[SPARK-5521] PCA wrapper for easy transform vectors

I implement a simple PCA wrapper for easy transform of vectors by PCA for example LabeledPoint or another complicated structure.

Example of usage:
```
  import org.apache.spark.mllib.regression.LinearRegressionWithSGD
  import org.apache.spark.mllib.regression.LabeledPoint
  import org.apache.spark.mllib.linalg.Vectors
  import org.apache.spark.mllib.feature.PCA

  val data = sc.textFile("data/mllib/ridge-data/lpsa.data").map { line =>
    val parts = line.split(',')
    LabeledPoint(parts(0).toDouble, Vectors.dense(parts(1).split(' ').map(_.toDouble)))
  }.cache()

  val splits = data.randomSplit(Array(0.6, 0.4), seed = 11L)
  val training = splits(0).cache()
  val test = splits(1)

  val pca = PCA.create(training.first().features.size/2, data.map(_.features))
  val training_pca = training.map(p => p.copy(features = pca.transform(p.features)))
  val test_pca = test.map(p => p.copy(features = pca.transform(p.features)))

  val numIterations = 100
  val model = LinearRegressionWithSGD.train(training, numIterations)
  val model_pca = LinearRegressionWithSGD.train(training_pca, numIterations)

  val valuesAndPreds = test.map { point =>
    val score = model.predict(point.features)
    (score, point.label)
  }

  val valuesAndPreds_pca = test_pca.map { point =>
    val score = model_pca.predict(point.features)
    (score, point.label)
  }

  val MSE = valuesAndPreds.map{case(v, p) => math.pow((v - p), 2)}.mean()
  val MSE_pca = valuesAndPreds_pca.map{case(v, p) => math.pow((v - p), 2)}.mean()

  println("Mean Squared Error = " + MSE)
  println("PCA Mean Squared Error = " + MSE_pca)
```

Author: Kirill A. Korinskiy <catap@catap.ru>
Author: Joseph K. Bradley <joseph@databricks.com>

Closes #4304 from catap/pca and squashes the following commits:

501bcd9 [Joseph K. Bradley] Small updates: removed k from Java-friendly PCA fit().  In PCASuite, converted results to set for comparison. Added an error message for bad k in PCA.
9dcc02b [Kirill A. Korinskiy] [SPARK-5521] fix scala style
1892a06 [Kirill A. Korinskiy] [SPARK-5521] PCA wrapper for easy transform vectors

1 files changed, 54 insertions, 1 deletions

diff --git a/docs/mllib-feature-extraction.md b/docs/mllib-feature-extraction.md
index 03fedd0101..f723cd6b9d 100644
--- a/docs/mllib-feature-extraction.md
+++ b/docs/mllib-feature-extraction.md
@@ -507,7 +507,6 @@ v_N
 
 This example below demonstrates how to load a simple vectors file, extract a set of vectors, then transform those vectors using a transforming vector value.
 
-
 <div class="codetabs">
 <div data-lang="scala">
 {% highlight scala %}
@@ -531,3 +530,57 @@ val transformedData2 = parsedData.map(x => transformer.transform(x))
 </div>
 
 
+## PCA
+
+A feature transformer that projects vectors to a low-dimensional space using PCA.
+Details you can read at [dimensionality reduction](mllib-dimensionality-reduction.html).
+
+### Example
+
+The following code demonstrates how to compute principal components on a `Vector`
+and use them to project the vectors into a low-dimensional space while keeping associated labels
+for calculation a [Linear Regression]((mllib-linear-methods.html))
+
+<div class="codetabs">
+<div data-lang="scala">
+{% highlight scala %}
+import org.apache.spark.mllib.regression.LinearRegressionWithSGD
+import org.apache.spark.mllib.regression.LabeledPoint
+import org.apache.spark.mllib.linalg.Vectors
+import org.apache.spark.mllib.feature.PCA
+
+val data = sc.textFile("data/mllib/ridge-data/lpsa.data").map { line =>
+  val parts = line.split(',')
+  LabeledPoint(parts(0).toDouble, Vectors.dense(parts(1).split(' ').map(_.toDouble)))
+}.cache()
+
+val splits = data.randomSplit(Array(0.6, 0.4), seed = 11L)
+val training = splits(0).cache()
+val test = splits(1)
+
+val pca = new PCA(training.first().features.size/2).fit(data.map(_.features))
+val training_pca = training.map(p => p.copy(features = pca.transform(p.features)))
+val test_pca = test.map(p => p.copy(features = pca.transform(p.features)))
+
+val numIterations = 100
+val model = LinearRegressionWithSGD.train(training, numIterations)
+val model_pca = LinearRegressionWithSGD.train(training_pca, numIterations)
+
+val valuesAndPreds = test.map { point =>
+  val score = model.predict(point.features)
+  (score, point.label)
+}
+
+val valuesAndPreds_pca = test_pca.map { point =>
+  val score = model_pca.predict(point.features)
+  (score, point.label)
+}
+
+val MSE = valuesAndPreds.map{case(v, p) => math.pow((v - p), 2)}.mean()
+val MSE_pca = valuesAndPreds_pca.map{case(v, p) => math.pow((v - p), 2)}.mean()
+
+println("Mean Squared Error = " + MSE)
+println("PCA Mean Squared Error = " + MSE_pca)
+{% endhighlight %}
+</div>
+</div>