[SPARK-1506][MLLIB] Documentation improvements for MLlib 1.0

Preview: http://54.82.240.23:4000/mllib-guide.html

Table of contents:

* Basics
  * Data types
  * Summary statistics
* Classification and regression
  * linear support vector machine (SVM)
  * logistic regression
  * linear linear squares, Lasso, and ridge regression
  * decision tree
  * naive Bayes
* Collaborative Filtering
  * alternating least squares (ALS)
* Clustering
  * k-means
* Dimensionality reduction
  * singular value decomposition (SVD)
  * principal component analysis (PCA)
* Optimization
  * stochastic gradient descent
  * limited-memory BFGS (L-BFGS)

Author: Xiangrui Meng <meng@databricks.com>

Closes #422 from mengxr/mllib-doc and squashes the following commits:

944e3a9 [Xiangrui Meng] merge master
f9fda28 [Xiangrui Meng] minor
9474065 [Xiangrui Meng] add alpha to ALS examples
928e630 [Xiangrui Meng] initialization_mode -> initializationMode
5bbff49 [Xiangrui Meng] add imports to labeled point examples
c17440d [Xiangrui Meng] fix python nb example
28f40dc [Xiangrui Meng] remove localhost:4000
369a4d3 [Xiangrui Meng] Merge branch 'master' into mllib-doc
7dc95cc [Xiangrui Meng] update linear methods
053ad8a [Xiangrui Meng] add links to go back to the main page
abbbf7e [Xiangrui Meng] update ALS argument names
648283e [Xiangrui Meng] level down statistics
14e2287 [Xiangrui Meng] add sample libsvm data and use it in guide
8cd2441 [Xiangrui Meng] minor updates
186ab07 [Xiangrui Meng] update section names
6568d65 [Xiangrui Meng] update toc, level up lr and svm
162ee12 [Xiangrui Meng] rename section names
5c1e1b1 [Xiangrui Meng] minor
8aeaba1 [Xiangrui Meng] wrap long lines
6ce6a6f [Xiangrui Meng] add summary statistics to toc
5760045 [Xiangrui Meng] claim beta
cc604bf [Xiangrui Meng] remove classification and regression
92747b3 [Xiangrui Meng] make section titles consistent
e605dd6 [Xiangrui Meng] add LIBSVM loader
f639674 [Xiangrui Meng] add python section to migration guide
c82ffb4 [Xiangrui Meng] clean optimization
31660eb [Xiangrui Meng] update linear algebra and stat
0a40837 [Xiangrui Meng] first pass over linear methods
1fc8271 [Xiangrui Meng] update toc
906ed0a [Xiangrui Meng] add a python example to naive bayes
5f0a700 [Xiangrui Meng] update collaborative filtering
656d416 [Xiangrui Meng] update mllib-clustering
86e143a [Xiangrui Meng] remove data types section from main page
8d1a128 [Xiangrui Meng] move part of linear algebra to data types and add Java/Python examples
d1b5cbf [Xiangrui Meng] merge master
72e4804 [Xiangrui Meng] one pass over tree guide
64f8995 [Xiangrui Meng] move decision tree guide to a separate file
9fca001 [Xiangrui Meng] add first version of linear algebra guide
53c9552 [Xiangrui Meng] update dependencies
f316ec2 [Xiangrui Meng] add migration guide
f399f6c [Xiangrui Meng] move linear-algebra to dimensionality-reduction
182460f [Xiangrui Meng] add guide for naive Bayes
137fd1d [Xiangrui Meng] re-organize toc
a61e434 [Xiangrui Meng] update mllib's toc

1 files changed, 476 insertions, 0 deletions

diff --git a/docs/mllib-basics.md b/docs/mllib-basics.md
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+---
+layout: global
+title: <a href="mllib-guide.html">MLlib</a> - Basics
+---
+
+* Table of contents
+{:toc}
+
+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 underly linear algebra operations are provided by
+[Breeze](http://www.scalanlp.org/) and [jblas](http://jblas.org/).
+A training example used in supervised learning is called "labeled point" in MLlib.
