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From pwend...@apache.org
Subject git commit: Merge pull request #552 from martinjaggi/master. Closes #552.
Date Sat, 08 Feb 2014 19:39:23 GMT
Updated Branches:

Merge pull request #552 from martinjaggi/master. Closes #552.

tex formulas in the documentation

using mathjax.
and spliting the MLlib documentation by techniques

see jira
https://spark-project.atlassian.net/browse/MLLIB-19
and
https://github.com/shivaram/spark/compare/mathjax

Author: Martin Jaggi <m.jaggi@gmail.com>

== Merge branch commits ==

commit 0364bfabbfc347f917216057a20c39b631842481
Author: Martin Jaggi <m.jaggi@gmail.com>
Date:   Fri Feb 7 03:19:38 2014 +0100

minor polishing, as suggested by @pwendell

Author: Martin Jaggi <m.jaggi@gmail.com>
Date:   Thu Feb 6 18:04:26 2014 +0100

enabling inline latex formulas with $.$

same mathjax configuration as used in math.stackexchange.com

sample usage in the linear algebra (SVD) documentation

commit bbafafd2b497a5acaa03a140bb9de1fbb7d67ffa
Author: Martin Jaggi <m.jaggi@gmail.com>
Date:   Thu Feb 6 17:31:29 2014 +0100

split MLlib documentation by techniques

and linked from the main mllib-guide.md site

commit d1c5212b93c67436543c2d8ddbbf610fdf0a26eb
Author: Martin Jaggi <m.jaggi@gmail.com>
Date:   Thu Feb 6 16:59:43 2014 +0100

enable mathjax formula in the .md documentation files

code by @shivaram

commit d73948db0d9bc36296054e79fec5b1a657b4eab4
Author: Martin Jaggi <m.jaggi@gmail.com>
Date:   Thu Feb 6 16:57:23 2014 +0100

minor update on how to compile the documentation

Project: http://git-wip-us.apache.org/repos/asf/incubator-spark/repo
Commit: http://git-wip-us.apache.org/repos/asf/incubator-spark/commit/fabf1749
Tree: http://git-wip-us.apache.org/repos/asf/incubator-spark/tree/fabf1749
Diff: http://git-wip-us.apache.org/repos/asf/incubator-spark/diff/fabf1749

Commit: fabf1749995103841e6a3975892572f376ee48d0
Parents: 3a9d82c
Author: Martin Jaggi <m.jaggi@gmail.com>
Authored: Sat Feb 8 11:39:13 2014 -0800
Committer: Patrick Wendell <pwendell@gmail.com>
Committed: Sat Feb 8 11:39:13 2014 -0800

----------------------------------------------------------------------
docs/_layouts/global.html               |  13 +
docs/css/main.css                       |   8 +
docs/mllib-classification-regression.md | 206 +++++++++++
docs/mllib-clustering.md                | 106 ++++++
docs/mllib-collaborative-filtering.md   | 130 +++++++
docs/mllib-guide.md                     | 490 ++-------------------------
docs/mllib-linear-algebra.md            |  61 ++++
docs/mllib-optimization.md              |  40 +++
9 files changed, 586 insertions(+), 472 deletions(-)
----------------------------------------------------------------------

----------------------------------------------------------------------
index dfcf753..cc09d6e 100644
@@ -10,9 +10,9 @@ We include the Spark documentation as part of the source (as opposed to using a

In this directory you will find textfiles formatted using Markdown, with an ".md" suffix. You can read those text files directly if you want. Start with index.md.

-To make things quite a bit prettier and make the links easier to follow, generate the html version of the documentation based on the src directory by running jekyll in the docs directory. Use the command SKIP_SCALADOC=1 jekyll to skip building and copying over the scaladoc which can be timely. To use the jekyll command, you will need to have Jekyll installed, the easiest way to do this is via a Ruby Gem, see the [jekyll installation instructions](https://github.com/mojombo/jekyll/wiki/install). This will create a directory called _site containing index.html as well as the rest of the compiled files. Read more about Jekyll at https://github.com/mojombo/jekyll/wiki.
+To make things quite a bit prettier and make the links easier to follow, generate the html version of the documentation based on the src directory by running jekyll build in the docs directory. Use the command SKIP_SCALADOC=1 jekyll build to skip building and copying over the scaladoc which can be timely. To use the jekyll command, you will need to have Jekyll installed, the easiest way to do this is via a Ruby Gem, see the [jekyll installation instructions](http://jekyllrb.com/docs/installation). This will create a directory called _site containing index.html as well as the rest of the compiled files. Read more about Jekyll at https://github.com/mojombo/jekyll/wiki.

-In addition to generating the site as html from the markdown files, jekyll can serve up the site via a webserver. To build and run a webserver use the command jekyll --server which (currently) runs the webserver on port 4000, then visit the site at http://localhost:4000.
+In addition to generating the site as html from the markdown files, jekyll can serve up the site via a webserver. To build and run a local webserver use the command jekyll serve (or the faster variant SKIP_SCALADOC=1 jekyll serve), which runs the webserver on port 4000, then visit the site at http://localhost:4000.

## Pygments

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/_layouts/global.html
----------------------------------------------------------------------
diff --git a/docs/_layouts/global.html b/docs/_layouts/global.html
index 3352595..b65686c 100755
--- a/docs/_layouts/global.html
+++ b/docs/_layouts/global.html
@@ -195,4 +195,17 @@
</script>

</body>
+    <!-- MathJax Section -->
+    <script type="text/javascript"
+     src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
+    <script>
+      MathJax.Hub.Config({
+        tex2jax: {
+          inlineMath: [ ["$", "$"], ["\\\$$","\\\$$"] ],
+          displayMath: [ ["$$","$$"], ["\$", "\$"] ],
+          processEscapes: true,
+          skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
+        }
+      });
+    </script>
</html>

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/css/main.css
----------------------------------------------------------------------
diff --git a/docs/css/main.css b/docs/css/main.css
index 8566400..f6fe7d5 100755
--- a/docs/css/main.css
+++ b/docs/css/main.css
.nav-tabs > li > a, .nav-tabs > li > a:hover {
color: #333;
}
+
+/**
+ * MathJax (embedded latex formulas)
+ */
+.MathJax .mo { color: inherit }
+.MathJax .mi { color: inherit }
+.MathJax .mf { color: inherit }
+.MathJax .mh { color: inherit }

