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From rawkintr...@apache.org
Subject [03/51] [partial] mahout git commit: WEBSITE rename old-site
Date Sat, 29 Apr 2017 05:08:24 GMT
http://git-wip-us.apache.org/repos/asf/mahout/blob/0e718ec9/website/oldsite/_site/users/algorithms/d-spca.html
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+
+    <h1 id="distributed-stochastic-pca">Distributed Stochastic PCA</h1>
+
+<h2 id="intro">Intro</h2>
+
+<p>Mahout has a distributed implementation of Stochastic PCA<a href="Lyubimov and Palumbo, [&quot;Apache Mahout: Beyond MapReduce; Distributed Algorithm Design&quot;](https://www.amazon.com/Apache-Mahout-MapReduce-Dmitriy-Lyubimov/dp/1523775785)">1</a>. This algorithm computes the exact equivalent of Mahout’s 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 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">\(\mathbf{A−1\mu^\top \approx U\Sigma V^\top}\)</code>:</p>
+
+<ol>
+  <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">\(\mathbf{Y_0 \leftarrow A\Omega − 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)</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 class="highlighter-rouge">\(\mathbf{(B_{i−1})_{∗j} \leftarrow (B_{i−1})_{∗j}−\mu_j s_Q}\)</code>.</li>
+      <li><code class="highlighter-rouge">\(\mathbf{Y_i \leftarrow A{B_{i−1}}^\top−1(s_B−\mu^\top \mu s_Q)^\top}\)</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 class="highlighter-rouge">\(\mathbf{C \triangleq s_Q {s_B}^\top}\)</code>. <code class="highlighter-rouge">\(\mathbf{M \leftarrow B_q {B_q}^\top − C − C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)</code>.</li>
+  <li>Compute an eigensolution of the small symmetric <code 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">\(\mathbf{V = B^\top \hat{U} \Sigma^{−1}}\)</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 class="highlighter-rouge">dspca(...)</code> is implemented in the mahout <code class="highlighter-rouge">math-scala</code> algebraic optimizer which translates Mahout’s 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 dspca[K](drmA: DrmLike[K], k: Int, p: Int = 15, q: Int = 0): 
+(DrmLike[K], DrmLike[Int], Vector) = {
+
+    // Some mapBlock() calls need it
+    implicit val ktag =  drmA.keyClassTag
+
+    val drmAcp = drmA.checkpoint()
+    implicit val ctx = drmAcp.context
+
+    val m = drmAcp.nrow
+	val n = drmAcp.ncol
+    assert(k &lt;= (m min n), "k cannot be greater than smaller of m, n.")
+    val pfxed = safeToNonNegInt((m min n) - k min p)
+
+    // Actual decomposition rank
+    val r = k + pfxed
+
+    // Dataset mean
+    val mu = drmAcp.colMeans
+
+    val mtm = mu dot mu
+
+    // We represent Omega by its seed.
+    val omegaSeed = RandomUtils.getRandom().nextInt()
+    val omega = Matrices.symmetricUniformView(n, r, omegaSeed)
+
+    // This done in front in a single-threaded fashion for now. Even though it doesn't require any
+    // memory beyond that is required to keep xi around, it still might be parallelized to backs
+    // for significantly big n and r. TODO
+    val s_o = omega.t %*% mu
+
+    val bcastS_o = drmBroadcast(s_o)
+    val bcastMu = drmBroadcast(mu)
+
+    var drmY = drmAcp.mapBlock(ncol = r) {
+        case (keys, blockA) ⇒
+            val s_o:Vector = bcastS_o
+            val blockY = blockA %*% Matrices.symmetricUniformView(n, r, omegaSeed)
+            for (row ← 0 until blockY.nrow) blockY(row, ::) -= s_o
+            keys → blockY
+    }
+            // Checkpoint Y
+            .checkpoint()
+
+    var drmQ = dqrThin(drmY, checkRankDeficiency = false)._1.checkpoint()
+
+    var s_q = drmQ.colSums()
+    var bcastVarS_q = drmBroadcast(s_q)
+
+    // This actually should be optimized as identically partitioned map-side A'B since A and Q should
+    // still be identically partitioned.
+    var drmBt = (drmAcp.t %*% drmQ).checkpoint()
+
+    var s_b = (drmBt.t %*% mu).collect(::, 0)
+    var bcastVarS_b = drmBroadcast(s_b)
+
+    for (i ← 0 until q) {
+
+        // These closures don't seem to live well with outside-scope vars. This doesn't record closure
+        // attributes correctly. So we create additional set of vals for broadcast vars to properly
+        // create readonly closure attributes in this very scope.