+
+## Local vector
+
+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 `[1.0, 0.0, 3.0]` or in sparse format as `(3, [0, 2], [1.0, 3.0])`, where `3` is the size
+of the vector.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+The base class of local vectors is
+[`Vector`](api/mllib/index.html#org.apache.spark.mllib.linalg.Vector), and we provide two
+implementations: [`DenseVector`](api/mllib/index.html#org.apache.spark.mllib.linalg.DenseVector) and
+[`SparseVector`](api/mllib/index.html#org.apache.spark.mllib.linalg.SparseVector).  We recommend
+using the factory methods implemented in
+[`Vectors`](api/mllib/index.html#org.apache.spark.mllib.linalg.Vector) to create local vectors.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.{Vector, Vectors}
+
+// Create a dense vector (1.0, 0.0, 3.0).
+val dv: Vector = Vectors.dense(1.0, 0.0, 3.0)
+// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.
+val sv1: Vector = Vectors.sparse(3, Array(0, 2), Array(1.0, 3.0))
+// Create a sparse vector (1.0, 0.0, 3.0) by specifying its nonzero entries.
+val sv2: Vector = Vectors.sparse(3, Seq((0, 1.0), (2, 3.0)))
+{% endhighlight %}
+
+***Note***
+
+Scala imports `scala.collection.immutable.Vector` by default, so you have to import
+`org.apache.spark.mllib.linalg.Vector` explicitly to use MLlib's `Vector`.
+
+</div>
+
+<div data-lang="java" markdown="1">
+
+The base class of local vectors is
+[`Vector`](api/mllib/index.html#org.apache.spark.mllib.linalg.Vector), and we provide two
+implementations: [`DenseVector`](api/mllib/index.html#org.apache.spark.mllib.linalg.DenseVector) and
+[`SparseVector`](api/mllib/index.html#org.apache.spark.mllib.linalg.SparseVector).  We recommend
+using the factory methods implemented in
+[`Vectors`](api/mllib/index.html#org.apache.spark.mllib.linalg.Vector) to create local vectors.
+
+{% highlight java %}
+import org.apache.spark.mllib.linalg.Vector;
+import org.apache.spark.mllib.linalg.Vectors;
+
+// Create a dense vector (1.0, 0.0, 3.0).
+Vector dv = Vectors.dense(1.0, 0.0, 3.0);
+// Create a sparse vector (1.0, 0.0, 3.0) by specifying its indices and values corresponding to nonzero entries.
+Vector sv = Vectors.sparse(3, new int[] {0, 2}, new double[] {1.0, 3.0});
+{% endhighlight %}
+</div>
+
+<div data-lang="python" markdown="1">
+MLlib recognizes the following types as dense vectors:
+
+* NumPy's [`array`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html)
+* Python's list, e.g., `[1, 2, 3]`
+
+and the following as sparse vectors:
+
+* MLlib's [`SparseVector`](api/pyspark/pyspark.mllib.linalg.SparseVector-class.html).
+* SciPy's
+  [`csc_matrix`](http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csc_matrix.html#scipy.sparse.csc_matrix)
+  with a single column
+
+We recommend using NumPy arrays over lists for efficiency, and using the factory methods implemented
+in [`Vectors`](api/pyspark/pyspark.mllib.linalg.Vectors-class.html) to create sparse vectors.
+
+{% highlight python %}
+import numpy as np
+import scipy.sparse as sps
+from pyspark.mllib.linalg import Vectors
+
+# Use a NumPy array as a dense vector.
+dv1 = np.array([1.0, 0.0, 3.0])
+# Use a Python list as a dense vector.
+dv2 = [1.0, 0.0, 3.0]
+# Create a SparseVector.
+sv1 = Vectors.sparse(3, [0, 2], [1.0, 3.0])
+# Use a single-column SciPy csc_matrix as a sparse vector.
+sv2 = sps.csc_matrix((np.array([1.0, 3.0]), np.array([0, 2]), np.array([0, 2])), shape = (3, 1))
+{% endhighlight %}
+
+</div>
+</div>
+
+## Labeled point
+
+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: $0, 1, 2, \ldots$.
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+
+A labeled point is represented by the case class
+[`LabeledPoint`](api/mllib/index.html#org.apache.spark.mllib.regression.LabeledPoint).