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-classification-regression.md
----------------------------------------------------------------------
diff --git a/docs/mllib-classification-regression.md b/docs/mllib-classification-regression.md
new file mode 100644
index 0000000..edb9338
--- /dev/null
+++ b/docs/mllib-classification-regression.md
@@ -0,0 +1,206 @@
+---
+layout: global
+title: MLlib - Classification and Regression
+---
+
+{:toc}
+
+
+# Binary Classification
+
+Binary classification is a supervised learning problem in which we want to
+classify entities into one of two distinct categories or labels, e.g.,
+predicting whether or not emails are spam.  This problem involves executing a
+learning *Algorithm* on a set of *labeled* examples, i.e., a set of entities
+represented via (numerical) features along with underlying category labels.
+The algorithm returns a trained *Model* that can predict the label for new
+entities for which the underlying label is unknown.
+
+MLlib currently supports two standard model families for binary classification,
+namely [Linear Support Vector Machines
+(SVMs)](http://en.wikipedia.org/wiki/Support_vector_machine) and [Logistic
+Regression](http://en.wikipedia.org/wiki/Logistic_regression), along with [L1
+and L2 regularized](http://en.wikipedia.org/wiki/Regularization_(mathematics))
+variants of each model family.  The training algorithms all leverage an
+[below](#gradient-descent-primitive)), and take as input a regularization
+parameter (*regParam*) along with various parameters associated with gradient
+descent (*stepSize*, *numIterations*, *miniBatchFraction*).
+
+Available algorithms for binary classification:
+
+* [SVMWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.SVMWithSGD)
+* [LogisticRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.LogisticRegressionWithSGD)
+
+# Linear Regression
+
+Linear regression is another classical supervised learning setting.  In this
+problem, each entity is associated with a real-valued label (as opposed to a
+binary label as in binary classification), and we want to predict labels as
+closely as possible given numerical features representing entities.  MLlib
+supports linear regression as well as L1
+([lasso](http://en.wikipedia.org/wiki/Lasso_(statistics)#Lasso_method)) and L2
+([ridge](http://en.wikipedia.org/wiki/Ridge_regression)) regularized variants.
+The regression algorithms in MLlib also leverage the underlying gradient
+descent primitive (described [below](#gradient-descent-primitive)), and have
+the same parameters as the binary classification algorithms described above.
+
+Available algorithms for linear regression:
+
+* [LinearRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.LinearRegressionWithSGD)
+* [RidgeRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.RidgeRegressionWithSGD)
+* [LassoWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.LassoWithSGD)
+
+Behind the scenes, all above methods use the SGD implementation from the
+gradient descent primitive in MLlib, see the
+<a href="mllib-optimization.html">optimization</a> part:
+
+
+
+# Usage in Scala
+
+Following code snippets can be executed in spark-shell.
+
+## Binary Classification
+
+The following code snippet illustrates how to load a sample dataset, execute a
+training algorithm on this training data using a static method in the algorithm
+object, and make predictions with the resulting model to compute the training
+error.
+
+{% highlight scala %}
+import org.apache.spark.SparkContext
+import org.apache.spark.mllib.classification.SVMWithSGD
+import org.apache.spark.mllib.regression.LabeledPoint
+
+// Load and parse the data file
+val data = sc.textFile("mllib/data/sample_svm_data.txt")
+val parsedData = data.map { line =>
+  val parts = line.split(' ')
+  LabeledPoint(parts(0).toDouble, parts.tail.map(x => x.toDouble).toArray)
+}
+
+// Run training algorithm to build the model
+val numIterations = 20
+val model = SVMWithSGD.train(parsedData, numIterations)
+
+// Evaluate model on training examples and compute training error
+val labelAndPreds = parsedData.map { point =>
+  val prediction = model.predict(point.features)
+  (point.label, prediction)
+}
+val trainErr = labelAndPreds.filter(r => r._1 != r._2).count.toDouble / parsedData.count
+println("Training Error = " + trainErr)
+{% endhighlight %}
+
+
+The SVMWithSGD.train() method by default performs L2 regularization with the
+regularization parameter set to 1.0. If we want to configure this algorithm, we
+can customize SVMWithSGD further by creating a new object directly and
+calling setter methods. All other MLlib algorithms support customization in
+this way as well. For example, the following code produces an L1 regularized
+variant of SVMs with regularization parameter set to 0.1, and runs the training
+algorithm for 200 iterations.
+
+{% highlight scala %}
+import org.apache.spark.mllib.optimization.L1Updater
+
+val svmAlg = new SVMWithSGD()
+svmAlg.optimizer.setNumIterations(200)
+  .setRegParam(0.1)
+  .setUpdater(new L1Updater)
+val modelL1 = svmAlg.run(parsedData)
+{% endhighlight %}
+
+## Linear Regression
+The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
+example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
+compute the Mean Squared Error at the end to evaluate
+[goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
+
+{% highlight scala %}
+import org.apache.spark.mllib.regression.LinearRegressionWithSGD
+import org.apache.spark.mllib.regression.LabeledPoint
+
+// Load and parse the data
+val data = sc.textFile("mllib/data/ridge-data/lpsa.data")
+val parsedData = data.map { line =>
+  val parts = line.split(',')
+  LabeledPoint(parts(0).toDouble, parts(1).split(' ').map(x => x.toDouble).toArray)
+}
+
+// Building the model
+val numIterations = 20
+val model = LinearRegressionWithSGD.train(parsedData, numIterations)
+
+// Evaluate model on training examples and compute training error
+val valuesAndPreds = parsedData.map { point =>
+  val prediction = model.predict(point.features)
+  (point.label, prediction)
+}
+val MSE = valuesAndPreds.map{ case(v, p) => math.pow((v - p), 2)}.reduce(_ + _)/valuesAndPreds.count
+println("training Mean Squared Error = " + MSE)
+{% endhighlight %}
+
+
+Similarly you can use RidgeRegressionWithSGD and LassoWithSGD and compare training
+[Mean Squared Errors](http://en.wikipedia.org/wiki/Mean_squared_error).
+
+
+# Usage in Java
+
+All of MLlib's methods use Java-friendly types, so you can import and call them there the same
+way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the
+Spark Java API uses a separate JavaRDD class. You can convert a Java RDD to a Scala one by
+calling .rdd() on your JavaRDD object.
+
+# Usage in Python
+Following examples can be tested in the PySpark shell.
+
+## Binary Classification
+The following example shows how to load a sample dataset, build Logistic Regression model,
+and make predictions with the resulting model to compute the training error.
+
+{% highlight python %}
+from pyspark.mllib.classification import LogisticRegressionWithSGD
+from numpy import array
+
+# Load and parse the data
+data = sc.textFile("mllib/data/sample_svm_data.txt")
+parsedData = data.map(lambda line: array([float(x) for x in line.split(' ')]))
+model = LogisticRegressionWithSGD.train(parsedData)
+
+# Build the model
+labelsAndPreds = parsedData.map(lambda point: (int(point.item(0)),
+        model.predict(point.take(range(1, point.size)))))
+
+# Evaluating the model on training data
+trainErr = labelsAndPreds.filter(lambda (v, p): v != p).count() / float(parsedData.count())
+print("Training Error = " + str(trainErr))
+{% endhighlight %}
+
+## Linear Regression
+The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
+example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
+compute the Mean Squared Error at the end to evaluate
+[goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
+
+{% highlight python %}
+from pyspark.mllib.regression import LinearRegressionWithSGD
+from numpy import array
+
+# Load and parse the data
+data = sc.textFile("mllib/data/ridge-data/lpsa.data")
+parsedData = data.map(lambda line: array([float(x) for x in line.replace(',', ' ').split(' ')]))
+
+# Build the model
+model = LinearRegressionWithSGD.train(parsedData)
+
+# Evaluate the model on training data
+valuesAndPreds = parsedData.map(lambda point: (point.item(0),
+        model.predict(point.take(range(1, point.size)))))
+MSE = valuesAndPreds.map(lambda (v, p): (v - p)**2).reduce(lambda x, y: x + y)/valuesAndPreds.count()
+print("Mean Squared Error = " + str(MSE))
+{% endhighlight %}