+        val bcastS_q = bcastVarS_q
+        val bcastMuInner = bcastMu
+
+        // Fix Bt as B' -= xi cross s_q
+        drmBt = drmBt.mapBlock() {
+            case (keys, block) ⇒
+                val s_q: Vector = bcastS_q
+                val mu: Vector = bcastMuInner
+                keys.zipWithIndex.foreach {
+                    case (key, idx) ⇒ block(idx, ::) -= s_q * mu(key)
+                }
+                keys → block
+        }
+
+        drmY.uncache()
+        drmQ.uncache()
+
+        val bCastSt_b = drmBroadcast(s_b -=: mtm * s_q)
+
+        drmY = (drmAcp %*% drmBt)
+            // Fix Y by subtracting st_b from each row of the AB'
+            .mapBlock() {
+            case (keys, block) ⇒
+                val st_b: Vector = bCastSt_b
+                block := { (_, c, v) ⇒ v - st_b(c) }
+                keys → block
+        }
+        // Checkpoint Y
+        .checkpoint()
+
+        drmQ = dqrThin(drmY, checkRankDeficiency = false)._1.checkpoint()
+
+        s_q = drmQ.colSums()
+        bcastVarS_q = drmBroadcast(s_q)
+
+        // This on the other hand should be inner-join-and-map A'B optimization since A and Q_i are not
+        // identically partitioned anymore.
+        drmBt = (drmAcp.t %*% drmQ).checkpoint()
+
+        s_b = (drmBt.t %*% mu).collect(::, 0)
+        bcastVarS_b = drmBroadcast(s_b)
+    }
+
+    val c = s_q cross s_b
+    val inCoreBBt = (drmBt.t %*% drmBt).checkpoint(CacheHint.NONE).collect -=:
+        c -=: c.t +=: mtm *=: (s_q cross s_q)
+    val (inCoreUHat, d) = eigen(inCoreBBt)
+    val s = d.sqrt
+
+    // Since neither drmU nor drmV are actually computed until actually used, we don't need the flags
+    // instructing compute (or not compute) either of the U,V outputs anymore. Neat, isn't it?
+    val drmU = drmQ %*% inCoreUHat
+    val drmV = drmBt %*% (inCoreUHat %*% diagv(1 / s))
+
+    (drmU(::, 0 until k), drmV(::, 0 until k), s(0 until k))
+}
+</code></pre>
+</div>
+
+<h2 id="usage">Usage</h2>
+
+<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>
+
+<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>
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+<h2 id="intro">Intro</h2>
+
+<p>Mahout has a distributed implementation of Stochastic Singular Value Decomposition <a href="[Mahout Scala and Mahout Spark Bindings for Linear Algebra Subroutines](http://mahout.apache.org/users/sparkbindings/ScalaSparkBindings.pdf)">1</a> using the parallelization strategy comprehensively defined in Nathan Halko’s dissertation <a href="http://amath.colorado.edu/faculty/martinss/Pubs/2012_halko_dissertation.pdf">“Randomized methods for computing low-rank approximations of matrices”</a> <a href="[Halko, Martinsson, Tropp](http://arxiv.org/abs/0909.4061)">2</a>.</p>
+
+<h2 id="modified-ssvd-algorithm">Modified SSVD Algorithm</h2>
+
+<p>Given an <code class="highlighter-rouge">\(m\times n\)</code>
+matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code>, a target rank <code class="highlighter-rouge">\(k\in\mathbb{N}_{1}\)</code>
+, an oversampling parameter <code class="highlighter-rouge">\(p\in\mathbb{N}_{1}\)</code>, 
+and the number of additional power iterations <code class="highlighter-rouge">\(q\in\mathbb{N}_{0}\)</code>, 
+this procedure computes an <code class="highlighter-rouge">\(m\times\left(k+p\right)\)</code>
+SVD <code class="highlighter-rouge">\(\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\)</code>:</p>
+
+<ol>
+  <li>
+    <p>Create seed for random <code class="highlighter-rouge">\(n\times\left(k+p\right)\)</code>
+  matrix <code class="highlighter-rouge">\(\boldsymbol{\Omega}\)</code>. The seed defines matrix <code class="highlighter-rouge">\(\mathbf{\Omega}\)</code>
+  using Gaussian unit vectors per one of suggestions in [Halko, Martinsson, Tropp].</p>
+  </li>
+  <li>
+    <p><code class="highlighter-rouge">\(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)</code></p>
+  </li>
+  <li>
+    <p>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y}\rightarrow\mathbf{Q}\)</code>
+  by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y}=\mathbf{Q}\mathbf{R}\)</code>.
