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authorPatrick Wendell <pwendell@apache.org>2014-07-11 17:23:23 +0000
committerPatrick Wendell <pwendell@apache.org>2014-07-11 17:23:23 +0000
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+
+ <h1 class="title"><a href="mllib-guide.html">MLlib</a> - Basics</h1>
+
+
+ <ul id="markdown-toc">
+ <li><a href="#local-vector">Local vector</a></li>
+ <li><a href="#labeled-point">Labeled point</a></li>
+ <li><a href="#local-matrix">Local matrix</a></li>
+ <li><a href="#distributed-matrix">Distributed matrix</a> <ul>
+ <li><a href="#rowmatrix">RowMatrix</a> <ul>
+ <li><a href="#multivariate-summary-statistics">Multivariate summary statistics</a></li>
+ </ul>
+ </li>
+ <li><a href="#indexedrowmatrix">IndexedRowMatrix</a></li>
+ <li><a href="#coordinatematrix">CoordinateMatrix</a></li>
+ </ul>
+ </li>
+</ul>
+
+<p>MLlib supports local vectors and matrices stored on a single machine,
+as well as distributed matrices backed by one or more RDDs.
+In the current implementation, local vectors and matrices are simple data models
+to serve public interfaces. The underlying linear algebra operations are provided by
+<a href="http://www.scalanlp.org/">Breeze</a> and <a href="http://jblas.org/">jblas</a>.
+A training example used in supervised learning is called &#8220;labeled point&#8221; in MLlib.</p>
+
+<h2 id="local-vector">Local vector</h2>
+
+<p>A local vector has integer-typed and 0-based indices and double-typed values, stored on a single
+machine. MLlib supports two types of local vectors: dense and sparse. A dense vector is backed by
+a double array representing its entry values, while a sparse vector is backed by two parallel
+arrays: indices and values. For example, a vector $(1.0, 0.0, 3.0)$ can be represented in dense
+format as <code>[1.0, 0.0, 3.0]</code> or in sparse format as <code>(3, [0, 2], [1.0, 3.0])</code>, where <code>3</code> is the size
+of the vector.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>The base class of local vectors is
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vector"><code>Vector</code></a>, and we provide two
+implementations: <a href="api/scala/index.html#org.apache.spark.mllib.linalg.DenseVector"><code>DenseVector</code></a> and
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.SparseVector"><code>SparseVector</code></a>. We recommend
+using the factory methods implemented in
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Vector"><code>Vectors</code></a> to create local vectors.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.</span><span class="o">{</span><span class="nc">Vector</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">}</span>
+
+<span class="c1">// Create a dense vector (1.0, 0.0, 3.0).</span>
+<span class="k">val</span> <span class="n">dv</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)</span>
+<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.</span>
+<span class="k">val</span> <span class="n">sv1</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">))</span>
+<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its nonzero entries.</span>
+<span class="k">val</span> <span class="n">sv2</span><span class="k">:</span> <span class="kt">Vector</span> <span class="o">=</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Seq</span><span class="o">((</span><span class="mi">0</span><span class="o">,</span> <span class="mf">1.0</span><span class="o">),</span> <span class="o">(</span><span class="mi">2</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)))</span>
+</code></pre></div>
+
+ <p><strong><em>Note</em></strong></p>
+
+ <p>Scala imports <code>scala.collection.immutable.Vector</code> by default, so you have to import
+<code>org.apache.spark.mllib.linalg.Vector</code> explicitly to use MLlib&#8217;s <code>Vector</code>.</p>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>The base class of local vectors is
+<a href="api/java/org/apache/spark/mllib/linalg/Vector.html"><code>Vector</code></a>, and we provide two
+implementations: <a href="api/java/org/apache/spark/mllib/linalg/DenseVector.html"><code>DenseVector</code></a> and
+<a href="api/java/org/apache/spark/mllib/linalg/SparseVector.html"><code>SparseVector</code></a>. We recommend
+using the factory methods implemented in
+<a href="api/java/org/apache/spark/mllib/linalg/Vector.html"><code>Vectors</code></a> to create local vectors.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span>
+
+<span class="c1">// Create a dense vector (1.0, 0.0, 3.0).</span>
+<span class="n">Vector</span> <span class="n">dv</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">);</span>
+<span class="c1">// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.</span>