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-clustering.md
----------------------------------------------------------------------
diff --git a/docs/mllib-clustering.md b/docs/mllib-clustering.md
new file mode 100644
index 0000000..65ed75b
--- /dev/null
+++ b/docs/mllib-clustering.md
@@ -0,0 +1,106 @@
+---
+layout: global
+title: MLlib - Clustering
+---
+
+{:toc}
+
+
+# Clustering
+
+Clustering is an unsupervised learning problem whereby we aim to group subsets
+of entities with one another based on some notion of similarity.  Clustering is
+often used for exploratory analysis and/or as a component of a hierarchical
+supervised learning pipeline (in which distinct classifiers or regression
+models are trained for each cluster). MLlib supports
+[k-means](http://en.wikipedia.org/wiki/K-means_clustering) clustering, one of
+the most commonly used clustering algorithms that clusters the data points into
+predfined number of clusters. The MLlib implementation includes a parallelized
+variant of the [k-means++](http://en.wikipedia.org/wiki/K-means%2B%2B) method
+called [kmeans||](http://theory.stanford.edu/~sergei/papers/vldb12-kmpar.pdf).
+The implementation in MLlib has the following parameters:
+
+* *k* is the number of desired clusters.
+* *maxIterations* is the maximum number of iterations to run.
+* *initializationMode* specifies either random initialization or
+initialization via k-means\|\|.
+* *runs* is the number of times to run the k-means algorithm (k-means is not
+guaranteed to find a globally optimal solution, and when run multiple times on
+a given dataset, the algorithm returns the best clustering result).
+* *initializiationSteps* determines the number of steps in the k-means\|\| algorithm.
+* *epsilon* determines the distance threshold within which we consider k-means to have converged.
+
+Available algorithms for clustering:
+
+* [KMeans](api/mllib/index.html#org.apache.spark.mllib.clustering.KMeans)
+
+
+
+# Usage in Scala
+
+Following code snippets can be executed in spark-shell.
+
+In the following example after loading and parsing data, we use the KMeans object to cluster the data
+into two clusters. The number of desired clusters is passed to the algorithm. We then compute Within
+Set Sum of Squared Error (WSSSE). You can reduce this error measure by increasing *k*. In fact the
+optimal *k* is usually one where there is an "elbow" in the WSSSE graph.
+
+{% highlight scala %}
+import org.apache.spark.mllib.clustering.KMeans
+
+// Load and parse the data
+val data = sc.textFile("kmeans_data.txt")
+val parsedData = data.map( _.split(' ').map(_.toDouble))
+
+// Cluster the data into two classes using KMeans
+val numIterations = 20
+val numClusters = 2
+val clusters = KMeans.train(parsedData, numClusters, numIterations)
+
+// Evaluate clustering by computing Within Set Sum of Squared Errors
+val WSSSE = clusters.computeCost(parsedData)
+println("Within Set Sum of Squared Errors = " + WSSSE)
+{% endhighlight %}
+
+
+# Usage in Java
+
+All of MLlib's methods use Java-friendly types, so you can import and call them there the same
+way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the
+Spark Java API uses a separate JavaRDD class. You can convert a Java RDD to a Scala one by
+calling .rdd() on your JavaRDD object.
+
+# Usage in Python
+Following examples can be tested in the PySpark shell.
+
+In the following example after loading and parsing data, we use the KMeans object to cluster the data
+into two clusters. The number of desired clusters is passed to the algorithm. We then compute Within
+Set Sum of Squared Error (WSSSE). You can reduce this error measure by increasing *k*. In fact the
+optimal *k* is usually one where there is an "elbow" in the WSSSE graph.
+
+{% highlight python %}
+from pyspark.mllib.clustering import KMeans
+from numpy import array
+from math import sqrt
+
+# Load and parse the data
+data = sc.textFile("kmeans_data.txt")
+parsedData = data.map(lambda line: array([float(x) for x in line.split(' ')]))
+
+# Build the model (cluster the data)
+clusters = KMeans.train(parsedData, 2, maxIterations=10,
+        runs=30, initialization_mode="random")
+
+# Evaluate clustering by computing Within Set Sum of Squared Errors
+def error(point):
+    center = clusters.centers[clusters.predict(point)]
+    return sqrt(sum([x**2 for x in (point - center)]))
+
+WSSSE = parsedData.map(lambda point: error(point)).reduce(lambda x, y: x + y)
+print("Within Set Sum of Squared Error = " + str(WSSSE))
+{% endhighlight %}
+
+Similarly you can use RidgeRegressionWithSGD and LassoWithSGD and compare training Mean Squared
+Errors.
+