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.Vectors
+import org.apache.spark.mllib.regression.LabeledPoint
+
+// Create a labeled point with a positive label and a dense feature vector.
+val pos = LabeledPoint(1.0, Vectors.dense(1.0, 0.0, 3.0))
+
+// Create a labeled point with a negative label and a sparse feature vector.
+val neg = LabeledPoint(0.0, Vectors.sparse(3, Array(0, 2), Array(1.0, 3.0)))
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+A labeled point is represented by
+[`LabeledPoint`](api/mllib/index.html#org.apache.spark.mllib.regression.LabeledPoint).
+
+{% highlight java %}
+import org.apache.spark.mllib.linalg.Vectors;
+import org.apache.spark.mllib.regression.LabeledPoint;
+
+// Create a labeled point with a positive label and a dense feature vector.
+LabeledPoint pos = new LabeledPoint(1.0, Vectors.dense(1.0, 0.0, 3.0));
+
+// Create a labeled point with a negative label and a sparse feature vector.
+LabeledPoint neg = new LabeledPoint(1.0, Vectors.sparse(3, new int[] {0, 2}, new double[] {1.0, 3.0}));
+{% endhighlight %}
+</div>
+
+<div data-lang="python" markdown="1">
+
+A labeled point is represented by
+[`LabeledPoint`](api/pyspark/pyspark.mllib.regression.LabeledPoint-class.html).
+
+{% highlight python %}
+from pyspark.mllib.linalg import SparseVector
+from pyspark.mllib.regression import LabeledPoint
+
+# Create a labeled point with a positive label and a dense feature vector.
+pos = LabeledPoint(1.0, [1.0, 0.0, 3.0])
+
+# Create a labeled point with a negative label and a sparse feature vector.
+neg = LabeledPoint(0.0, SparseVector(3, [0, 2], [1.0, 3.0]))
+{% endhighlight %}
+</div>
+</div>
+
+***Sparse data***
+
+It is very common in practice to have sparse training data.  MLlib supports reading training
+examples stored in `LIBSVM` format, which is the default format used by
+[`LIBSVM`](http://www.csie.ntu.edu.tw/~cjlin/libsvm/) and
+[`LIBLINEAR`](http://www.csie.ntu.edu.tw/~cjlin/liblinear/).  It is a text format.  Each line
+represents a labeled sparse feature vector using the following format:
+
+~~~
+label index1:value1 index2:value2 ...
+~~~
+
+where the indices are one-based and in ascending order. 
+After loading, the feature indices are converted to zero-based.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+[`MLUtils.loadLibSVMData`](api/mllib/index.html#org.apache.spark.mllib.util.MLUtils$) reads training
+examples stored in LIBSVM format.
+
+{% highlight scala %}
+import org.apache.spark.mllib.regression.LabeledPoint
+import org.apache.spark.mllib.util.MLUtils
+import org.apache.spark.rdd.RDD
+
+val training: RDD[LabeledPoint] = MLUtils.loadLibSVMData(sc, "mllib/data/sample_libsvm_data.txt")
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+[`MLUtils.loadLibSVMData`](api/mllib/index.html#org.apache.spark.mllib.util.MLUtils$) reads training
+examples stored in LIBSVM format.
+
+{% highlight java %}
+import org.apache.spark.mllib.regression.LabeledPoint;
+import org.apache.spark.mllib.util.MLUtils;
+import org.apache.spark.rdd.RDDimport;
+
+RDD[LabeledPoint] training = MLUtils.loadLibSVMData(sc, "mllib/data/sample_libsvm_data.txt")
+{% endhighlight %}
+</div>
+</div>
+
+## Local matrix
+
+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 `\[ \begin{pmatrix}
+1.0 & 2.0 \\
+3.0 & 4.0 \\
+5.0 & 6.0
+\end{pmatrix}
+\]`
+is stored in a one-dimensional array `[1.0, 3.0, 5.0, 2.0, 4.0, 6.0]` with the matrix size `(3, 2)`.
+We are going to add sparse matrix in the next release.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+The base class of local matrices is
+[`Matrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.Matrix), and we provide one
+implementation: [`DenseMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.DenseMatrix).
