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</ul> - </div> - </div> -</div> + <!--</ul>--> + </div><!-- /.navbar-collapse --> </div> +</nav> + + <div class="container mt-5 pb-4"> + + <div class="row"> - <div class="col-md-8"> - <div class="page-header"> - <h1>Distributed Cholesky QR </h1> - </div> - <h2 id="intro">Intro</h2> + <div class="col-lg-8"> + <h2 id="intro">Intro</h2> <p>Mahout has a distributed implementation of QR decomposition for tall thin matrices<a href="[Mahout Scala and Mahout Spark Bindings for Linear Algebra Subroutines](http://mahout.apache.org/users/sparkbindings/ScalaSparkBindings.pdf)">1</a>.</p> <h2 id="algorithm">Algorithm</h2> -<p>For the classic QR decomposition of the form <code>\(\mathbf{A}=\mathbf{QR},\mathbf{A}\in\mathbb{R}^{m\times n}\)</code> a distributed version is fairly easily achieved if <code>\(\mathbf{A}\)</code> is tall and thin such that <code>\(\mathbf{A}^{\top}\mathbf{A}\)</code> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, only <code>\(\mathbf{A}\)</code> and <code>\(\mathbf{Q}\)</code> are distributed matrices and <code>\(\mathbf{A^{\top}A}\)</code> and <code>\(\mathbf{R}\)</code> are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of <code>\(\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}\)</code>. After that we take <code>\(\mathbf{R}= \mathbf{L}^{\top}\)</code> and <code>\(\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. The latter is easily achieved by multiplying each vertical block of <code>\(\mathbf{A}\)</code> by <code>\(\left(\mathbf{L}^{\top}\right)^{-1}\)</code> . (There is no actual matrix inversion happening).</p> +<p>For the classic QR decomposition of the form <script type="math/tex">\mathbf{A}=\mathbf{QR},\mathbf{A}\in\mathbb{R}^{m\times n}</script> +a distributed version is fairly easily achieved if <script type="math/tex">\mathbf{A}</script> is tall and thin such that +<script type="math/tex">\mathbf{A}^{\top}\mathbf{A}</script> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, +only <script type="math/tex">\mathbf{A}</script> and <script type="math/tex">\mathbf{Q}</script> are distributed matrices and <script type="math/tex">\mathbf{A^{\top}A}</script> and +<script type="math/tex">\mathbf{R}</script> are in-core products. We just compute the in-core version of the Cholesky decomposition +in the form of <script type="math/tex">\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}</script>. After that we take <script type="math/tex">\mathbf{R}= \mathbf{L}^{\top}</script> + and <script type="math/tex">\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}</script>. The latter is easily achieved by multiplying each + vertical block of <script type="math/tex">\mathbf{A}</script> by <script type="math/tex">\left(\mathbf{L}^{\top}\right)^{-1}</script>. (There is no actual matrix inversion + happening).</p> <h2 id="implementation">Implementation</h2> -<p>Mahout <code>dqrThin(...)</code> is implemented in the mahout <code>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> +<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> -<pre><code>def dqrThin[K: ClassTag](A: DrmLike[K], checkRankDeficiency: Boolean = true): (DrmLike[K], Matrix) = { +<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 @@ -335,48 +200,43 @@ Q -> inCoreR } </code></pre> +</div> <h2 id="usage">Usage</h2> -<p>The scala <code>dqrThin(...)</code> method can easily be called in any Spark or H2O application built with the <code>math-scala</code> library and the corresponding <code>Spark</code> or <code>H2O</code> engine module as follows:</p> +<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> -<pre><code>import org.apache.mahout.math._ +<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> - - -</div> -<div id="footer"> - <div class="container"> - <p>© 2017 Apache Mahout - with help from <a href="http://jekyllbootstrap.com"; target="_blank" title="The Definitive Jekyll Blogging Framework">Jekyll Bootstrap</a> - and <a href="http://getbootstrap.com"; target="_blank">Bootstrap</a> - </p> </div> -</div> - - +</div> + <footer class="footer bg-light"> + <div class="container text-center small"> + Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + </div> +</footer> + <script src="/assets/vendor/jquery/jquery-slim.min.js"></script> + <script src="/assets/vendor/popper/popper.min.js"></script> + <script src="/assets/vendor/bootstrap/js/bootstrap.min.js"></script> + <script src="/assets/header.js"></script> + <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML"; type="text/javascript"></script> -<!-- Latest compiled and minified JavaScript, requires jQuery 1.x (2.x not supported in IE8) --> -<!-- Placed at the end of the document so the pages load faster --> -<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js";></script> -<script src="/assets/themes/mahout3/js/bootstrap.min.js"></script> </body> -</html> +</html>