+<span class="n">Vector</span> <span class="n">sv</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">});</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="python">
+ <p>MLlib recognizes the following types as dense vectors:</p>
+
+ <ul>
+ <li>NumPy&#8217;s <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html"><code>array</code></a></li>
+ <li>Python&#8217;s list, e.g., <code>[1, 2, 3]</code></li>
+ </ul>
+
+ <p>and the following as sparse vectors:</p>
+
+ <ul>
+ <li>MLlib&#8217;s <a href="api/python/pyspark.mllib.linalg.SparseVector-class.html"><code>SparseVector</code></a>.</li>
+ <li>SciPy&#8217;s
+<a href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csc_matrix.html#scipy.sparse.csc_matrix"><code>csc_matrix</code></a>
+with a single column</li>
+ </ul>
+
+ <p>We recommend using NumPy arrays over lists for efficiency, and using the factory methods implemented
+in <a href="api/python/pyspark.mllib.linalg.Vectors-class.html"><code>Vectors</code></a> to create sparse vectors.</p>
+
+ <div class="highlight"><pre><code class="python"><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">scipy.sparse</span> <span class="kn">as</span> <span class="nn">sps</span>
+<span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">Vectors</span>
+
+<span class="c"># Use a NumPy array as a dense vector.</span>
+<span class="n">dv1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>
+<span class="c"># Use a Python list as a dense vector.</span>
+<span class="n">dv2</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]</span>
+<span class="c"># Create a SparseVector.</span>
+<span class="n">sv1</span> <span class="o">=</span> <span class="n">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>
+<span class="c"># Use a single-column SciPy csc_matrix as a sparse vector.</span>
+<span class="n">sv2</span> <span class="o">=</span> <span class="n">sps</span><span class="o">.</span><span class="n">csc_matrix</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])),</span> <span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+</code></pre></div>
+
+ </div>
+</div>
+
+<h2 id="labeled-point">Labeled point</h2>
+
+<p>A labeled point is a local vector, either dense or sparse, associated with a label/response.
+In MLlib, labeled points are used in supervised learning algorithms.
+We use a double to store a label, so we can use labeled points in both regression and classification.
+For binary classification, label should be either $0$ (negative) or $1$ (positive).
+For multiclass classification, labels should be class indices staring from zero: $0, 1, 2, \ldots$.</p>
+
+<div class="codetabs">
+
+<div data-lang="scala">
+
+ <p>A labeled point is represented by the case class
+<a href="api/scala/index.html#org.apache.spark.mllib.regression.LabeledPoint"><code>LabeledPoint</code></a>.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>
+
+<span class="c1">// Create a labeled point with a positive label and a dense feature vector.</span>
+<span class="k">val</span> <span class="n">pos</span> <span class="k">=</span> <span class="nc">LabeledPoint</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">))</span>
+
+<span class="c1">// Create a labeled point with a negative label and a sparse feature vector.</span>
+<span class="k">val</span> <span class="n">neg</span> <span class="k">=</span> <span class="nc">LabeledPoint</span><span class="o">(</span><span class="mf">0.0</span><span class="o">,</span> <span class="nc">Vectors</span><span class="o">.</span><span class="n">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">),</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">)))</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>A labeled point is represented by
+<a href="api/java/org/apache/spark/mllib/regression/LabeledPoint.html"><code>LabeledPoint</code></a>.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vectors</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>
+
+<span class="c1">// Create a labeled point with a positive label and a dense feature vector.</span>
+<span class="n">LabeledPoint</span> <span class="n">pos</span> <span class="o">=</span> <span class="k">new</span> <span class="n">LabeledPoint</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">0.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">));</span>
+
+<span class="c1">// Create a labeled point with a negative label and a sparse feature vector.</span>
+<span class="n">LabeledPoint</span> <span class="n">neg</span> <span class="o">=</span> <span class="k">new</span> <span class="n">LabeledPoint</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="n">Vectors</span><span class="o">.</span><span class="na">sparse</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="k">new</span> <span class="kt">int</span><span class="o">[]</span> <span class="o">{</span><span class="mi">0</span><span class="o">,</span> <span class="mi">2</span><span class="o">},</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">}));</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="python">