+Sparse matrix will be added in the next release.  We recommend using the factory methods implemented
+in [`Matrices`](api/mllib/index.html#org.apache.spark.mllib.linalg.Matrices) to create local
+matrices.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.{Matrix, Matrices}
+
+// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))
+val dm: Matrix = Matrices.dense(3, 2, Array(1.0, 3.0, 5.0, 2.0, 4.0, 6.0))
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+The base class of local matrices is
+[`Matrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.Matrix), and we provide one
+implementation: [`DenseMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.DenseMatrix).
+Sparse matrix will be added in the next release.  We recommend using the factory methods implemented
+in [`Matrices`](api/mllib/index.html#org.apache.spark.mllib.linalg.Matrices) to create local
+matrices.
+
+{% highlight java %}
+import org.apache.spark.mllib.linalg.Matrix;
+import org.apache.spark.mllib.linalg.Matrices;
+
+// Create a dense matrix ((1.0, 2.0), (3.0, 4.0), (5.0, 6.0))
+Matrix dm = Matrices.dense(3, 2, new double[] {1.0, 3.0, 5.0, 2.0, 4.0, 6.0});
+{% endhighlight %}
+</div>
+
+</div>
+
+## Distributed matrix
+
+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.
+
+***Note***
+
+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.
+
+### RowMatrix
+
+A `RowMatrix` 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 `data matrix` 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.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+A [`RowMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix) can be
+created from an `RDD[Vector]` instance.  Then we can compute its column summary statistics.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.Vector
+import org.apache.spark.mllib.linalg.distributed.RowMatrix
+
+val rows: RDD[Vector] = ... // an RDD of local vectors
+// Create a RowMatrix from an RDD[Vector].
+val mat: RowMatrix = new RowMatrix(rows)
+
+// Get its size.
+val m = mat.numRows()
+val n = mat.numCols()
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+A [`RowMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.RowMatrix) can be
+created from a `JavaRDD<Vector>` instance.  Then we can compute its column summary statistics.
+
+{% highlight java %}
+import org.apache.spark.mllib.linalg.Vector;
+import org.apache.spark.mllib.linalg.distributed.RowMatrix;
+
+JavaRDD<Vector> rows = ... // a JavaRDD of local vectors
+// Create a RowMatrix from an JavaRDD<Vector>.
+RowMatrix mat = new RowMatrix(rows.rdd());
+
+// Get its size.
+long m = mat.numRows();
+long n = mat.numCols();
+{% endhighlight %}
+</div>
+</div>
+
+#### Multivariate summary statistics
+
+We provide column summary statistics for `RowMatrix`. 
+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.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+`RowMatrix#computeColumnSummaryStatistics` returns an instance of
+[`MultivariateStatisticalSummary`](api/mllib/index.html#org.apache.spark.mllib.stat.MultivariateStatisticalSummary),
+which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
+total count.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.Matrix
+import org.apache.spark.mllib.linalg.distributed.RowMatrix
+import org.apache.spark.mllib.stat.MultivariateStatisticalSummary
+
+val mat: RowMatrix = ... // a RowMatrix
+
+// Compute column summary statistics.
+val summary: MultivariateStatisticalSummary = mat.computeColumnSummaryStatistics()
+println(summary.mean) // a dense vector containing the mean value for each column
+println(summary.variance) // column-wise variance
+println(summary.numNonzers) // number of nonzeros in each column
+
+// Compute the covariance matrix.
+val Cov: Matrix = mat.computeCovariance()
+{% endhighlight %}
+</div>
+</div>
+
+### IndexedRowMatrix
+
+An `IndexedRowMatrix` is similar to a `RowMatrix` 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.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+An
+[`IndexedRowMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix)
+can be created from an `RDD[IndexedRow]` instance, where
+[`IndexedRow`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRow) is a
+wrapper over `(Long, Vector)`.  An `IndexedRowMatrix` can be converted to a `RowMatrix` by dropping
+its row indices.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.distributed.{IndexedRow, IndexedRowMatrix, RowMatrix}
+
+val rows: RDD[IndexedRow] = ... // an RDD of indexed rows
+// Create an IndexedRowMatrix from an RDD[IndexedRow].