+  Also, <code class="highlighter-rouge">\(\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</code>; denoted as <code class="highlighter-rouge">\(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)</code></p>
+  </li>
+  <li>
+    <p><code class="highlighter-rouge">\(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times n}\)</code>.</p>
+  </li>
+  <li>
+    <p>If <code class="highlighter-rouge">\(q&gt;0\)</code>
+  repeat: for <code class="highlighter-rouge">\(i=1..q\)</code>: 
+  <code class="highlighter-rouge">\(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\)</code>
+  (power iterations step).</p>
+  </li>
+  <li>
+    <p>Compute Eigensolution of a small Hermitian <code class="highlighter-rouge">\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\)</code>,
+  <code class="highlighter-rouge">\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</code>.</p>
+  </li>
+  <li>
+    <p>Singular values <code class="highlighter-rouge">\(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\)</code>,
+  or, in other words, <code class="highlighter-rouge">\(s_{i}=\sqrt{\sigma_{i}}\)</code>.</p>
+  </li>
+  <li>
+    <p>If needed, compute <code class="highlighter-rouge">\(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\)</code>.</p>
+  </li>
+  <li>
+    <p>If needed, compute <code class="highlighter-rouge">\(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\)</code>.
+Another way is <code class="highlighter-rouge">\(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\)</code>.</p>
+  </li>
+</ol>
+
+<h2 id="implementation">Implementation</h2>
+
+<p>Mahout <code class="highlighter-rouge">dssvd(...)</code> is implemented in the mahout <code class="highlighter-rouge">math-scala</code> algebraic optimizer which translates Mahout’s 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 dssvd[K: ClassTag](drmA: DrmLike[K], k: Int, p: Int = 15, q: Int = 0):
+    (DrmLike[K], DrmLike[Int], Vector) = {
+
+    val drmAcp = drmA.checkpoint()
+
+    val m = drmAcp.nrow
+    val n = drmAcp.ncol
+    assert(k &lt;= (m min n), "k cannot be greater than smaller of m, n.")
+    val pfxed = safeToNonNegInt((m min n) - k min p)
+
+    // Actual decomposition rank
+    val r = k + pfxed
+
+    // We represent Omega by its seed.
+    val omegaSeed = RandomUtils.getRandom().nextInt()
+
+    // Compute Y = A*Omega.  
+    var drmY = drmAcp.mapBlock(ncol = r) {
+        case (keys, blockA) =&gt;
+            val blockY = blockA %*% Matrices.symmetricUniformView(n, r, omegaSeed)
+        keys -&gt; blockY
+    }
+
+    var drmQ = dqrThin(drmY.checkpoint())._1
+
+    // Checkpoint Q if last iteration
+    if (q == 0) drmQ = drmQ.checkpoint()
+
+    var drmBt = drmAcp.t %*% drmQ
+    
+    // Checkpoint B' if last iteration
+    if (q == 0) drmBt = drmBt.checkpoint()
+
+    for (i &lt;- 0  until q) {
+        drmY = drmAcp %*% drmBt
+        drmQ = dqrThin(drmY.checkpoint())._1            
+        
+        // Checkpoint Q if last iteration
+        if (i == q - 1) drmQ = drmQ.checkpoint()
+        
+        drmBt = drmAcp.t %*% drmQ
+        
+        // Checkpoint B' if last iteration
+        if (i == q - 1) drmBt = drmBt.checkpoint()
+    }
+
+    val (inCoreUHat, d) = eigen(drmBt.t %*% drmBt)
+    val s = d.sqrt
+
+    // Since neither drmU nor drmV are actually computed until actually used
+    // we don't need the flags instructing compute (or not compute) either of the U,V outputs 
+    val drmU = drmQ %*% inCoreUHat
+    val drmV = drmBt %*% (inCoreUHat %*%: diagv(1 /: s))
+
+    (drmU(::, 0 until k), drmV(::, 0 until k), s(0 until k))
+}
+</code></pre>
+</div>
+
+<p>Note: As a side effect of checkpointing, U and V values are returned as logical operators (i.e. they are neither checkpointed nor computed).  Therefore there is no physical work actually done to compute <code class="highlighter-rouge">\(\mathbf{U}\)</code> or <code class="highlighter-rouge">\(\mathbf{V}\)</code> until they are used in a subsequent expression.</p>
+
+<h2 id="usage">Usage</h2>
+
+<p>The scala <code class="highlighter-rouge">dssvd(...)</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(drmU, drmV, s) = dssvd(drma, k = 40, q = 1)
+</code></pre>
+</div>
+
+<h2 id="references">References</h2>
+
+<p>approximations of matrices](http://amath.colorado.edu/faculty/martinss/Pubs/2012_halko_dissertation.pdf)</p>
+
+
+   </div>
+  </div>     
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