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-collaborative-filtering.md
----------------------------------------------------------------------
diff --git a/docs/mllib-collaborative-filtering.md b/docs/mllib-collaborative-filtering.md
new file mode 100644
index 0000000..aa22f67
--- /dev/null
+++ b/docs/mllib-collaborative-filtering.md
@@ -0,0 +1,130 @@
+---
+layout: global
+title: MLlib - Collaborative Filtering
+---
+
+{:toc}
+
+# Collaborative Filtering
+
+[Collaborative filtering](http://en.wikipedia.org/wiki/Recommender_system#Collaborative_filtering)
+is commonly used for recommender systems.  These techniques aim to fill in the
+missing entries of a user-item association matrix.  MLlib currently supports
+model-based collaborative filtering, in which users and products are described
+by a small set of latent factors that can be used to predict missing entries.
+In particular, we implement the [alternating least squares
+(ALS)](http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf)
+algorithm to learn these latent factors. The implementation in MLlib has the
+following parameters:
+
+* *numBlocks* is the number of blacks used to parallelize computation (set to -1 to auto-configure).
+* *rank* is the number of latent factors in our model.
+* *iterations* is the number of iterations to run.
+* *lambda* specifies the regularization parameter in ALS.
+* *implicitPrefs* specifies whether to use the *explicit feedback* ALS variant or one adapted for *implicit feedback* data
+* *alpha* is a parameter applicable to the implicit feedback variant of ALS that governs the *baseline* confidence in preference observations
+
+## Explicit vs Implicit Feedback
+
+The standard approach to matrix factorization based collaborative filtering treats
+the entries in the user-item matrix as *explicit* preferences given by the user to the item.
+
+It is common in many real-world use cases to only have access to *implicit feedback*
+(e.g. views, clicks, purchases, likes, shares etc.). The approach used in MLlib to deal with
+such data is taken from
+[Collaborative Filtering for Implicit Feedback Datasets](http://www2.research.att.com/~yifanhu/PUB/cf.pdf).
+Essentially instead of trying to model the matrix of ratings directly, this approach treats the data as
+a combination of binary preferences and *confidence values*. The ratings are then related
+to the level of confidence in observed user preferences, rather than explicit ratings given to items.
+The model then tries to find latent factors that can be used to predict the expected preference of a user
+for an item.
+
+Available algorithms for collaborative filtering:
+
+* [ALS](api/mllib/index.html#org.apache.spark.mllib.recommendation.ALS)
+
+
+# Usage in Scala
+
+Following code snippets can be executed in spark-shell.
+
+In the following example we load rating data. Each row consists of a user, a product and a rating.
+We use the default ALS.train() method which assumes ratings are explicit. We evaluate the recommendation
+model by measuring the Mean Squared Error of rating prediction.
+
+{% highlight scala %}
+import org.apache.spark.mllib.recommendation.ALS
+import org.apache.spark.mllib.recommendation.Rating
+
+// Load and parse the data
+val data = sc.textFile("mllib/data/als/test.data")
+val ratings = data.map(_.split(',') match {
+    case Array(user, item, rate) =>  Rating(user.toInt, item.toInt, rate.toDouble)
+})
+
+// Build the recommendation model using ALS
+val numIterations = 20
+val model = ALS.train(ratings, 1, 20, 0.01)
+
+// Evaluate the model on rating data
+val usersProducts = ratings.map{ case Rating(user, product, rate)  => (user, product)}
+val predictions = model.predict(usersProducts).map{
+    case Rating(user, product, rate) => ((user, product), rate)
+}
+val ratesAndPreds = ratings.map{
+    case Rating(user, product, rate) => ((user, product), rate)
+}.join(predictions)
+val MSE = ratesAndPreds.map{
+    case ((user, product), (r1, r2)) =>  math.pow((r1- r2), 2)
+}.reduce(_ + _)/ratesAndPreds.count
+println("Mean Squared Error = " + MSE)
+{% endhighlight %}
+
+If the rating matrix is derived from other source of information (i.e., it is inferred from
+other signals), you can use the trainImplicit method to get better results.
+
+{% highlight scala %}
+val model = ALS.trainImplicit(ratings, 1, 20, 0.01)
+{% endhighlight %}
+
+# Usage in Java
+
+All of MLlib's methods use Java-friendly types, so you can import and call them there the same
+way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the
+Spark Java API uses a separate JavaRDD class. You can convert a Java RDD to a Scala one by
+calling .rdd() on your JavaRDD object.
+
+# Usage in Python
+Following examples can be tested in the PySpark shell.
+
+In the following example we load rating data. Each row consists of a user, a product and a rating.
+We use the default ALS.train() method which assumes ratings are explicit. We evaluate the
+recommendation by measuring the Mean Squared Error of rating prediction.
+
+{% highlight python %}
+from pyspark.mllib.recommendation import ALS
+from numpy import array
+
+# Load and parse the data
+data = sc.textFile("mllib/data/als/test.data")
+ratings = data.map(lambda line: array([float(x) for x in line.split(',')]))
+
+# Build the recommendation model using Alternating Least Squares
+model = ALS.train(ratings, 1, 20)
+
+# Evaluate the model on training data
+testdata = ratings.map(lambda p: (int(p[0]), int(p[1])))
+predictions = model.predictAll(testdata).map(lambda r: ((r[0], r[1]), r[2]))
+ratesAndPreds = ratings.map(lambda r: ((r[0], r[1]), r[2])).join(predictions)
+MSE = ratesAndPreds.map(lambda r: (r[1][0] - r[1][1])**2).reduce(lambda x, y: x + y)/ratesAndPreds.count()
+print("Mean Squared Error = " + str(MSE))
+{% endhighlight %}
+
+If the rating matrix is derived from other source of information (i.e., it is inferred from other
+signals), you can use the trainImplicit method to get better results.
+
+{% highlight python %}
+# Build the recommendation model using Alternating Least Squares based on implicit ratings
+model = ALS.trainImplicit(ratings, 1, 20)
+{% endhighlight %}

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-guide.md
----------------------------------------------------------------------
diff --git a/docs/mllib-guide.md b/docs/mllib-guide.md
index 0e34da4..76308ec 100644
--- a/docs/mllib-guide.md
+++ b/docs/mllib-guide.md
@@ -3,16 +3,32 @@ layout: global
title: Machine Learning Library (MLlib)
---