+val mat: IndexedRowMatrix = new IndexedRowMatrix(rows)
+
+// Get its size.
+val m = mat.numRows()
+val n = mat.numCols()
+
+// Drop its row indices.
+val rowMat: RowMatrix = mat.toRowMatrix()
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+An
+[`IndexedRowMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix)
+can be created from an `JavaRDD<IndexedRow>` instance, where
+[`IndexedRow`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.IndexedRow) is a
+wrapper over `(long, Vector)`.  An `IndexedRowMatrix` can be converted to a `RowMatrix` by dropping
+its row indices.
+
+{% highlight java %}
+import org.apache.spark.mllib.linalg.distributed.IndexedRow;
+import org.apache.spark.mllib.linalg.distributed.IndexedRowMatrix;
+import org.apache.spark.mllib.linalg.distributed.RowMatrix;
+
+JavaRDD[IndexedRow] rows = ... // a JavaRDD of indexed rows
+// Create an IndexedRowMatrix from a JavaRDD<IndexedRow>.
+IndexedRowMatrix mat = new IndexedRowMatrix(rows.rdd());
+
+// Get its size.
+long m = mat.numRows();
+long n = mat.numCols();
+
+// Drop its row indices.
+RowMatrix rowMat = mat.toRowMatrix();
+{% endhighlight %}
+</div></div>
+
+### CoordinateMatrix
+
+A `CoordinateMatrix` is a distributed matrix backed by an RDD of its entries.  Each entry is a tuple
+of `(i: Long, j: Long, value: Double)`, where `i` is the row index, `j` is the column index, and
+`value` is the entry value.  A `CoordinateMatrix` should be used only in the case when both
+dimensions of the matrix are huge and the matrix is very sparse.
+
+<div class="codetabs">
+<div data-lang="scala" markdown="1">
+
+A
+[`CoordinateMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.CoordinateMatrix)
+can be created from an `RDD[MatrixEntry]` instance, where
+[`MatrixEntry`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.MatrixEntry) is a
+wrapper over `(Long, Long, Double)`.  A `CoordinateMatrix` can be converted to a `IndexedRowMatrix`
+with sparse rows by calling `toIndexedRowMatrix`.  In this release, we do not provide other
+computation for `CoordinateMatrix`.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.distributed.{CoordinateMatrix, MatrixEntry}
+
+val entries: RDD[MatrixEntry] = ... // an RDD of matrix entries
+// Create a CoordinateMatrix from an RDD[MatrixEntry].
+val mat: CoordinateMatrix = new CoordinateMatrix(entries)
+
+// Get its size.
+val m = mat.numRows()
+val n = mat.numCols()
+
+// Convert it to an IndexRowMatrix whose rows are sparse vectors.
+val indexedRowMatrix = mat.toIndexedRowMatrix()
+{% endhighlight %}
+</div>
+
+<div data-lang="java" markdown="1">
+
+A
+[`CoordinateMatrix`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.CoordinateMatrix)
+can be created from a `JavaRDD<MatrixEntry>` instance, where
+[`MatrixEntry`](api/mllib/index.html#org.apache.spark.mllib.linalg.distributed.MatrixEntry) is a
+wrapper over `(long, long, double)`.  A `CoordinateMatrix` can be converted to a `IndexedRowMatrix`
+with sparse rows by calling `toIndexedRowMatrix`.
+
+{% highlight scala %}
+import org.apache.spark.mllib.linalg.distributed.CoordinateMatrix;
+import org.apache.spark.mllib.linalg.distributed.MatrixEntry;
+
+JavaRDD<MatrixEntry> entries = ... // a JavaRDD of matrix entries
+// Create a CoordinateMatrix from a JavaRDD<MatrixEntry>.
+CoordinateMatrix mat = new CoordinateMatrix(entries);
+
+// Get its size.
+long m = mat.numRows();
+long n = mat.numCols();
+
+// Convert it to an IndexRowMatrix whose rows are sparse vectors.
+IndexedRowMatrix indexedRowMatrix = mat.toIndexedRowMatrix();
+{% endhighlight %}
+</div>
+</div>