+
+ <p>A labeled point is represented by
+<a href="api/python/pyspark.mllib.regression.LabeledPoint-class.html"><code>LabeledPoint</code></a>.</p>
+
+ <div class="highlight"><pre><code class="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.linalg</span> <span class="kn">import</span> <span class="n">SparseVector</span>
+<span class="kn">from</span> <span class="nn">pyspark.mllib.regression</span> <span class="kn">import</span> <span class="n">LabeledPoint</span>
+
+<span class="c"># Create a labeled point with a positive label and a dense feature vector.</span>
+<span class="n">pos</span> <span class="o">=</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">])</span>
+
+<span class="c"># Create a labeled point with a negative label and a sparse feature vector.</span>
+<span class="n">neg</span> <span class="o">=</span> <span class="n">LabeledPoint</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">SparseVector</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]))</span>
+</code></pre></div>
+
+ </div>
+</div>
+
+<p><strong><em>Sparse data</em></strong></p>
+
+<p>It is very common in practice to have sparse training data. MLlib supports reading training
+examples stored in <code>LIBSVM</code> format, which is the default format used by
+<a href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/"><code>LIBSVM</code></a> and
+<a href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/"><code>LIBLINEAR</code></a>. It is a text format. Each line
+represents a labeled sparse feature vector using the following format:</p>
+
+<pre><code>label index1:value1 index2:value2 ...
+</code></pre>
+
+<p>where the indices are one-based and in ascending order.
+After loading, the feature indices are converted to zero-based.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p><a href="api/scala/index.html#org.apache.spark.mllib.util.MLUtils$"><code>MLUtils.loadLibSVMFile</code></a> reads training
+examples stored in LIBSVM format.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.rdd.RDD</span>
+
+<span class="k">val</span> <span class="n">examples</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">LabeledPoint</span><span class="o">]</span> <span class="k">=</span> <span class="nc">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="o">(</span><span class="n">sc</span><span class="o">,</span> <span class="s">&quot;mllib/data/sample_libsvm_data.txt&quot;</span><span class="o">)</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+ <p><a href="api/java/org/apache/spark/mllib/util/MLUtils.html"><code>MLUtils.loadLibSVMFile</code></a> reads training
+examples stored in LIBSVM format.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.regression.LabeledPoint</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.util.MLUtils</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
+
+<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">LabeledPoint</span><span class="o">&gt;</span> <span class="n">examples</span> <span class="o">=</span>
+ <span class="n">MLUtils</span><span class="o">.</span><span class="na">loadLibSVMFile</span><span class="o">(</span><span class="n">jsc</span><span class="o">.</span><span class="na">sc</span><span class="o">(),</span> <span class="s">&quot;mllib/data/sample_libsvm_data.txt&quot;</span><span class="o">).</span><span class="na">toJavaRDD</span><span class="o">();</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="python">
+ <p><a href="api/python/pyspark.mllib.util.MLUtils-class.html"><code>MLUtils.loadLibSVMFile</code></a> reads training
+examples stored in LIBSVM format.</p>
+
+ <div class="highlight"><pre><code class="python"><span class="kn">from</span> <span class="nn">pyspark.mllib.util</span> <span class="kn">import</span> <span class="n">MLUtils</span>
+
+<span class="n">examples</span> <span class="o">=</span> <span class="n">MLUtils</span><span class="o">.</span><span class="n">loadLibSVMFile</span><span class="p">(</span><span class="n">sc</span><span class="p">,</span> <span class="s">&quot;mllib/data/sample_libsvm_data.txt&quot;</span><span class="p">)</span>
+</code></pre></div>
+
+ </div>
+</div>
+
+<h2 id="local-matrix">Local matrix</h2>
+
+<p>A local matrix has integer-typed row and column indices and double-typed values, stored on a single
+machine. MLlib supports dense matrix, whose entry values are stored in a single double array in
+column major. For example, the following matrix <code>\[ \begin{pmatrix}
+1.0 &amp; 2.0 \\
+3.0 &amp; 4.0 \\
+5.0 &amp; 6.0
+\end{pmatrix}
+\]</code>
+is stored in a one-dimensional array <code>[1.0, 3.0, 5.0, 2.0, 4.0, 6.0]</code> with the matrix size <code>(3, 2)</code>.
