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From build...@apache.org
Subject svn commit: r1006168 - in /websites/staging/mahout/trunk/content: ./ users/algorithms/d-spca.html
Date Fri, 03 Feb 2017 23:35:18 GMT
Author: buildbot
Date: Fri Feb  3 23:35:18 2017
New Revision: 1006168

Log:
Staging update by buildbot for mahout

Modified:
    websites/staging/mahout/trunk/content/   (props changed)
    websites/staging/mahout/trunk/content/users/algorithms/d-spca.html

Propchange: websites/staging/mahout/trunk/content/
------------------------------------------------------------------------------
--- cms:source-revision (original)
+++ cms:source-revision Fri Feb  3 23:35:18 2017
@@ -1 +1 @@
-1781619
+1781623

Modified: websites/staging/mahout/trunk/content/users/algorithms/d-spca.html
==============================================================================
--- websites/staging/mahout/trunk/content/users/algorithms/d-spca.html (original)
+++ websites/staging/mahout/trunk/content/users/algorithms/d-spca.html Fri Feb  3 23:35:18
2017
@@ -288,24 +288,24 @@ h2:hover > .headerlink, h3:hover > .head
 <li>Create seed for random <em>n</em> <code>\(\times\)</code>
<em>(k+p)</em> matrix <code>\(\Omega\)</code>.</li>
 <li><code>\(s_\Omega \leftarrow \Omega^\top \mu\)</code>.</li>
 <li><code>\(\mathbf{Y_0 \leftarrow A\Omega − 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)</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>\(s_Q \leftarrow Q^\top 1\)</code>.</li>
-<li><code>\(\mathbf{B_0} \leftarrow Q^\top A: B \in \mathbb{R}^((k+p)\times n)\)</code>.</li>
-<li><code>\(s_B \leftarrow (B_0)^\top \mu\)</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>For <em>i</em> in 1..<em>q</em> repeat (power iterations):<ul>
-<li>For <em>j</em> in 1..<em>n</em> <code>\(apply(B_(i−1))_(∗j)
\leftarrow (B_(i−1))_(∗j)−\mu_j s_Q\)</code>.</li>
-<li><code>\(\mathbf{Y_i) \leftarrow \mathbf{(AB_(i−1)^\top)−1(s_B−\mu^\top
\mu s_Q^\top)}\)</code>.</li>
+<li>For <em>j</em> in 1..<em>n</em> apply <code>\(\mathbf{(B_{i−1})_{∗j}
\leftarrow (B_{i−1})_{∗j}−\mu_j s_Q}\)</code>.</li>
+<li><code>\(\mathbf{Y_i \leftarrow (A{B_{i−1}}^\top)−1(s_B−\mu^\top
\mu s_Q)^\top)}\)</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><code>\(\mathbf{s_B \leftarrow {B_i}^\top \mu}\)</code>.</li>
 </ul>
 </li>
-<li>Let <code>\(\mathbf{C \triangleq s_Q (s_B)^\top}\)</code>. <code>\(\mathbf{M
\leftarrow B_q (B_q)^\top − C − C^\top + \mu^\top \mu s_Q (s_Q)^\top}\)</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>Let <code>\(\mathbf{C \triangleq s_Q {s_B}^\top}\)</code>. <code>\(\mathbf{M
\leftarrow B_q {B_q}^\top − C − C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)</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>\(\mathbf{V = B^\top \hat{U} \Sigma^(−1)}\)</code>.
Another way is `(\mathbf{V = A^\top U\Sigma^(−1)})1.</li>
+<li>If needed, compute <code>\(\mathbf{V = B^\top \hat{U} \Sigma^{−1}}\)</code>.
Another way is `(\mathbf{V = A^\top U\Sigma^{−1}}).</li>
 <li>If needed, items converted to the PCA space can be computed as <code>\(\mathbf{U\Sigma}\)</code>.</li>
 </ol>
 <h2 id="implementation">Implementation<a class="headerlink" href="#implementation"
title="Permanent link">&para;</a></h2>



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