-{:toc}

MLlib is a Spark implementation of some common machine learning (ML)
functionality, as well associated tests and data generators.  MLlib
currently supports four common types of machine learning problem settings,
namely, binary classification, regression, clustering and collaborative
filtering, as well as an underlying gradient descent optimization primitive.
-This guide will outline the functionality supported in MLlib and also provides
-an example of invoking MLlib.
+
+# Available Methods
+The following links provide a detailed explanation of the methods and usage examples for each of them:
+
+* <a href="mllib-classification-regression.html">Classification and Regression</a>
+  * Binary Classification
+    * SVM (L1 and L2 regularized)
+    * Logistic Regression (L1 and L2 regularized)
+  * Linear Regression
+    * Least Squares
+    * Lasso
+    * Ridge Regression
+* <a href="mllib-clustering.html">Clustering</a>
+  * k-Means
+* <a href="mllib-collaborative-filtering.html">Collaborative Filtering</a>
+  * Matrix Factorization using Alternating Least Squares
+* <a href="mllib-optimization.html">Optimization</a>
+* <a href="mllib-linear-algebra.html">Linear Algebra</a>
+  * Singular Value Decomposition

# Dependencies
MLlib uses the [jblas](https://github.com/mikiobraun/jblas) linear algebra library, which itself
@@ -24,469 +40,3 @@ detect these libraries automatically.
To use MLlib in Python, you will need [NumPy](http://www.numpy.org) version 1.7 or newer
and Python 2.7.