+We are going to add sparse matrix in the next release.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>The base class of local matrices is
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrix"><code>Matrix</code></a>, and we provide one
+implementation: <a href="api/scala/index.html#org.apache.spark.mllib.linalg.DenseMatrix"><code>DenseMatrix</code></a>.
+Sparse matrix will be added in the next release. We recommend using the factory methods implemented
+in <a href="api/scala/index.html#org.apache.spark.mllib.linalg.Matrices"><code>Matrices</code></a> to create local
+matrices.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.</span><span class="o">{</span><span class="nc">Matrix</span><span class="o">,</span> <span class="nc">Matrices</span><span class="o">}</span>
+
+<span class="c1">// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))</span>
+<span class="k">val</span> <span class="n">dm</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="nc">Matrices</span><span class="o">.</span><span class="n">dense</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="nc">Array</span><span class="o">(</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">,</span> <span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">))</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>The base class of local matrices is
+<a href="api/java/org/apache/spark/mllib/linalg/Matrix.html"><code>Matrix</code></a>, and we provide one
+implementation: <a href="api/java/org/apache/spark/mllib/linalg/DenseMatrix.html"><code>DenseMatrix</code></a>.
+Sparse matrix will be added in the next release. We recommend using the factory methods implemented
+in <a href="api/java/org/apache/spark/mllib/linalg/Matrices.html"><code>Matrices</code></a> to create local
+matrices.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrices</span><span class="o">;</span>
+
+<span class="c1">// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))</span>
+<span class="n">Matrix</span> <span class="n">dm</span> <span class="o">=</span> <span class="n">Matrices</span><span class="o">.</span><span class="na">dense</span><span class="o">(</span><span class="mi">3</span><span class="o">,</span> <span class="mi">2</span><span class="o">,</span> <span class="k">new</span> <span class="kt">double</span><span class="o">[]</span> <span class="o">{</span><span class="mf">1.0</span><span class="o">,</span> <span class="mf">3.0</span><span class="o">,</span> <span class="mf">5.0</span><span class="o">,</span> <span class="mf">2.0</span><span class="o">,</span> <span class="mf">4.0</span><span class="o">,</span> <span class="mf">6.0</span><span class="o">});</span>
+</code></pre></div>
+
+ </div>
+
+</div>
+
+<h2 id="distributed-matrix">Distributed matrix</h2>
+
+<p>A distributed matrix has long-typed row and column indices and double-typed values, stored
+distributively in one or more RDDs. It is very important to choose the right format to store large
+and distributed matrices. Converting a distributed matrix to a different format may require a
+global shuffle, which is quite expensive. We implemented three types of distributed matrices in
+this release and will add more types in the future.</p>
+
+<p>The basic type is called <code>RowMatrix</code>. A <code>RowMatrix</code> is a row-oriented distributed
+matrix without meaningful row indices, e.g., a collection of feature vectors.
+It is backed by an RDD of its rows, where each row is a local vector.
+We assume that the number of columns is not huge for a <code>RowMatrix</code>.
+An <code>IndexedRowMatrix</code> is similar to a <code>RowMatrix</code> but with row indices,
+which can be used for identifying rows and joins.
+A <code>CoordinateMatrix</code> is a distributed matrix stored in <a href="https://en.wikipedia.org/wiki/Sparse_matrix">coordinate list (COO)</a> format,
+backed by an RDD of its entries.</p>
+
+<p><strong><em>Note</em></strong></p>
+
+<p>The underlying RDDs of a distributed matrix must be deterministic, because we cache the matrix size.
