Merge pull request #460 from srowen/RandomInitialALSVectors

Choose initial user/item vectors uniformly on the unit sphere

...rather than within the unit square to possibly avoid bias in the initial state and improve convergence.

The current implementation picks the N vector elements uniformly at random from [0,1). This means they all point into one quadrant of the vector space. As N gets just a little large, the vector tend strongly to point into the "corner", towards (1,1,1...,1). The vectors are not unit vectors either.

I suggest choosing the elements as Gaussian ~ N(0,1) and normalizing. This gets you uniform random choices on the unit sphere which is more what's of interest here. It has worked a little better for me in the past.

This is pretty minor but wanted to warm up suggesting a few tweaks to ALS.
Please excuse my Scala, pretty new to it.

Author: Sean Owen <sowen@cloudera.com>

== Merge branch commits ==

commit 492b13a7469e5a4ed7591ee8e56d8bd7570dfab6
Author: Sean Owen <sowen@cloudera.com>
Date:   Mon Jan 27 08:05:25 2014 +0000

    Style: spaces around binary operators

commit ce2b5b5a4fefa0356875701f668f01f02ba4d87e
Author: Sean Owen <sowen@cloudera.com>
Date:   Sun Jan 19 22:50:03 2014 +0000

    Generate factors with all positive components, per discussion in https://github.com/apache/incubator-spark/pull/460

commit b6f7a8a61643a8209e8bc662e8e81f2d15c710c7
Author: Sean Owen <sowen@cloudera.com>
Date:   Sat Jan 18 15:54:42 2014 +0000

    Choose initial user/item vectors uniformly on the unit sphere rather than within the unit square to possibly avoid bias in the initial state and improve convergence

mllib/src/main/scala/org/apache/spark/mllib/recommendation/ALS.scala

@@ -18,6 +18,7 @@
 package org.apache.spark.mllib.recommendation
 
 import scala.collection.mutable.{ArrayBuffer, BitSet}
+import scala.math.{abs, sqrt}
 import scala.util.Random
 import scala.util.Sorting
 
@@ -301,7 +302,14 @@ class ALS private (var numBlocks: Int, var rank: Int, var iterations: Int, var l
    * Make a random factor vector with the given random.
    */
   private def randomFactor(rank: Int, rand: Random): Array[Double] = {
-    Array.fill(rank)(rand.nextDouble)
+    // Choose a unit vector uniformly at random from the unit sphere, but from the
+    // "first quadrant" where all elements are nonnegative. This can be done by choosing
+    // elements distributed as Normal(0,1) and taking the absolute value, and then normalizing.
+    // This appears to create factorizations that have a slightly better reconstruction
+    // (<1%) compared picking elements uniformly at random in [0,1].
+    val factor = Array.fill(rank)(abs(rand.nextGaussian()))
+    val norm = sqrt(factor.map(x => x * x).sum)
+    factor.map(x => x / norm)
   }
 
   /**