-# Binary Classification
-
-Binary classification is a supervised learning problem in which we want to
-classify entities into one of two distinct categories or labels, e.g.,
-predicting whether or not emails are spam.  This problem involves executing a
-learning *Algorithm* on a set of *labeled* examples, i.e., a set of entities
-represented via (numerical) features along with underlying category labels.
-The algorithm returns a trained *Model* that can predict the label for new
-entities for which the underlying label is unknown.
-
-MLlib currently supports two standard model families for binary classification,
-namely [Linear Support Vector Machines
-(SVMs)](http://en.wikipedia.org/wiki/Support_vector_machine) and [Logistic
-Regression](http://en.wikipedia.org/wiki/Logistic_regression), along with [L1
-and L2 regularized](http://en.wikipedia.org/wiki/Regularization_(mathematics))
-variants of each model family.  The training algorithms all leverage an
-[below](#gradient-descent-primitive)), and take as input a regularization
-parameter (*regParam*) along with various parameters associated with gradient
-descent (*stepSize*, *numIterations*, *miniBatchFraction*).
-
-Available algorithms for binary classification:
-
-* [SVMWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.SVMWithSGD)
-* [LogisticRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.classification.LogisticRegressionWithSGD)
-
-# Linear Regression
-
-Linear regression is another classical supervised learning setting.  In this
-problem, each entity is associated with a real-valued label (as opposed to a
-binary label as in binary classification), and we want to predict labels as
-closely as possible given numerical features representing entities.  MLlib
-supports linear regression as well as L1
-([lasso](http://en.wikipedia.org/wiki/Lasso_(statistics)#Lasso_method)) and L2
-([ridge](http://en.wikipedia.org/wiki/Ridge_regression)) regularized variants.
-The regression algorithms in MLlib also leverage the underlying gradient
-descent primitive (described [below](#gradient-descent-primitive)), and have
-the same parameters as the binary classification algorithms described above.
-
-Available algorithms for linear regression:
-
-* [LinearRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.LinearRegressionWithSGD)
-* [RidgeRegressionWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.RidgeRegressionWithSGD)
-* [LassoWithSGD](api/mllib/index.html#org.apache.spark.mllib.regression.LassoWithSGD)
-
-# Clustering
-
-Clustering is an unsupervised learning problem whereby we aim to group subsets
-of entities with one another based on some notion of similarity.  Clustering is
-often used for exploratory analysis and/or as a component of a hierarchical
-supervised learning pipeline (in which distinct classifiers or regression
-models are trained for each cluster). MLlib supports
-[k-means](http://en.wikipedia.org/wiki/K-means_clustering) clustering, one of
-the most commonly used clustering algorithms that clusters the data points into
-predfined number of clusters. The MLlib implementation includes a parallelized
-variant of the [k-means++](http://en.wikipedia.org/wiki/K-means%2B%2B) method
-called [kmeans||](http://theory.stanford.edu/~sergei/papers/vldb12-kmpar.pdf).
-The implementation in MLlib has the following parameters:
-
-* *k* is the number of desired clusters.
-* *maxIterations* is the maximum number of iterations to run.
-* *initializationMode* specifies either random initialization or
-initialization via k-means\|\|.
-* *runs* is the number of times to run the k-means algorithm (k-means is not
-guaranteed to find a globally optimal solution, and when run multiple times on
-a given dataset, the algorithm returns the best clustering result).
-* *initializiationSteps* determines the number of steps in the k-means\|\| algorithm.
-* *epsilon* determines the distance threshold within which we consider k-means to have converged.
-
-Available algorithms for clustering:
-
-* [KMeans](api/mllib/index.html#org.apache.spark.mllib.clustering.KMeans)
-
-# Collaborative Filtering
-
-[Collaborative filtering](http://en.wikipedia.org/wiki/Recommender_system#Collaborative_filtering)
-is commonly used for recommender systems.  These techniques aim to fill in the
-missing entries of a user-item association matrix.  MLlib currently supports
-model-based collaborative filtering, in which users and products are described
-by a small set of latent factors that can be used to predict missing entries.
-In particular, we implement the [alternating least squares
-(ALS)](http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf)
-algorithm to learn these latent factors. The implementation in MLlib has the
-following parameters:
-
-* *numBlocks* is the number of blacks used to parallelize computation (set to -1 to auto-configure).
-* *rank* is the number of latent factors in our model.
-* *iterations* is the number of iterations to run.
-* *lambda* specifies the regularization parameter in ALS.
-* *implicitPrefs* specifies whether to use the *explicit feedback* ALS variant or one adapted for *implicit feedback* data
-* *alpha* is a parameter applicable to the implicit feedback variant of ALS that governs the *baseline* confidence in preference observations
-
-## Explicit vs Implicit Feedback
-
-The standard approach to matrix factorization based collaborative filtering treats
-the entries in the user-item matrix as *explicit* preferences given by the user to the item.
-
-It is common in many real-world use cases to only have access to *implicit feedback*
-(e.g. views, clicks, purchases, likes, shares etc.). The approach used in MLlib to deal with
-such data is taken from
-[Collaborative Filtering for Implicit Feedback Datasets](http://www2.research.att.com/~yifanhu/PUB/cf.pdf).
-Essentially instead of trying to model the matrix of ratings directly, this approach treats the data as
-a combination of binary preferences and *confidence values*. The ratings are then related
-to the level of confidence in observed user preferences, rather than explicit ratings given to items.
-The model then tries to find latent factors that can be used to predict the expected preference of a user
-for an item.
-
-Available algorithms for collaborative filtering:
-
-* [ALS](api/mllib/index.html#org.apache.spark.mllib.recommendation.ALS)
-
-
-stochastic variants thereof) are first-order optimization methods that are
-well-suited for large-scale and distributed computation. Gradient descent
-methods aim to find a local minimum of a function by iteratively taking steps
-in the direction of the negative gradient of the function at the current point,
-i.e., the current parameter value. Gradient descent is included as a low-level
-primitive in MLlib, upon which various ML algorithms are developed, and has the
-following parameters:
-
-* *gradient* is a class that computes the stochastic gradient of the function
-being optimized, i.e., with respect to a single training example, at the
-current parameter value. MLlib includes gradient classes for common loss
-functions, e.g., hinge, logistic, least-squares.  The gradient class takes as
-input a training example, its label, and the current parameter value.
-* *updater* is a class that updates weights in each iteration of gradient
-descent. MLlib includes updaters for cases without regularization, as well as
-L1 and L2 regularizers.
-* *stepSize* is a scalar value denoting the initial step size for gradient
-descent. All updaters in MLlib use a step size at the t-th step equal to
-stepSize / sqrt(t).
-* *numIterations* is the number of iterations to run.
-* *regParam* is the regularization parameter when using L1 or L2 regularization.
-* *miniBatchFraction* is the fraction of the data used to compute the gradient
-at each iteration.
-
-
-
-# Using MLLib in Scala
-
-Following code snippets can be executed in spark-shell.
-
-## Binary Classification
-
-The following code snippet illustrates how to load a sample dataset, execute a
-training algorithm on this training data using a static method in the algorithm
-object, and make predictions with the resulting model to compute the training
-error.
-
-{% highlight scala %}
-import org.apache.spark.SparkContext
-import org.apache.spark.mllib.classification.SVMWithSGD
-import org.apache.spark.mllib.regression.LabeledPoint
-
-// Load and parse the data file
-val data = sc.textFile("mllib/data/sample_svm_data.txt")
-val parsedData = data.map { line =>
-  val parts = line.split(' ')
-  LabeledPoint(parts(0).toDouble, parts.tail.map(x => x.toDouble).toArray)
-}
-
-// Run training algorithm to build the model
-val numIterations = 20
-val model = SVMWithSGD.train(parsedData, numIterations)
-
-// Evaluate model on training examples and compute training error
-val labelAndPreds = parsedData.map { point =>
-  val prediction = model.predict(point.features)
-  (point.label, prediction)
-}
-val trainErr = labelAndPreds.filter(r => r._1 != r._2).count.toDouble / parsedData.count
-println("Training Error = " + trainErr)
-{% endhighlight %}
-
-
-The SVMWithSGD.train() method by default performs L2 regularization with the
-regularization parameter set to 1.0. If we want to configure this algorithm, we
-can customize SVMWithSGD further by creating a new object directly and
-calling setter methods. All other MLlib algorithms support customization in
-this way as well. For example, the following code produces an L1 regularized
-variant of SVMs with regularization parameter set to 0.1, and runs the training
-algorithm for 200 iterations.
-
-{% highlight scala %}
-import org.apache.spark.mllib.optimization.L1Updater
-
-val svmAlg = new SVMWithSGD()
-svmAlg.optimizer.setNumIterations(200)
-  .setRegParam(0.1)
-  .setUpdater(new L1Updater)
-val modelL1 = svmAlg.run(parsedData)
-{% endhighlight %}
-
-## Linear Regression
-The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
-example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
-compute the Mean Squared Error at the end to evaluate
-[goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
-
-{% highlight scala %}
-import org.apache.spark.mllib.regression.LinearRegressionWithSGD
-import org.apache.spark.mllib.regression.LabeledPoint
-
-// Load and parse the data
-val data = sc.textFile("mllib/data/ridge-data/lpsa.data")
-val parsedData = data.map { line =>
-  val parts = line.split(',')
-  LabeledPoint(parts(0).toDouble, parts(1).split(' ').map(x => x.toDouble).toArray)
-}
-
-// Building the model
-val numIterations = 20
-val model = LinearRegressionWithSGD.train(parsedData, numIterations)
-
-// Evaluate model on training examples and compute training error
-val valuesAndPreds = parsedData.map { point =>
-  val prediction = model.predict(point.features)
-  (point.label, prediction)
-}
-val MSE = valuesAndPreds.map{ case(v, p) => math.pow((v - p), 2)}.reduce(_ + _)/valuesAndPreds.count
-println("training Mean Squared Error = " + MSE)
-{% endhighlight %}
-
-
-Similarly you can use RidgeRegressionWithSGD and LassoWithSGD and compare training
-[Mean Squared Errors](http://en.wikipedia.org/wiki/Mean_squared_error).
-
-## Clustering
-In the following example after loading and parsing data, we use the KMeans object to cluster the data
-into two clusters. The number of desired clusters is passed to the algorithm. We then compute Within
-Set Sum of Squared Error (WSSSE). You can reduce this error measure by increasing *k*. In fact the
-optimal *k* is usually one where there is an "elbow" in the WSSSE graph.
-
-{% highlight scala %}
-import org.apache.spark.mllib.clustering.KMeans
-
-// Load and parse the data
-val data = sc.textFile("kmeans_data.txt")
-val parsedData = data.map( _.split(' ').map(_.toDouble))
-
-// Cluster the data into two classes using KMeans
-val numIterations = 20
-val numClusters = 2
-val clusters = KMeans.train(parsedData, numClusters, numIterations)
-
-// Evaluate clustering by computing Within Set Sum of Squared Errors
-val WSSSE = clusters.computeCost(parsedData)
-println("Within Set Sum of Squared Errors = " + WSSSE)
-{% endhighlight %}
-
-
-## Collaborative Filtering
-In the following example we load rating data. Each row consists of a user, a product and a rating.
-We use the default ALS.train() method which assumes ratings are explicit. We evaluate the recommendation
-model by measuring the Mean Squared Error of rating prediction.
-
-{% highlight scala %}
-import org.apache.spark.mllib.recommendation.ALS
-import org.apache.spark.mllib.recommendation.Rating
-
-// Load and parse the data
-val data = sc.textFile("mllib/data/als/test.data")
-val ratings = data.map(_.split(',') match {
-    case Array(user, item, rate) =>  Rating(user.toInt, item.toInt, rate.toDouble)
-})
-
-// Build the recommendation model using ALS
-val numIterations = 20
-val model = ALS.train(ratings, 1, 20, 0.01)
-
-// Evaluate the model on rating data
-val usersProducts = ratings.map{ case Rating(user, product, rate)  => (user, product)}
-val predictions = model.predict(usersProducts).map{
-    case Rating(user, product, rate) => ((user, product), rate)
-}
-val ratesAndPreds = ratings.map{
-    case Rating(user, product, rate) => ((user, product), rate)
-}.join(predictions)
-val MSE = ratesAndPreds.map{
-    case ((user, product), (r1, r2)) =>  math.pow((r1- r2), 2)
-}.reduce(_ + _)/ratesAndPreds.count
-println("Mean Squared Error = " + MSE)
-{% endhighlight %}
-
-If the rating matrix is derived from other source of information (i.e., it is inferred from
-other signals), you can use the trainImplicit method to get better results.
-
-{% highlight scala %}
-val model = ALS.trainImplicit(ratings, 1, 20, 0.01)
-{% endhighlight %}
-
-# Using MLLib in Java
-
-All of MLlib's methods use Java-friendly types, so you can import and call them there the same
-way you do in Scala. The only caveat is that the methods take Scala RDD objects, while the
-Spark Java API uses a separate JavaRDD class. You can convert a Java RDD to a Scala one by
-calling .rdd() on your JavaRDD object.
-
-# Using MLLib in Python
-Following examples can be tested in the PySpark shell.
-
-## Binary Classification
-The following example shows how to load a sample dataset, build Logistic Regression model,
-and make predictions with the resulting model to compute the training error.
-
-{% highlight python %}
-from pyspark.mllib.classification import LogisticRegressionWithSGD
-from numpy import array
-
-# Load and parse the data
-data = sc.textFile("mllib/data/sample_svm_data.txt")
-parsedData = data.map(lambda line: array([float(x) for x in line.split(' ')]))
-model = LogisticRegressionWithSGD.train(parsedData)
-
-# Build the model
-labelsAndPreds = parsedData.map(lambda point: (int(point.item(0)),
-        model.predict(point.take(range(1, point.size)))))
-
-# Evaluating the model on training data
-trainErr = labelsAndPreds.filter(lambda (v, p): v != p).count() / float(parsedData.count())
-print("Training Error = " + str(trainErr))
-{% endhighlight %}
-
-## Linear Regression
-The following example demonstrate how to load training data, parse it as an RDD of LabeledPoint. The
-example then uses LinearRegressionWithSGD to build a simple linear model to predict label values. We
-compute the Mean Squared Error at the end to evaluate
-[goodness of fit](http://en.wikipedia.org/wiki/Goodness_of_fit)
-
-{% highlight python %}
-from pyspark.mllib.regression import LinearRegressionWithSGD
-from numpy import array
-
-# Load and parse the data
-data = sc.textFile("mllib/data/ridge-data/lpsa.data")
-parsedData = data.map(lambda line: array([float(x) for x in line.replace(',', ' ').split(' ')]))
-
-# Build the model
-model = LinearRegressionWithSGD.train(parsedData)
-
-# Evaluate the model on training data
-valuesAndPreds = parsedData.map(lambda point: (point.item(0),
-        model.predict(point.take(range(1, point.size)))))
-MSE = valuesAndPreds.map(lambda (v, p): (v - p)**2).reduce(lambda x, y: x + y)/valuesAndPreds.count()
-print("Mean Squared Error = " + str(MSE))
-{% endhighlight %}
-
-
-## Clustering
-In the following example after loading and parsing data, we use the KMeans object to cluster the data
-into two clusters. The number of desired clusters is passed to the algorithm. We then compute Within
-Set Sum of Squared Error (WSSSE). You can reduce this error measure by increasing *k*. In fact the
-optimal *k* is usually one where there is an "elbow" in the WSSSE graph.
-
-{% highlight python %}
-from pyspark.mllib.clustering import KMeans
-from numpy import array
-from math import sqrt
-
-# Load and parse the data
-data = sc.textFile("kmeans_data.txt")
-parsedData = data.map(lambda line: array([float(x) for x in line.split(' ')]))
-
-# Build the model (cluster the data)
-clusters = KMeans.train(parsedData, 2, maxIterations=10,
-        runs=30, initialization_mode="random")
-
-# Evaluate clustering by computing Within Set Sum of Squared Errors
-def error(point):
-    center = clusters.centers[clusters.predict(point)]
-    return sqrt(sum([x**2 for x in (point - center)]))
-
-WSSSE = parsedData.map(lambda point: error(point)).reduce(lambda x, y: x + y)
-print("Within Set Sum of Squared Error = " + str(WSSSE))
-{% endhighlight %}
-
-Similarly you can use RidgeRegressionWithSGD and LassoWithSGD and compare training Mean Squared
-Errors.
-
-## Collaborative Filtering
-In the following example we load rating data. Each row consists of a user, a product and a rating.
-We use the default ALS.train() method which assumes ratings are explicit. We evaluate the
-recommendation by measuring the Mean Squared Error of rating prediction.
-
-{% highlight python %}
-from pyspark.mllib.recommendation import ALS
-from numpy import array
-
-# Load and parse the data
-data = sc.textFile("mllib/data/als/test.data")
-ratings = data.map(lambda line: array([float(x) for x in line.split(',')]))
-
-# Build the recommendation model using Alternating Least Squares
-model = ALS.train(ratings, 1, 20)
-
-# Evaluate the model on training data
-testdata = ratings.map(lambda p: (int(p[0]), int(p[1])))
-predictions = model.predictAll(testdata).map(lambda r: ((r[0], r[1]), r[2]))
-ratesAndPreds = ratings.map(lambda r: ((r[0], r[1]), r[2])).join(predictions)
-MSE = ratesAndPreds.map(lambda r: (r[1][0] - r[1][1])**2).reduce(lambda x, y: x + y)/ratesAndPreds.count()
-print("Mean Squared Error = " + str(MSE))
-{% endhighlight %}
-
-If the rating matrix is derived from other source of information (i.e., it is inferred from other
-signals), you can use the trainImplicit method to get better results.
-
-{% highlight python %}
-# Build the recommendation model using Alternating Least Squares based on implicit ratings
-model = ALS.trainImplicit(ratings, 1, 20)
-{% endhighlight %}
-
-
-# Singular Value Decomposition
-Singular Value Decomposition for Tall and Skinny matrices.
-Given an *m x n* matrix *A*, we can compute matrices *U, S, V* such that
-
-*A = U * S * V^T*
-
-There is no restriction on m, but we require n^2 doubles to
-fit in memory locally on one machine.
-Further, n should be less than m.
-
-The decomposition is computed by first computing *A^TA = V S^2 V^T*,
-computing SVD locally on that (since n x n is small),
-from which we recover S and V.
-Then we compute U via easy matrix multiplication
-as *U =  A * V * S^-1*
-
-Only singular vectors associated with largest k singular values
-are recovered. If there are k
-such values, then the dimensions of the return will be:
-
-* *S* is *k x k* and diagonal, holding the singular values on diagonal.
-* *U* is *m x k* and satisfies U^T*U = eye(k).
-* *V* is *n x k* and satisfies V^TV = eye(k).
-
-All input and output is expected in sparse matrix format, 0-indexed
-as tuples of the form ((i,j),value) all in
-SparseMatrix RDDs. Below is example usage.
-
-{% highlight scala %}
-
-import org.apache.spark.SparkContext
-import org.apache.spark.mllib.linalg.SVD
-import org.apache.spark.mllib.linalg.SparseMatrix
-import org.apache.spark.mllib.linalg.MatrixEntry
-
-// Load and parse the data file
-val data = sc.textFile("mllib/data/als/test.data").map { line =>
-  val parts = line.split(',')
-  MatrixEntry(parts(0).toInt, parts(1).toInt, parts(2).toDouble)
-}
-val m = 4
-val n = 4
-val k = 1
-
-// recover largest singular vector
-val decomposed = SVD.sparseSVD(SparseMatrix(data, m, n), k)
-val = decomposed.S.data
-
-println("singular values = " + s.toArray.mkString)
-{% endhighlight %}
\ No newline at end of file