+It is always error-prone to have non-deterministic RDDs.</p>
+
+<h3 id="rowmatrix">RowMatrix</h3>
+
+<p>A <code>RowMatrix</code> is a row-oriented distributed matrix without meaningful row indices, backed by an RDD
+of its rows, where each row is a local vector. This is similar to <code>data matrix</code> in the context of
+multivariate statistics. Since each row is represented by a local vector, the number of columns is
+limited by the integer range but it should be much smaller in practice.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>A <a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix</code></a> can be
+created from an <code>RDD[Vector]</code> instance. Then we can compute its column summary statistics.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span>
+
+<span class="k">val</span> <span class="n">rows</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">Vector</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of local vectors</span>
+<span class="c1">// Create a RowMatrix from an RDD[Vector].</span>
+<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">)</span>
+
+<span class="c1">// Get its size.</span>
+<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
+<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>A <a href="api/java/org/apache/spark/mllib/linalg/distributed/RowMatrix.html"><code>RowMatrix</code></a> can be
+created from a <code>JavaRDD&lt;Vector&gt;</code> instance. Then we can compute its column summary statistics.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Vector</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span>
+
+<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">Vector</span><span class="o">&gt;</span> <span class="n">rows</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of local vectors</span>
+<span class="c1">// Create a RowMatrix from an JavaRDD&lt;Vector&gt;.</span>
+<span class="n">RowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="n">RowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
+
+<span class="c1">// Get its size.</span>
+<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
+<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>
+</code></pre></div>
+
+ </div>
+</div>
+
+<h4 id="multivariate-summary-statistics">Multivariate summary statistics</h4>
+
+<p>We provide column summary statistics for <code>RowMatrix</code>.
+If the number of columns is not large, say, smaller than 3000, you can also compute
+the covariance matrix as a local matrix, which requires $\mathcal{O}(n^2)$ storage where $n$ is the
+number of columns. The total CPU time is $\mathcal{O}(m n^2)$, where $m$ is the number of rows,
+which could be faster if the rows are sparse.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p><a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix"><code>RowMatrix#computeColumnSummaryStatistics</code></a> returns an instance of
+<a href="api/scala/index.html#org.apache.spark.mllib.stat.MultivariateStatisticalSummary"><code>MultivariateStatisticalSummary</code></a>,
+which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
+total count.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span>
+<span class="k">import</span> <span class="nn">org.apache.spark.mllib.stat.MultivariateStatisticalSummary</span>
+
+<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a RowMatrix</span>
+
+<span class="c1">// Compute column summary statistics.</span>
+<span class="k">val</span> <span class="n">summary</span><span class="k">:</span> <span class="kt">MultivariateStatisticalSummary</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">computeColumnSummaryStatistics</span><span class="o">()</span>
+<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">mean</span><span class="o">)</span> <span class="c1">// a dense vector containing the mean value for each column</span>
+<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">variance</span><span class="o">)</span> <span class="c1">// column-wise variance</span>
+<span class="n">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="n">numNonzeros</span><span class="o">)</span> <span class="c1">// number of nonzeros in each column</span>
+
+<span class="c1">// Compute the covariance matrix.</span>
+<span class="k">val</span> <span class="n">cov</span><span class="k">:</span> <span class="kt">Matrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">computeCovariance</span><span class="o">()</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p><a href="api/java/org/apache/spark/mllib/linalg/distributed/RowMatrix.html#computeColumnSummaryStatistics()"><code>RowMatrix#computeColumnSummaryStatistics</code></a> returns an instance of
+<a href="api/java/org/apache/spark/mllib/stat/MultivariateStatisticalSummary.html"><code>MultivariateStatisticalSummary</code></a>,
+which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
