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>\(\mathbf{Y_0 \leftarrow A\Omega â 1 {s_\Omega}^\top, Y \in
\mathbb{R}^{m\times(k+p)}}\)</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>\(s_Q \leftarrow Q^\top 1\)</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>\(apply(B_(iâ1))_(âj) \leftarrow
(B_(iâ1))_(âj)â\mu_j s_Q\)</code>.</li>
-<li><code>\(\mathbf{Y_i) \leftarrow
\mathbf{(AB_(iâ1)^\top)â1(s_Bâ\mu^\top \mu s_Q^\top)}\)</code>.</li>
+<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>\(\mathbf{Y_i \leftarrow (A{B_{iâ1}}^\top)â1(s_Bâ\mu^\top \mu
s_Q)^\top)}\)</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><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>\(\mathbf{M \leftarrow B_q (B_q)^\top â C â C^\top + \mu^\top \mu s_Q
(s_Q)^\top}\)</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>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>. Another way is `(\mathbf{V = A^\top
U\Sigma^(â1)})1.</li>
+<li>If needed, compute <code>\(\mathbf{V = B^\top \hat{U}
\Sigma^{â1}}\)</code>. Another way is `(\mathbf{V = A^\top
U\Sigma^{â1}}).</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">¶</a></h2>
