) t + if (checkRankDeficiency && !ch.isPositiveDefinite) + throw new IllegalArgumentException("R is rank-deficient.") + val bcastAtA = sc.broadcast(inCoreAtA) + val Q = A.mapBlock() { + case (keys, block) => keys -> chol(bcastAtA).solveRight(block) + } + Q -> inCoreR +} +</code></pre> +</div> + +<h2 id="usage">Usage</h2> + +<p>The scala <code class="highlighter-rouge">dqrThin(...)</code> method can easily be called in any Spark or H2O application built with the <code class="highlighter-rouge">math-scala</code> library and the corresponding <code class="highlighter-rouge">Spark</code> or <code class="highlighter-rouge">H2O</code> engine module as follows:</p> + +<div class="highlighter-rouge"><pre class="highlight"><code>import org.apache.mahout.math._ +import decompositions._ +import drm._ + +val(drmQ, inCoreR) = dqrThin(drma) +</code></pre> +</div> + +<h2 id="references">References</h2> + + + </div> + </div> +</div> + <footer class="footer" align="center"> + <div class="container"> + <p> + Copyright © 2014-2016 The Apache Software Foundation, Licensed under + the <a href="http://www.apache.org/licenses/LICENSE-2.0";>Apache License, Version 2.0</a>. + <br /> + Apache Mahout, Mahout, Apache, the Apache feather logo, and the elephant rider logo are either registered trademarks or trademarks of <a href="http://www.apache.org/foundation/marks/";>The Apache Software Foundation</a> in the United States and other countries. + </p> + </div> + </footer> + + <script src="/assets/themes/mahout-retro/js/jquery-1.9.1.min.js"></script> + <script src="/assets/themes/mahout-retro/js/bootstrap.min.js"></script> + <script> + (function() { + var cx = '012254517474945470291:vhsfv7eokdc'; + var gcse = document.createElement('script'); + gcse.type = 'text/javascript'; + gcse.async = true; + gcse.src = (document.location.protocol == 'https:' ? 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class="highlighter-rouge">\(\mathbf{A}^{\top}\mathbf{A}\)</code> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, only <code class="highlighter-rouge">\(\mathbf{A}\)</code> and <code class="highlighter-rouge">\(\mathbf{Q}\)</code> are distributed matrices and <code class="highlighter-rouge">\(\mathbf{A^{\top}A}\)</code> and <code class="highlighter-rouge">\(\mathbf{R}\)</code> are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of <code class="highlighter-rouge">\(\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}\)</code>. After that we take <code class="highlighter-rouge">\(\mathbf{R}= \mathbf{L}^{\top}\)</cod e> and <code class="highlighter-rouge">\(\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. The latter is easily achieved by multiplying each vertical block of <code class="highlighter-rouge">\(\mathbf{A}\)</code> by <code class="highlighter-rouge">\(\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. (There is no actual matrix inversion happening).</p> + +<h2 id="implementation">Implementation</h2> + +<p>Mahout <code class="highlighter-rouge">dqrThin(...)</code> is implemented in the mahout <code class="highlighter-rouge">math-scala</code> algebraic optimizer which translates Mahoutâs R-like linear algebra operators into a physical plan for both Spark and H2O distributed engines.</p> + +<div class="highlighter-rouge"><pre class="highlight"><code>def dqrThin[K: ClassTag](A: DrmLike[K], checkRankDeficiency: Boolean = true): (DrmLike[K], Matrix) = { + if (drmA.ncol > 5000) + log.warn("A is too fat. A'A must fit in memory and easily broadcasted.") + implicit val ctx = drmA.context + val AtA = (drmA.t %*% drmA).checkpoint() + val inCoreAtA = AtA.collect + val ch = chol(inCoreAtA) + val inCoreR = (ch.getL cloned) t + if (checkRankDeficiency && !ch.isPositiveDefinite) + throw new IllegalArgumentException("R is rank-deficient.") + val bcastAtA = sc.broadcast(inCoreAtA) + val Q = A.mapBlock() { + case (keys, block) => keys -> chol(bcastAtA).solveRight(block) + } + Q -> inCoreR +} +</code></pre> +</div> + +<h2 id="usage">Usage</h2> + +<p>The scala <code class="highlighter-rouge">dqrThin(...)</code> method can easily be called in any Spark or H2O application built with the <code class="highlighter-rouge">math-scala</code> library and the corresponding <code class="highlighter-rouge">Spark</code> or <code class="highlighter-rouge">H2O</code> engine module as follows:</p> + +<div class="highlighter-rouge"><pre class="highlight"><code>import org.apache.mahout.math._ +import decompositions._ +import drm._ + +val(drmQ, inCoreR) = dqrThin(drma) +</code></pre> +</div> + +<h2 id="references">References</h2> + + + </div> + </div> +</div> + <footer class="footer" align="center"> + <div class="container"> + <p> + Copyright © 2014-2016 The Apache Software Foundation, Licensed under + the <a href="http://www.apache.org/licenses/LICENSE-2.0";>Apache License, Version 2.0</a>. + <br /> + Apache Mahout, Mahout, Apache, the Apache feather logo, and the elephant rider logo are either registered trademarks or trademarks of <a href="http://www.apache.org/foundation/marks/";>The Apache Software Foundation</a> in the United States and other countries. + </p> + </div> + </footer> + + <script src="/assets/themes/mahout-retro/js/jquery-1.9.1.min.js"></script> + <script src="/assets/themes/mahout-retro/js/bootstrap.min.js"></script> + <script> + (function() { + var cx = '012254517474945470291:vhsfv7eokdc'; + var gcse = document.createElement('script'); + gcse.type = 'text/javascript'; + gcse.async = true; + gcse.src = (document.location.protocol == 'https:' ? 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