+total count.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.Matrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.stat.MultivariateStatisticalSummary</span><span class="o">;</span>
+
+<span class="n">RowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a RowMatrix</span>
+
+<span class="c1">// Compute column summary statistics.</span>
+<span class="n">MultivariateStatisticalSummary</span> <span class="n">summary</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">computeColumnSummaryStatistics</span><span class="o">();</span>
+<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">mean</span><span class="o">());</span> <span class="c1">// a dense vector containing the mean value for each column</span>
+<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">variance</span><span class="o">());</span> <span class="c1">// column-wise variance</span>
+<span class="n">System</span><span class="o">.</span><span class="na">out</span><span class="o">.</span><span class="na">println</span><span class="o">(</span><span class="n">summary</span><span class="o">.</span><span class="na">numNonzeros</span><span class="o">());</span> <span class="c1">// number of nonzeros in each column</span>
+
+<span class="c1">// Compute the covariance matrix.</span>
+<span class="n">Matrix</span> <span class="n">cov</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">computeCovariance</span><span class="o">();</span>
+</code></pre></div>
+
+ </div>
+</div>
+
+<h3 id="indexedrowmatrix">IndexedRowMatrix</h3>
+
+<p>An <code>IndexedRowMatrix</code> is similar to a <code>RowMatrix</code> but with meaningful row indices. It is backed by
+an RDD of indexed rows, which each row is represented by its index (long-typed) and a local vector.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>An
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix"><code>IndexedRowMatrix</code></a>
+can be created from an <code>RDD[IndexedRow]</code> instance, where
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRow"><code>IndexedRow</code></a> is a
+wrapper over <code>(Long, Vector)</code>. An <code>IndexedRowMatrix</code> can be converted to a <code>RowMatrix</code> by dropping
+its row indices.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.</span><span class="o">{</span><span class="nc">IndexedRow</span><span class="o">,</span> <span class="nc">IndexedRowMatrix</span><span class="o">,</span> <span class="nc">RowMatrix</span><span class="o">}</span>
+
+<span class="k">val</span> <span class="n">rows</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">IndexedRow</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of indexed rows</span>
+<span class="c1">// Create an IndexedRowMatrix from an RDD[IndexedRow].</span>
+<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">IndexedRowMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">IndexedRowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">)</span>
+
+<span class="c1">// Get its size.</span>
+<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
+<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>
+
+<span class="c1">// Drop its row indices.</span>
+<span class="k">val</span> <span class="n">rowMat</span><span class="k">:</span> <span class="kt">RowMatrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toRowMatrix</span><span class="o">()</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>An
+<a href="api/java/org/apache/spark/mllib/linalg/distributed/IndexedRowMatrix.html"><code>IndexedRowMatrix</code></a>
+can be created from an <code>JavaRDD&lt;IndexedRow&gt;</code> instance, where
+<a href="api/java/org/apache/spark/mllib/linalg/distributed/IndexedRow.html"><code>IndexedRow</code></a> is a
+wrapper over <code>(long, Vector)</code>. An <code>IndexedRowMatrix</code> can be converted to a <code>RowMatrix</code> by dropping
+its row indices.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRow</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.RowMatrix</span><span class="o">;</span>
+
+<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">IndexedRow</span><span class="o">&gt;</span> <span class="n">rows</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of indexed rows</span>
+<span class="c1">// Create an IndexedRowMatrix from a JavaRDD&lt;IndexedRow&gt;.</span>
+<span class="n">IndexedRowMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="n">IndexedRowMatrix</span><span class="o">(</span><span class="n">rows</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
+
+<span class="c1">// Get its size.</span>
+<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
+<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>
+
+<span class="c1">// Drop its row indices.</span>
+<span class="n">RowMatrix</span> <span class="n">rowMat</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">toRowMatrix</span><span class="o">();</span>
+</code></pre></div>
+
+ </div></div>
+
+<h3 id="coordinatematrix">CoordinateMatrix</h3>
+