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-linear-algebra.md
----------------------------------------------------------------------
diff --git a/docs/mllib-linear-algebra.md b/docs/mllib-linear-algebra.md
new file mode 100644
index 0000000..cc203d8
--- /dev/null
+++ b/docs/mllib-linear-algebra.md
@@ -0,0 +1,61 @@
+---
+layout: global
+title: MLlib - Linear Algebra
+---
+
+{:toc}
+
+
+# Singular Value Decomposition
+Singular Value Decomposition for Tall and Skinny matrices.
+Given an $m \times n$ matrix $A$, we can compute matrices $U,S,V$ such that
+
+$+ A = U \cdot S \cdot V^T +$
+
+There is no restriction on m, but we require n^2 doubles to
+fit in memory locally on one machine.
+Further, n should be less than m.
+
+The decomposition is computed by first computing $A^TA = V S^2 V^T$,
+computing SVD locally on that (since $n \times n$ is small),
+from which we recover $S$ and $V$.
+Then we compute U via easy matrix multiplication
+as $U = A \cdot V \cdot S^{-1}$.
+
+Only singular vectors associated with largest k singular values
+are recovered. If there are k
+such values, then the dimensions of the return will be:
+
+* $S$ is $k \times k$ and diagonal, holding the singular values on diagonal.
+* $U$ is $m \times k$ and satisfies $U^T U = \mathop{eye}(k)$.
+* $V$ is $n \times k$ and satisfies $V^T V = \mathop{eye}(k)$.
+
+All input and output is expected in sparse matrix format, 0-indexed
+as tuples of the form ((i,j),value) all in
+SparseMatrix RDDs. Below is example usage.
+
+{% highlight scala %}
+
+import org.apache.spark.SparkContext
+import org.apache.spark.mllib.linalg.SVD
+import org.apache.spark.mllib.linalg.SparseMatrix
+import org.apache.spark.mllib.linalg.MatrixEntry
+
+// Load and parse the data file
+val data = sc.textFile("mllib/data/als/test.data").map { line =>
+  val parts = line.split(',')
+  MatrixEntry(parts(0).toInt, parts(1).toInt, parts(2).toDouble)
+}
+val m = 4
+val n = 4
+val k = 1
+
+// recover largest singular vector
+val decomposed = SVD.sparseSVD(SparseMatrix(data, m, n), k)
+val = decomposed.S.data
+
+println("singular values = " + s.toArray.mkString)
+{% endhighlight %}