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</ul> - </div> - </div> -</div> + + <!--</ul>--> + </div><!-- /.navbar-collapse --> </div> +</nav> + + <div class="container mt-5 pb-4"> - <div class="col-md-8"> - <div class="page-header"> - <h1>Distributed Stochastic PCA </h1> - </div> - <h2 id="intro">Intro</h2> + <div class="row"> -<p>Mahout has a distributed implementation of Stochastic PCA<a href="Lyubimov and Palumbo, ["Apache Mahout: Beyond MapReduce; Distributed Algorithm Design"](https://www.amazon.com/Apache-Mahout-MapReduce-Dmitriy-Lyubimov/dp/1523775785)">1</a>. This algorithm computes the exact equivalent of Mahoutâs dssvd(<code>\(\mathbf{A-1\mu^\top}\)</code>) by modifying the <code>dssvd</code> algorithm so as to avoid forming <code>\(\mathbf{A-1\mu^\top}\)</code>, which would densify a sparse input. Thus, it is suitable for work with both dense and sparse inputs.</p> + <div class="col-lg-8"> + <h2 id="intro">Intro</h2> + +<p>Mahout has a distributed implementation of Stochastic PCA<a href="Lyubimov and Palumbo, ["Apache Mahout: Beyond MapReduce; Distributed Algorithm Design"](https://www.amazon.com/Apache-Mahout-MapReduce-Dmitriy-Lyubimov/dp/1523775785)">1</a>. This algorithm computes the exact equivalent of Mahoutâs dssvd(<code class="highlighter-rouge">\(\mathbf{A-1\mu^\top}\)</code>) by modifying the <code class="highlighter-rouge">dssvd</code> algorithm so as to avoid forming <code class="highlighter-rouge">\(\mathbf{A-1\mu^\top}\)</code>, which would densify a sparse input. Thus, it is suitable for work with both dense and sparse inputs.</p> <h2 id="algorithm">Algorithm</h2> -<p>Given an <em>m</em> <code>\(\times\)</code> <em>n</em> matrix <code>\(\mathbf{A}\)</code>, a target rank <em>k</em>, and an oversampling parameter <em>p</em>, this procedure computes a <em>k</em>-rank PCA by finding the unknowns in <code>\(\mathbf{Aâ1\mu^\top \approx U\Sigma V^\top}\)</code>:</p> +<p>Given an <em>m</em> <code class="highlighter-rouge">\(\times\)</code> <em>n</em> matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code>, a target rank <em>k</em>, and an oversampling parameter <em>p</em>, this procedure computes a <em>k</em>-rank PCA by finding the unknowns in <code class="highlighter-rouge">\(\mathbf{Aâ1\mu^\top \approx U\Sigma V^\top}\)</code>:</p> <ol> - <li>Create seed for random <em>n</em> <code>\(\times\)</code> <em>(k+p)</em> matrix <code>\(\Omega\)</code>.</li> - <li><code>\(\mathbf{s_\Omega \leftarrow \Omega^\top \mu}\)</code>.</li> - <li><code>\(\mathbf{Y_0 \leftarrow A\Omega â 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)</code>.</li> - <li>Column-orthonormalize <code>\(\mathbf{Y_0} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code>\(\mathbf{Y_0} = \mathbf{QR}\)</code>. Also, <code>\(\mathbf{Q}\in\mathbb{R}^{m\times(k+p)}, \mathbf{R}\in\mathbb{R}^{(k+p)\times(k+p)}\)</code>.</li> - <li><code>\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> - <li><code>\(\mathbf{B_0 \leftarrow Q^\top A: B \in \mathbb{R}^{(k+p)\times n}}\)</code>.</li> - <li><code>\(\mathbf{s_B \leftarrow {B_0}^\top \mu}\)</code>.</li> + <li>Create seed for random <em>n</em> <code class="highlighter-rouge">\(\times\)</code> <em>(k+p)</em> matrix <code class="highlighter-rouge">\(\Omega\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_\Omega \leftarrow \Omega^\top \mu}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{Y_0 \leftarrow A\Omega â 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)</code>.</li> + <li>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y_0} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y_0} = \mathbf{QR}\)</code>. Also, <code class="highlighter-rouge">\(\mathbf{Q}\in\mathbb{R}^{m\times(k+p)}, \mathbf{R}\in\mathbb{R}^{(k+p)\times(k+p)}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{B_0 \leftarrow Q^\top A: B \in \mathbb{R}^{(k+p)\times n}}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_B \leftarrow {B_0}^\top \mu}\)</code>.</li> <li>For <em>i</em> in 1..<em>q</em> repeat (power iterations): <ul> - <li>For <em>j</em> in 1..<em>n</em> apply <code>\(\mathbf{(B_{iâ1})_{âj} \leftarrow (B_{iâ1})_{âj}â\mu_j s_Q}\)</code>.