+<p>A <code>CoordinateMatrix</code> is a distributed matrix backed by an RDD of its entries. Each entry is a tuple
+of <code>(i: Long, j: Long, value: Double)</code>, where <code>i</code> is the row index, <code>j</code> is the column index, and
+<code>value</code> is the entry value. A <code>CoordinateMatrix</code> should be used only in the case when both
+dimensions of the matrix are huge and the matrix is very sparse.</p>
+
+<div class="codetabs">
+<div data-lang="scala">
+
+ <p>A
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.CoordinateMatrix"><code>CoordinateMatrix</code></a>
+can be created from an <code>RDD[MatrixEntry]</code> instance, where
+<a href="api/scala/index.html#org.apache.spark.mllib.linalg.distributed.MatrixEntry"><code>MatrixEntry</code></a> is a
+wrapper over <code>(Long, Long, Double)</code>. A <code>CoordinateMatrix</code> can be converted to a <code>IndexedRowMatrix</code>
+with sparse rows by calling <code>toIndexedRowMatrix</code>. In this release, we do not provide other
+computation for <code>CoordinateMatrix</code>.</p>
+
+ <div class="highlight"><pre><code class="scala"><span class="k">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.</span><span class="o">{</span><span class="nc">CoordinateMatrix</span><span class="o">,</span> <span class="nc">MatrixEntry</span><span class="o">}</span>
+
+<span class="k">val</span> <span class="n">entries</span><span class="k">:</span> <span class="kt">RDD</span><span class="o">[</span><span class="kt">MatrixEntry</span><span class="o">]</span> <span class="k">=</span> <span class="o">...</span> <span class="c1">// an RDD of matrix entries</span>
+<span class="c1">// Create a CoordinateMatrix from an RDD[MatrixEntry].</span>
+<span class="k">val</span> <span class="n">mat</span><span class="k">:</span> <span class="kt">CoordinateMatrix</span> <span class="o">=</span> <span class="k">new</span> <span class="nc">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">)</span>
+
+<span class="c1">// Get its size.</span>
+<span class="k">val</span> <span class="n">m</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numRows</span><span class="o">()</span>
+<span class="k">val</span> <span class="n">n</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">numCols</span><span class="o">()</span>
+
+<span class="c1">// Convert it to an IndexRowMatrix whose rows are sparse vectors.</span>
+<span class="k">val</span> <span class="n">indexedRowMatrix</span> <span class="k">=</span> <span class="n">mat</span><span class="o">.</span><span class="n">toIndexedRowMatrix</span><span class="o">()</span>
+</code></pre></div>
+
+ </div>
+
+<div data-lang="java">
+
+ <p>A
+<a href="api/java/org/apache/spark/mllib/linalg/distributed/CoordinateMatrix.html"><code>CoordinateMatrix</code></a>
+can be created from a <code>JavaRDD&lt;MatrixEntry&gt;</code> instance, where
+<a href="api/java/org/apache/spark/mllib/linalg/distributed/MatrixEntry.html"><code>MatrixEntry</code></a> is a
+wrapper over <code>(long, long, double)</code>. A <code>CoordinateMatrix</code> can be converted to a <code>IndexedRowMatrix</code>
+with sparse rows by calling <code>toIndexedRowMatrix</code>.</p>
+
+ <div class="highlight"><pre><code class="java"><span class="kn">import</span> <span class="nn">org.apache.spark.api.java.JavaRDD</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.CoordinateMatrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix</span><span class="o">;</span>
+<span class="kn">import</span> <span class="nn">org.apache.spark.mllib.linalg.distributed.MatrixEntry</span><span class="o">;</span>
+
+<span class="n">JavaRDD</span><span class="o">&lt;</span><span class="n">MatrixEntry</span><span class="o">&gt;</span> <span class="n">entries</span> <span class="o">=</span> <span class="o">...</span> <span class="c1">// a JavaRDD of matrix entries</span>
+<span class="c1">// Create a CoordinateMatrix from a JavaRDD&lt;MatrixEntry&gt;.</span>
+<span class="n">CoordinateMatrix</span> <span class="n">mat</span> <span class="o">=</span> <span class="k">new</span> <span class="n">CoordinateMatrix</span><span class="o">(</span><span class="n">entries</span><span class="o">.</span><span class="na">rdd</span><span class="o">());</span>
+
+<span class="c1">// Get its size.</span>
+<span class="kt">long</span> <span class="n">m</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numRows</span><span class="o">();</span>
+<span class="kt">long</span> <span class="n">n</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">numCols</span><span class="o">();</span>
+
+<span class="c1">// Convert it to an IndexRowMatrix whose rows are sparse vectors.</span>
+<span class="n">IndexedRowMatrix</span> <span class="n">indexedRowMatrix</span> <span class="o">=</span> <span class="n">mat</span><span class="o">.</span><span class="na">toIndexedRowMatrix</span><span class="o">();</span>
+</code></pre></div>
+
+ </div>
+</div>
+
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