http://git-wip-us.apache.org/repos/asf/incubator-spark/blob/fabf1749/docs/mllib-optimization.md
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diff --git a/docs/mllib-optimization.md b/docs/mllib-optimization.md
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+---
+layout: global
+title: MLlib - Optimization
+---
+
+{:toc}
+
+
+
+stochastic variants thereof) are first-order optimization methods that are
+well-suited for large-scale and distributed computation. Gradient descent
+methods aim to find a local minimum of a function by iteratively taking steps
+in the direction of the negative gradient of the function at the current point,
+i.e., the current parameter value. Gradient descent is included as a low-level
+primitive in MLlib, upon which various ML algorithms are developed, and has the
+following parameters:
+
+* *gradient* is a class that computes the stochastic gradient of the function
+being optimized, i.e., with respect to a single training example, at the
+current parameter value. MLlib includes gradient classes for common loss
+functions, e.g., hinge, logistic, least-squares.  The gradient class takes as
+input a training example, its label, and the current parameter value.
+* *updater* is a class that updates weights in each iteration of gradient
+descent. MLlib includes updaters for cases without regularization, as well as
+L1 and L2 regularizers.
+* *stepSize* is a scalar value denoting the initial step size for gradient
+descent. All updaters in MLlib use a step size at the t-th step equal to
+stepSize / sqrt(t).
+* *numIterations* is the number of iterations to run.
+* *regParam* is the regularization parameter when using L1 or L2 regularization.
+* *miniBatchFraction* is the fraction of the data used to compute the gradient
+at each iteration.
+