</li> - <li><code>\(\mathbf{Y_i \leftarrow A{B_{iâ1}}^\topâ1(s_Bâ\mu^\top \mu s_Q)^\top}\)</code>.</li> - <li>Column-orthonormalize <code>\(\mathbf{Y_i} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code>\(\mathbf{Y_i = QR}\)</code>.</li> - <li><code>\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> - <li><code>\(\mathbf{B_i \leftarrow Q^\top A}\)</code>.</li> - <li><code>\(\mathbf{s_B \leftarrow {B_i}^\top \mu}\)</code>.</li> + <li>For <em>j</em> in 1..<em>n</em> apply <code class="highlighter-rouge">\(\mathbf{(B_{iâ1})_{âj} \leftarrow (B_{iâ1})_{âj}â\mu_j s_Q}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{Y_i \leftarrow A{B_{iâ1}}^\topâ1(s_Bâ\mu^\top \mu s_Q)^\top}\)</code>.</li> + <li>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y_i} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y_i = QR}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{B_i \leftarrow Q^\top A}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_B \leftarrow {B_i}^\top \mu}\)</code>.</li> </ul> </li> - <li>Let <code>\(\mathbf{C \triangleq s_Q {s_B}^\top}\)</code>. <code>\(\mathbf{M \leftarrow B_q {B_q}^\top â C â C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)</code>.</li> - <li>Compute an eigensolution of the small symmetric <code>\(\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^{(k+p)\times(k+p)}}\)</code>.</li> - <li>The singular values <code>\(\Sigma = \Lambda^{\circ 0.5}\)</code>, or, in other words, <code>\(\mathbf{\sigma_i= \sqrt{\lambda_i}}\)</code>.</li> - <li>If needed, compute <code>\(\mathbf{U = Q\hat{U}}\)</code>.</li> - <li>If needed, compute <code>\(\mathbf{V = B^\top \hat{U} \Sigma^{â1}}\)</code>.</li> - <li>If needed, items converted to the PCA space can be computed as <code>\(\mathbf{U\Sigma}\)</code>.</li> + <li>Let <code class="highlighter-rouge">\(\mathbf{C \triangleq s_Q {s_B}^\top}\)</code>. <code class="highlighter-rouge">\(\mathbf{M \leftarrow B_q {B_q}^\top â C â C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)</code>.</li> + <li>Compute an eigensolution of the small symmetric <code class="highlighter-rouge">\(\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^{(k+p)\times(k+p)}}\)</code>.</li> + <li>The singular values <code class="highlighter-rouge">\(\Sigma = \Lambda^{\circ 0.5}\)</code>, or, in other words, <code class="highlighter-rouge">\(\mathbf{\sigma_i= \sqrt{\lambda_i}}\)</code>.</li> + <li>If needed, compute <code class="highlighter-rouge">\(\mathbf{U = Q\hat{U}}\)</code>.</li> + <li>If needed, compute <code class="highlighter-rouge">\(\mathbf{V = B^\top \hat{U} \Sigma^{â1}}\)</code>.</li> + <li>If needed, items converted to the PCA space can be computed as <code class="highlighter-rouge">\(\mathbf{U\Sigma}\)</code>.</li> </ol> <h2 id="implementation">Implementation</h2> -<p>Mahout <code>dspca(...)</code> is implemented in the mahout <code>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> +<p>Mahout <code class="highlighter-rouge">dspca(...)</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> -<pre><code>def dspca[K](drmA: DrmLike[K], k: Int, p: Int = 15, q: Int = 0): +<div class="highlighter-rouge"><pre class="highlight"><code>def dspca[K](drmA: DrmLike[K], k: Int, p: Int = 15, q: Int = 0): (DrmLike[K], DrmLike[Int], Vector) = { // Some mapBlock() calls need it @@ -462,17 +319,19 @@ (drmU(::, 0 until k), drmV(::, 0 until k), s(0 until k)) } </code></pre> +</div> <h2 id="usage">Usage</h2> -<p>The scala <code>dspca(...)</code> method can easily be called in any Spark, Flink, or H2O application built with the <code>math-scala</code> library and the corresponding <code>Spark</code>, <code>Flink</code>, or <code>H2O</code> engine module as follows:</p> +<p>The scala <code class="highlighter-rouge">dspca(...)</code> method can easily be called in any Spark, Flink, or H2O application built with the <code class="highlighter-rouge">math-scala</code> library and the corresponding <code class="highlighter-rouge">Spark</code>, <code class="highlighter-rouge">Flink</code>, or <code class="highlighter-rouge">H2O</code> engine module as follows:</p> -<pre><code>import org.apache.mahout.math._ +<div class="highlighter-rouge"><pre class="highlight"><code>import org.apache.mahout.math._ import decompositions._ import drm._ val (drmU, drmV, s) = dspca(drmA, k=200, q=1) </code></pre> +</div> <p>Note the parameter is optional and its default value is zero.</p> @@ -480,32 +339,25 @@ val (drmU, drmV, s) = dspca(drmA, k=200, q=1) </div> -</div> - 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